prompt_id stringlengths 64 64 | question stringlengths 23 5.98k | response stringlengths 439 33.7k | teacher_model stringclasses 1
value | thinking_enabled bool 1
class | generation_attempts int64 1 3 | finish_reason stringclasses 1
value | output_tokens int64 181 12.3k | accepted bool 1
class | stop_reason stringclasses 0
values | rejection_reasons listlengths 0 0 | sampling_seed int64 40.4k 4.29B | teacher_revision stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
cab09a021e4eefebe9824b4d5f33a1d2175a31ca9c1c5b22e71daa2833668a86 | Compute
\[\prod_{k = 1}^\infty \left(1 + \frac{1}{k^2 + k}\right).\] | We are asked to compute the infinite product:
$$
\prod_{k = 1}^\infty \left(1 + \frac{1}{k^2 + k}\right).
$$
---
### Step 1: Simplify the term inside the product
We start by simplifying the term inside the product:
$$
1 + \frac{1}{k^2 + k} = \frac{k^2 + k + 1}{k^2 + k}.
$$
So the product becomes:
$$
\prod_{k = 1... | Qwen/Qwen3-4B | false | 1 | stop | 4,052 | true | null | [] | 486,664,709 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.479480+00:00 |
9e765d0a531eddbb9f601800daa8d2e42da09700aea4bc8e8290e872233b2146 | In the figure below, points $A$ and $B$ lie on a circle whose diameter passes through the vertex $V$ of isosceles triangle $\triangle ABC$. If $BA = BC$, the measure of the largest angle formed by drawing line segments connecting points $A$, $B$, and $C$ to point $V$ equals $\tfrac{135}{2}$ degrees, and the measure of ... | We are given a geometric configuration involving an **isosceles triangle** $ \triangle ABC $ with:
- $ BA = BC $
- The **diameter** of a circle passes through vertex $ V $ of the triangle
- $ \angle CBA = \frac{\pi}{12} $ radians
- The **largest angle** formed by connecting points $ A, B, C $ to $ V $ is $ \frac{135}{... | Qwen/Qwen3-4B | false | 1 | stop | 2,623 | true | null | [] | 1,696,414,707 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.335195+00:00 |
0e86a1c95eff2937baa00378dcdf51a99ecff632766f5e4b8c73acfab460dc7b | In the system of congruences
$$\begin{align*}
2x + 3y &\equiv 1 \pmod 5,\\
3x - 2y &\equiv 4 \pmod 7,
\end{align*}$$
find the smallest non-negative integer $n$ for which $x^2 + xy + y^2 + n$ is divisible by $35$. | We are given a system of congruences:
$$
\begin{align*}
2x + 3y &\equiv 1 \pmod{5}, \\
3x - 2y &\equiv 4 \pmod{7},
\end{align*}
$$
and we are to find the **smallest non-negative integer $ n $** such that:
$$
x^2 + xy + y^2 + n \equiv 0 \pmod{35}.
$$
---
### Step 1: Solve the system of congruences
We will solve th... | Qwen/Qwen3-4B | false | 1 | stop | 2,839 | true | null | [] | 2,716,232,281 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:59.904827+00:00 |
df24be5ee2cf8f0ae2e6a89d43712fad2d41acba8fa584027a2d20f94613e4fe | A square pyramid with a base side length of 8 cm and a height of 10 cm is inscribed in a sphere. The pyramid's apex touches the sphere's surface, and its base is tangent to the sphere at the center of the base. What is the radius of the sphere? Express your answer to the nearest whole number. | We are given a **square pyramid** with:
- **Base side length** = 8 cm
- **Height** = 10 cm
- The pyramid is **inscribed in a sphere**, with the **apex touching the sphere** and the **base tangent to the sphere at the center of the base**.
We are to find the **radius of the sphere**, rounded to the nearest whole n... | Qwen/Qwen3-4B | false | 1 | stop | 3,099 | true | null | [] | 1,761,047,145 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:51.841858+00:00 |
a655e8a65a15dc1cb04804bd235c8dae97e8f9613a775fce023943119f863b71 | The function $f(x)$ is defined as follows:
$f(x^2)=xf(x)+\ds\frac{x}{1+x}$ for all $x\in\mathbb{R}$,
while $f(x)$ is an even function.
Determine $f(4)$. | We are given a functional equation and the fact that $ f(x) $ is an **even function**:
---
### Given:
1. $ f(x^2) = x f(x) + \frac{x}{1 + x} $ for all $ x \in \mathbb{R} $.
2. $ f(x) $ is an **even function**, i.e., $ f(-x) = f(x) $ for all $ x \in \mathbb{R} $.
We are to **determine** $ f(4) $.
---
### Step 1: Us... | Qwen/Qwen3-4B | false | 1 | stop | 2,849 | true | null | [] | 3,403,819,902 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.623724+00:00 |
a7eb37b4c8ee1dc717c71006165a05949f4c85f149f5cc603f442db0fb59ee8a | Solve the equation $\frac{x^3-2x^2-x+1}{x^2-2x-1}=\sqrt{\frac{x^2-2x-1}{x+1}}$. | We are given the equation:
$$
\frac{x^3 - 2x^2 - x + 1}{x^2 - 2x - 1} = \sqrt{\frac{x^2 - 2x - 1}{x + 1}}
$$
---
### **Step 1: Domain Restrictions**
We need to ensure that all expressions are defined.
- The denominator $x^2 - 2x - 1 \ne 0$
- The expression under the square root must be non-negative:
$$
\frac{x... | Qwen/Qwen3-4B | false | 1 | stop | 4,635 | true | null | [] | 4,001,629,361 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.117132+00:00 |
f1dd789f8082cba39043e11601cee5a241d63826a51323aab20339fd39318020 | Find the value of $x$ for which the graph of the function $f(x) = \frac{x^2+1}{x^3-2x^2+x+1}$ has a hole. | We are given the function:
$$
f(x) = \frac{x^2 + 1}{x^3 - 2x^2 + x + 1}
$$
We are asked to find the value of $ x $ for which the graph of this function has a **hole**.
---
### Step 1: Understand what a **hole** means
A **hole** in the graph of a rational function occurs when there is a **common factor** in the **n... | Qwen/Qwen3-4B | false | 1 | stop | 714 | true | null | [] | 1,470,096,670 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.832759+00:00 |
3edbddf40e9d47e3c4ac3dfbbb30bf6ed31fe43e7651ef73d1e4b635021fe8cb | Find all points that are 3 units from the parabola $y=x^2$ and 2 units from the point $(1,3)$. What is the sum of the $x$-coordinates of all these points? | We are given two conditions for points $(x, y)$:
1. The point is **3 units from the parabola** $y = x^2$.
2. The point is **2 units from the point** $(1, 3)$.
We are to **find all such points** and then **sum their $x$-coordinates**.
---
### Step 1: Distance from a point to a parabola
The distance from a point $(x... | Qwen/Qwen3-4B | false | 1 | stop | 2,274 | true | null | [] | 2,160,606,547 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:02.958284+00:00 |
5cc0d01e431747e81fd2dd7534e83164f7e5bf53e96a2046fba817e27ce79bdb | Given that
$\ds\sqrt {\f{x^2-3x-2}{x+2}}=x-3$
where $x$ is positive,
determine $\sqrt x$. | We are given the equation:
$$
\sqrt{\frac{x^2 - 3x - 2}{x + 2}} = x - 3
$$
We are to **determine** $\sqrt{x}$, assuming $x$ is **positive**.
---
### Step 1: Square both sides to eliminate the square root
$$
\left( \sqrt{\frac{x^2 - 3x - 2}{x + 2}} \right)^2 = (x - 3)^2
$$
$$
\frac{x^2 - 3x - 2}{x + 2} = x^2 - 6x ... | Qwen/Qwen3-4B | false | 1 | stop | 10,157 | true | null | [] | 3,572,146,232 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.513716+00:00 |
6ed8127f03ff07a5eddc5e572bd548056d0cf68cf0d2c5cfd56f6a8676932798 | In trapezoid $ABCD$, the diagonals intersect at $E$, and $EF$ is the angle bisector from vertex $E$ that intersects side $AD$ at $F$. If $AB = 6$, $CD = 8$, and $EF = 3 \sqrt{3}$, then find the area of triangle $AEF$. | We are given a trapezoid $ABCD$ with the following:
- $AB = 6$
- $CD = 8$
- Diagonals intersect at $E$
- $EF$ is the angle bisector from $E$, intersecting side $AD$ at $F$
- $EF = 3\sqrt{3}$
We are to **find the area of triangle $AEF$**.
---
### Step 1: Use the properties of trapezoid and diagonals
In a trapezoid ... | Qwen/Qwen3-4B | false | 1 | stop | 2,753 | true | null | [] | 1,247,182,954 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.406049+00:00 |
a74a6639355bfe431655522ef98aee96075bcfb9fcef3b32f679ae2999851b98 | Find the largest positive real number $y$ such that
\[\lceil y^3 \rceil - y^2 \lceil y \rceil = 12.\] | We are given the equation:
$$
\lceil y^3 \rceil - y^2 \lceil y \rceil = 12
$$
We are to **find the largest positive real number $ y $** that satisfies this equation.
---
### Step 1: Understand the ceiling function
The **ceiling function** $ \lceil x \rceil $ is the **smallest integer greater than or equal to $ x $... | Qwen/Qwen3-4B | false | 3 | stop | 8,276 | true | null | [] | 320,877,101 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:16:10.168238+00:00 |
98e5b2d886c9800264194feb1fbb602141ad992986cac0b0a2a6a85a0f8a6162 | Let $g : \mathbb{N} \to \mathbb{N}$ be a function such that
\[g(g(x) + g(y)) = g(x) + g(y) + g(1)\]for all $x,$ $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $m \times t.$ | We are given a function $ g : \mathbb{N} \to \mathbb{N} $ satisfying the functional equation:
$$
g(g(x) + g(y)) = g(x) + g(y) + g(1) \quad \text{for all } x, y \in \mathbb{N}.
$$
We are to find:
- $ m $: the number of possible values of $ g(1) $,
- $ t $: the sum of all possible values of $ g(1) $,
- and compute $ m... | Qwen/Qwen3-4B | false | 3 | stop | 4,550 | true | null | [] | 4,115,140,755 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:16:27.747593+00:00 |
431d3d1f1211d739dee2464c7fe92f2196a56c45e4ad9b428b1abc0135b7da22 | In quadrilateral $ABCD$, $AB=6$, $BC=8$, $CD=4$ and $\angle C=90^\circ$. The quadrilateral is inscribed in a circle centered at $O$. If the diagonals $AC$ and $BD$ intersect at $E$, and $\angle AED=90^\circ$, then what is the length of $OE$? | We are given a **cyclic quadrilateral** $ABCD$ with:
- $AB = 6$
- $BC = 8$
- $CD = 4$
- $\angle C = 90^\circ$
- Diagonals $AC$ and $BD$ intersect at point $E$, and $\angle AED = 90^\circ$
- The quadrilateral is inscribed in a circle centered at $O$
We are to find the length of $OE$.
---
### Step 1: Understand the G... | Qwen/Qwen3-4B | false | 1 | stop | 2,525 | true | null | [] | 1,343,601,450 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.275852+00:00 |
98948231653e1b2a141f8355a8b84603475b4887553c3a24cff5642a3ff9417a | What is the value of $b$ if $x^2+bx+1<0$ only when $x\in (0, 2)\cup(4,6)$? | We are given the inequality:
$$
x^2 + bx + 1 < 0
$$
and told that this inequality **only holds** when:
$$
x \in (0, 2) \cup (4, 6)
$$
This tells us that the quadratic expression $ x^2 + bx + 1 $ is **negative** only on the intervals $ (0, 2) $ and $ (4, 6) $, and **non-negative** outside of those intervals.
---
#... | Qwen/Qwen3-4B | false | 1 | stop | 3,135 | true | null | [] | 1,483,133,845 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.844569+00:00 |
41e9074b04f55ba6420b7a5502486ed59b298ffafe2815598a6ec2e5408864f6 | The function $f(x),$ defined for $0 \le x \le 1,$ has the following properties:
(i) $f(0) = 0.$
(ii) If $0 \le x < y \le 1,$ then $f(x) \le f(y).$
(iii) $f(1 - x) = 1 - f(x)$ for all $0 \le x \le 1.$
(iv) $f \left( \frac{x}{3} \right) = \frac{f(x)}{2}$ for $0 \le x \le 1.$
Find $f \left( \frac{2}{7} \right).$
Anothe... | We are given two problems with similar structures. Both involve functions defined on the interval $[0,1]$ with symmetry and scaling properties. We'll analyze and solve each one step by step.
---
### **First Problem: Find $ f\left(\frac{2}{7}\right) $, given:**
1. $ f(0) = 0 $
2. $ f(x) \le f(y) $ if $ 0 \le x < y \l... | Qwen/Qwen3-4B | false | 1 | stop | 2,874 | true | null | [] | 4,010,899,979 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.024942+00:00 |
c331742a7a6513ec8cb34c6bc8eac9f6d1252b8821e90a6bcf90117f46e58fd2 | Consider two functions, $f(x)$ and $g(x)$, such that their graphs intersect at $(1,1)$, $(3,4)$, and $(5,7)$. The graphs of $y=f(x+1)$ and $y=g(x-1)$ intersect at two points. What is the difference between the $y$-coordinates of these two points? | We are given two functions $ f(x) $ and $ g(x) $, whose graphs intersect at the points:
- $ (1, 1) $
- $ (3, 4) $
- $ (5, 7) $
This means:
$$
f(1) = 1,\quad g(1) = 1 \\
f(3) = 4,\quad g(3) = 4 \\
f(5) = 7,\quad g(5) = 7
$$
So, both functions pass through the same three points.
Now, we are told:
- The graphs of $ ... | Qwen/Qwen3-4B | false | 1 | stop | 2,187 | true | null | [] | 1,820,754,744 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.550752+00:00 |
c906888fc94bdbed15c46adc841503362020ad0ab876a6bbc8077b486b3a19f1 | Find the slope of the line that passes through the intersection points of the curves $y = \frac{1}{x^2}$ and $y = x^3$. Express your answer as a common fraction. | We are given two curves:
$$
y = \frac{1}{x^2} \quad \text{and} \quad y = x^3
$$
We are asked to find the **slope of the line that passes through the intersection points** of these two curves.
---
### **Step 1: Find the intersection points**
To find the points of intersection, set the two expressions for $ y $ equa... | Qwen/Qwen3-4B | false | 1 | stop | 1,708 | true | null | [] | 2,766,980,275 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.470224+00:00 |
c6c3f692d17734c039313f2d588ff9ba1f09b9168ff56c7f54b6e855100e2610 | Let $a,$ $b,$ and $c$ be real numbers such that
\[a^4 + b^4 + c^4 - 4abc(a + b + c) = 2.\]Find the minimum value of $a^2 + b^2 + c^2.$ | We are given the equation:
$$
a^4 + b^4 + c^4 - 4abc(a + b + c) = 2
$$
and we are to **find the minimum value of** $ a^2 + b^2 + c^2 $.
---
### Step 1: Use symmetry and assume symmetry in the variables
Let’s try to simplify the problem by assuming that the minimum occurs when $ a = b = c $. This is a common techni... | Qwen/Qwen3-4B | false | 1 | stop | 1,446 | true | null | [] | 4,115,865,990 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:59.951172+00:00 |
6e7c8713d656ed0b87aeac84030af5ea3509023675aba7ebb0b1a61198a075b3 | Solve for $x$: $\sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + 27}}} = 3.$ | We are given the equation:
$$
\sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + 27}}} = 3
$$
We are to **solve for $ x $**.
---
### Step 1: Let’s define a nested expression
Let’s denote:
$$
y = \sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + 27}}}
$$
We are told that $ y = 3 $.
So:
$$
\sqrt[3]{x + \sqrt[3]{x + \sqrt[3]{x + 27}}} ... | Qwen/Qwen3-4B | false | 1 | stop | 2,015 | true | null | [] | 2,550,865,468 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.699408+00:00 |
ddc2ff91ef2849b2164047e3fcf19bbbae18bdab698d73e84c500bdc5340eab5 | A set of six distinct positive integers is such that the sum of any three of them is a multiple of 11. What is the smallest possible sum of all six numbers? | We are given a set of **six distinct positive integers** such that the **sum of any three** of them is a **multiple of 11**. We are to find the **smallest possible sum** of all six numbers.
---
### Step 1: Understanding the Condition
Let the six numbers be $ a_1, a_2, a_3, a_4, a_5, a_6 $, all **distinct positive in... | Qwen/Qwen3-4B | false | 1 | stop | 1,575 | true | null | [] | 4,072,202,130 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.301833+00:00 |
f19aa52ecfb02d7a5092b0b20f255144b439bedbfae293b00a6faf0523226edf | A circle with center at the origin intersects the parabola $y^2 = 2x$ at two points $A$ and $B.$ The tangent to the circle at the point $(\sqrt{2}, 0)$ intersects the line $y = x$ at point $P.$ Find the length of the segment $AP$ multiplied by the length of the segment $BP.$ | We are given:
- A **circle** with center at the **origin** (i.e., $(0, 0)$).
- The **circle intersects the parabola** $ y^2 = 2x $ at two points $ A $ and $ B $.
- The **tangent to the circle at the point** $ (\sqrt{2}, 0) $ **intersects the line** $ y = x $ at point $ P $.
- We are to **find** the product $ AP \cdot ... | Qwen/Qwen3-4B | false | 1 | stop | 2,706 | true | null | [] | 2,985,876,877 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.873940+00:00 |
975f22428b7d38b62ab5ce8566b078ef09892a67ee372329d94a6e08c9956191 | Consider a convex quadrilateral $ABCD$ such that $AB=5$, $BC=4$, and $CD=6$. Find all possible values of the length of $AD$ so that the given quadrilateral can be inscribed in a circle. | We are given a **convex quadrilateral** $ABCD$ with sides:
- $AB = 5$
- $BC = 4$
- $CD = 6$
We are to **find all possible values of $AD$** such that the quadrilateral can be **inscribed in a circle** (i.e., it is **cyclic**).
---
### Step 1: Use the Cyclic Quadrilateral Property
A **cyclic quadrilateral** is one w... | Qwen/Qwen3-4B | false | 1 | stop | 2,924 | true | null | [] | 738,714,274 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.033953+00:00 |
55d7c611b7e548524df0f839fd6969ba51896ec93ee48e3b6dac39deb8777bce | An ellipse of semi-major axis $a$ and semi-minor axis $b$ is rotated about its minor axis to form an elliptical surface of revolution. The largest sphere that can be placed inside the surface is centered on the axis of rotation and has radius $r$. Find the volume of the sphere when $a=3$ and $b=2$. | We are given an **ellipse** with semi-major axis $ a = 3 $ and semi-minor axis $ b = 2 $. This ellipse is **rotated about its minor axis**, forming an **elliptical surface of revolution**. Inside this surface, we want to find the **volume of the largest sphere** that can be placed, assuming the sphere is **centered on ... | Qwen/Qwen3-4B | false | 1 | stop | 1,950 | true | null | [] | 1,767,576,597 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.559197+00:00 |
e3b95ca58a823dd8632494b8b3fd37fa51e6e5caa8124a687e4dc59098bb6a1f | In a right pyramid with a square base, the centers of the four triangular faces are the vertices of a smaller square pyramid. The ratio of the surface area of the smaller pyramid to that of the larger is $a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | We are given a **right pyramid with a square base**, and the **centers of the four triangular faces** form a **smaller square pyramid**. We are to find the **ratio of the surface areas** of the smaller pyramid to the larger one, expressed as a fraction $ \frac{a}{b} $, and then compute $ a + b $.
---
### Step 1: Unde... | Qwen/Qwen3-4B | false | 1 | stop | 2,721 | true | null | [] | 1,280,895,223 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.520948+00:00 |
68b6c4756474c7e851bc4f54aa2e20d2c50def5d097e13e6f2f8462b9410555d | In the accompanying diagram, two circles with radii $10$ and $8$ are drawn with their centers $15$ units apart. At $T$, one of the points of intersection, a line is drawn in such a way that the chords $ST$ and $TU$ have equal length. Find the square of the length of $ST$.
[asy]
size(160);
defaultpen(linewidth(.8pt)+fo... | We are given:
- Two circles with radii $ r_1 = 10 $ and $ r_2 = 8 $,
- Their centers are 15 units apart,
- A line passes through the point of intersection $ T $ such that the chords $ ST $ and $ TU $ have equal length,
- We are to find the **square of the length of $ ST $**.
---
### Step 1: Understand the Geometry
... | Qwen/Qwen3-4B | false | 1 | stop | 3,035 | true | null | [] | 3,030,701,962 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:59.991505+00:00 |
15a816a29bbb47d81e6ed22000c31c45d773b8d5c7ce3f3a048809ae4181dd1d | Find the product of the complex roots of $x^4 + 2x^2 + 3x - 5 = 0.$ | We are given the polynomial:
$$
x^4 + 2x^2 + 3x - 5 = 0
$$
We are asked to find the **product of the complex roots** of this equation.
---
### Step 1: Understand the polynomial
This is a quartic (degree 4) polynomial. Let's denote the roots of the polynomial as $ r_1, r_2, r_3, r_4 $, and suppose that two of them ... | Qwen/Qwen3-4B | false | 2 | stop | 4,534 | true | null | [] | 4,042,009,444 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:12:56.294386+00:00 |
31f5a0ab7d3c73edda5833ee9693762201eb8f7e59eed261b856cd129defe5bc | Let $p(x)$ be a monic polynomial of degree $n$ with integer coefficients, and let $M$ be the maximum value of $|p(x)|$ on the interval $[0, 1]$. Suppose that $M > 1$ and that $p(x)$ has no integer roots in the interval $[0, M]$. Show that $p(x)$ has at most $n - 1$ integer roots. | We are given:
- $ p(x) $ is a **monic polynomial** of degree $ n $ with **integer coefficients**.
- Let $ M = \max_{x \in [0,1]} |p(x)| $, and $ M > 1 $.
- $ p(x) $ has **no integer roots** in the interval $ [0, M] $.
- We are to **show** that $ p(x) $ has at most $ n - 1 $ **integer roots**.
---
### Step 1: Underst... | Qwen/Qwen3-4B | false | 1 | stop | 2,403 | true | null | [] | 2,303,609,586 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.752430+00:00 |
12781e57d9d9230b0875daa22079cb745d4986619b9f80c6f13928837b15317a | Triangle $DEF$ has positive integer side lengths with $DE=DF$. Let $G$ be the intersection of the bisectors of $\angle E$ and $\angle F$. Suppose $EG=12$. Find the smallest possible perimeter of $\triangle DEF$. | We are given a triangle $ \triangle DEF $ with the following properties:
- $ DE = DF $, so the triangle is **isosceles** with $ DE = DF $.
- Let $ DE = DF = a $, and $ EF = b $, so the sides are $ a, a, b $.
- $ G $ is the **intersection of the angle bisectors** of $ \angle E $ and $ \angle F $.
- $ EG = 12 $.
- We ar... | Qwen/Qwen3-4B | false | 1 | stop | 5,595 | true | null | [] | 2,170,105,323 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.381592+00:00 |
1799a53b2cf002799b4e940134d25151de526f8e9ebb555af55325e229c53d10 | In a certain geometric figure, a square with a side length of 8 units is rotated counterclockwise about its upper right corner by an angle of 90 degrees. The rotated square intersects the original square at a point that is 6 units away from the center of the square. What is the total length, in units, of the segments t... | We are given a **square of side length 8 units** that is **rotated counterclockwise by 90 degrees** about its **upper right corner**. The **rotated square intersects the original square** at a point that is **6 units from the center** of the square. We are to find the **total length of the segments** that form the **bo... | Qwen/Qwen3-4B | false | 1 | stop | 1,884 | true | null | [] | 1,641,030,438 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.877789+00:00 |
88719dfea4303c0fe67b77b8d15b04146b0171b7f8c5c73ae3bdc6b9589cdf20 | In triangle $ABC$, point $D$ lies on side $BC$, point $E$ lies on side $AC$, and point $F$ lies on side $AB$. Lines $AD$, $BE$, and $CF$ are concurrent at point $P$. Given that $BP:PE = 2:3$ and $CP:PF = 4:5$, find $\frac{CD}{DB}$. | We are given a triangle $ ABC $ with points $ D $, $ E $, and $ F $ on sides $ BC $, $ AC $, and $ AB $, respectively. The lines $ AD $, $ BE $, and $ CF $ are **concurrent** at point $ P $. We are also given:
- $ BP:PE = 2:3 $
- $ CP:PF = 4:5 $
We are to find the ratio $ \frac{CD}{DB} $.
---
### Step 1: Use Ceva’s... | Qwen/Qwen3-4B | false | 1 | stop | 1,864 | true | null | [] | 1,143,555,242 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.056790+00:00 |
621dca678249674974935e48df4850dd75491e04a72dabb7e6a830979cab2fe6 | Pirate Pete and Pirate Paul decide to split their treasure into two piles, one for each of them. They start by putting one coin in each pile, then they add two coins to each pile, then three coins, and so on, until they add $x$ coins to each pile. However, Pete gets a 10% bonus for being the captain, so he adds 10% mor... | We are given a problem involving two pirates, **Pirate Pete** and **Pirate Paul**, who are splitting treasure in a specific way.
---
### **Understanding the Process:**
They start by putting **1 coin in each pile**, then **2 coins in each pile**, then **3 coins in each pile**, and so on, **until they add $ x $ coins ... | Qwen/Qwen3-4B | false | 1 | stop | 2,804 | true | null | [] | 1,819,999,752 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:51.994353+00:00 |
8fe7cbf887fd79283b4b875f950bfa164ff62ed0b9db31fb0800f175917d6051 | A student is hosting a party and wants to make sure that each of his guests meets at least one new person. If he invites 10 friends and seats them around a circular table, how many different ways can he arrange the seating so that no two people who are already acquainted are sitting next to each other? | We are given:
- A student invites **10 friends**.
- These 10 friends are to be seated **around a circular table**.
- The goal is to **arrange the seating so that no two people who are already acquainted are sitting next to each other**.
- The key assumption here is that **the 10 friends are all acquainted with each ot... | Qwen/Qwen3-4B | false | 1 | stop | 1,097 | true | null | [] | 2,933,862,328 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.755894+00:00 |
c193d3a0d1e9169f9d894fc69443b99a79cba9e2bb9af13a111bc827e26a3e90 | Let $a,$ $b,$ $c$ be distinct, nonzero real numbers such that
\[a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}.\]
Find $|abc|.$
Here is another problem inspired by this one:
Let $x,$ $y,$ $z$ be distinct, nonzero real numbers such that
\[x^2 + y^2 = y^2 + z^2 = z^2 + x^2.\]
Find $|xyz|.$ | We are given that:
$$
x^2 + y^2 = y^2 + z^2 = z^2 + x^2
$$
Let’s denote this common value as $ k $, so:
$$
x^2 + y^2 = k \\
y^2 + z^2 = k \\
z^2 + x^2 = k
$$
We are to find $ |xyz| $, given that $ x, y, z $ are distinct, nonzero real numbers.
---
### Step 1: Subtract equations to eliminate $ k $
From the first t... | Qwen/Qwen3-4B | false | 1 | stop | 1,623 | true | null | [] | 3,220,560,794 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.465515+00:00 |
bdaa054381b27a80434b7e71755bad4739bad353cfbd0a42bcee301253d3bf06 | In trapezoid $ABCD$, $AD$ is parallel to $BC$, and the diagonals $AC$ and $BD$ intersect at $P$. If $AP = BP = 2PD$, and $CP = 5$, find $AD + BC$. | We are given a trapezoid $ABCD$ with $AD \parallel BC$, and the diagonals $AC$ and $BD$ intersect at point $P$. We are also given:
- $AP = BP = 2PD$
- $CP = 5$
We are to find $AD + BC$.
---
### Step 1: Use the given ratios
Let’s denote the lengths in terms of a variable.
Let $PD = x$, then:
- $AP = BP = 2x$
- $P... | Qwen/Qwen3-4B | false | 1 | stop | 1,803 | true | null | [] | 2,989,611,933 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.906426+00:00 |
dc9c3e6ab2b831aef0251bfd2b5e65542bc6c0182b2fd73b16abfaf33a4db6f0 | Let f(x) = ax^3+bx^2+cx+d be a cubic polynomial for which holds that the sum of the coefficients equals to the sum of the roots, the product of the coefficients equals to the product of the roots and the sum of the squares of the coefficients equals to the sum of the squares of the roots. If we know the roots of the po... | We are given a cubic polynomial:
$$
f(x) = ax^3 + bx^2 + cx + d
$$
Let the roots of the polynomial be in **geometric progression**. Let the roots be:
$$
\alpha, \alpha r, \alpha r^2
$$
Then the polynomial can be written as:
$$
f(x) = a(x - \alpha)(x - \alpha r)(x - \alpha r^2)
$$
We are told:
1. The **sum of the... | Qwen/Qwen3-4B | false | 2 | stop | 3,725 | true | null | [] | 2,260,016,330 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:16.954612+00:00 |
58bdab7727c352c65682a29647df71b1f8b09a36b36dc57d9babefd0aae8ef20 | Let $f(x) = \frac{2x-1}{x+3}$ and $g(x) = 3x^2 + 2x - 5$. Find constants $A$ and $B$ such that $Af(x) + Bg(x) = 6x^2 - 4x - 7$. What is the value of $AB$? | We are given:
- $ f(x) = \frac{2x - 1}{x + 3} $
- $ g(x) = 3x^2 + 2x - 5 $
- We are to find constants $ A $ and $ B $ such that:
$$
A f(x) + B g(x) = 6x^2 - 4x - 7
$$
We are to find $ AB $.
---
### Step 1: Express the equation
$$
A \cdot \frac{2x - 1}{x + 3} + B \cdot (3x^2 + 2x - 5) = 6x^2 - 4x - 7
$$
This is a... | Qwen/Qwen3-4B | false | 2 | stop | 2,408 | true | null | [] | 4,221,046,836 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:10:43.754213+00:00 |
b312e3cac58efa5927d01b78d2dbf358901d3f7834729046cb0afd2ebfbf702b | The coordinates of the center of the circle with equation $x^2+y^2=mx+ny-49$ are $(p,q)$. The distance between $(p,q)$ and the point $(5,7)$ is $\sqrt{65}$. Find $p+q$. | We are given the equation of a circle:
$$
x^2 + y^2 = mx + ny - 49
$$
We are to find the sum $ p + q $, where $ (p, q) $ is the **center** of this circle, and we are also told that the distance between the center $ (p, q) $ and the point $ (5, 7) $ is $ \sqrt{65} $.
---
### Step 1: Rewrite the equation in standard ... | Qwen/Qwen3-4B | false | 1 | stop | 1,851 | true | null | [] | 2,809,310,687 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.635910+00:00 |
a07654664e58767b57d5beaea3952bd58458c42e4b707ccb3270884eb5a28ea3 | Find the number of positive integers $n$ less than 100 such that the decimal representation of $\frac{1}{n}$ has a non-repeating block of digits (i.e., the decimal representation is either terminating or repeating, but with a repeating block of length greater than 1). | We are asked to find the number of **positive integers $ n < 100 $** such that the **decimal representation of $ \frac{1}{n} $** has a **non-repeating block of digits**. That is, the decimal is either **terminating** or has a **repeating block of length greater than 1**.
---
### Step 1: Understand the decimal represe... | Qwen/Qwen3-4B | false | 1 | stop | 1,085 | true | null | [] | 3,343,237,821 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:00.181215+00:00 |
ed74e8fd258661bf926d68150655a53b6eb15765abf7417054ee87facf7e5f73 | In the coordinate plane, a square with sides parallel to the axes is to be constructed so that its diagonals intersect on the line $y = x + 1$. Additionally, one of the vertices of the square must lie on the parabola $y = x^2 + 2$. How many different squares can be constructed under these conditions? | We are given the following conditions for constructing a square in the coordinate plane:
1. The **diagonals** of the square intersect on the line $ y = x + 1 $.
2. One of the **vertices** of the square lies on the **parabola** $ y = x^2 + 2 $.
3. The square has **sides parallel to the axes**.
We are to determine **ho... | Qwen/Qwen3-4B | false | 1 | stop | 3,975 | true | null | [] | 940,343,391 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:52.092968+00:00 |
90ff54acc208f4478d0745b6d1f5f5ec18ba7d0415ff37a416e56f147471aaa1 | Let $g(n)$ be the product of all the proper divisors of a positive integer $n$. If $g(g(n)) = 2n$, then call $n$ hyperabundant. How many hyperabundant positive integers are there? | We are given a function $ g(n) $, which is defined as the **product of all the proper divisors** of a positive integer $ n $. A **proper divisor** of $ n $ is any positive divisor of $ n $ **other than** $ n $ itself.
We are told that a number $ n $ is **hyperabundant** if:
$$
g(g(n)) = 2n
$$
We are to find how many... | Qwen/Qwen3-4B | false | 1 | stop | 3,966 | true | null | [] | 3,577,466,665 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.515745+00:00 |
ddb7792ce11408f50dd188757e78c017f814a943bdcc02945e14739c1cd6c1b8 | In a regular hexagon, each side is divided into 3 equal parts. What is the probability that a randomly chosen point inside the hexagon is closer to the center of the hexagon than to any of the vertices? Express your answer as a common fraction. | We are given a **regular hexagon**, and each side is divided into **3 equal parts**. We are to find the **probability** that a **randomly chosen point inside the hexagon** is **closer to the center** of the hexagon than to **any of the vertices**.
---
### Step 1: Understand the Geometry
Let’s consider a **regular he... | Qwen/Qwen3-4B | false | 1 | stop | 1,743 | true | null | [] | 694,633,071 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.911162+00:00 |
9f5ed4f58648ea9fd839a96c6d14ed8c33ce89481578cbb6d8ea8674e656189c | Find the minimum value of the constant $c$ for which the ellipse $\frac{x^2}{c} + y^2 = 1$ and the parabola $x^2 - y + 1 = 0$ intersect. | We are given:
- An **ellipse**: $\frac{x^2}{c} + y^2 = 1$
- A **parabola**: $x^2 - y + 1 = 0$
We are to **find the minimum value of the constant $ c $** such that these two curves **intersect**.
---
### Step 1: Express the parabola in terms of $ y $
From the parabola equation:
$$
x^2 - y + 1 = 0 \Rightarrow y = x... | Qwen/Qwen3-4B | false | 1 | stop | 1,577 | true | null | [] | 2,858,782,513 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.074230+00:00 |
b5a11fecab50922dfd996237e8ccb3a47474216fc77a48cd709f41c1e821c245 | Let $(b_1,b_2,b_3,\ldots,b_{12})$ be a sequence of 12 distinct positive integers such that the sum of the terms in the first half is equal to the sum of the terms in the second half, and each term in the second half is 3 more than the corresponding term in the first half. Find the number of such sequences. | We are given a sequence of 12 **distinct positive integers**:
$$
(b_1, b_2, b_3, \ldots, b_{12})
$$
such that:
1. The **sum of the first 6 terms** equals the **sum of the last 6 terms**.
2. Each term in the **second half** is **3 more** than the corresponding term in the **first half**.
We are to **find the number ... | Qwen/Qwen3-4B | false | 1 | stop | 803 | true | null | [] | 1,002,294,640 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.659042+00:00 |
0c43ce611a57af3bb11a107bae32b3bfe899839e1fda5f5777af91d3586a66c7 | The sides of a hexagon are all parallel to the sides of a rectangle with sides of length 6 and 5. The hexagon shares a side with the rectangle. Prove that the perimeter of the hexagon is 22 units. | We are given a **hexagon** whose **sides are all parallel to the sides of a rectangle with sides of length 6 and 5**, and the hexagon **shares one side with the rectangle**. We are to **prove that the perimeter of the hexagon is 22 units**.
---
### Step 1: Understand the Geometry
Let’s denote the rectangle as **ABCD... | Qwen/Qwen3-4B | false | 1 | stop | 1,271 | true | null | [] | 3,422,817,740 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.661985+00:00 |
ee593da1055efda3f594438931d8d279fb9b1800026168dca377e4836c57dbce | Points $M$, $N$, $P$, $Q$, and $R$ are located in 3-dimensional space, with points $M$, $N$, and $P$ lying in the same plane. It is given that $\angle PMQ = \angle QNR = 90^\circ$, and that $MP = MQ = NQ = NR = RP = 1$. What is the volume of pyramid $MNQR$? | We are given a geometric configuration in 3D space with the following points:
- Points $ M, N, P, Q, R $ are in 3D space.
- Points $ M, N, P $ lie in the same plane.
- $ \angle PMQ = \angle QNR = 90^\circ $
- $ MP = MQ = NQ = NR = RP = 1 $
We are to find the **volume of pyramid $ MNQR $**.
---
### Step 1: Understan... | Qwen/Qwen3-4B | false | 2 | stop | 3,720 | true | null | [] | 2,358,266,515 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:13:17.530854+00:00 |
b3740aa2369f23a8609e3d547afc57af77df902dbd508662355d4357fadea429 | A circle inscribed inside the isosceles trapezoid $ABCD$ touches the midpoint $E$ of the base $BC$.
Determine the length of the base of the trapezoid $EFCD$ in terms of the distance $AE$. | We are given an **isosceles trapezoid** $ABCD$ with an **inscribed circle** (i.e., a **tangential trapezoid**) that **touches the midpoint $E$ of the base $BC$**. We are to determine the **length of the base $EFCD$** (which is the base $CD$) in terms of the **distance $AE$**.
---
### Step 1: Understand the Geometry
... | Qwen/Qwen3-4B | false | 1 | stop | 4,997 | true | null | [] | 469,872,864 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:52.110404+00:00 |
5dc9b1d897e731a797203125b7b5b55eaa31fa09a14837d3505f50cc7f4ebc62 | What is the greatest possible number of zeros at the end of the product of two whole numbers whose sum of digits is 18? | We are given two **whole numbers** whose **sum of digits is 18**, and we are to find the **greatest possible number of zeros at the end of their product**.
---
### Step 1: Understand what makes a number end in zeros
A number ends in a zero if it is **divisible by 10**. A number is divisible by 10 if it is divisible ... | Qwen/Qwen3-4B | false | 1 | stop | 1,945 | true | null | [] | 3,144,113,364 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.877077+00:00 |
d202db701a31b26da61b568ea52549c6201970705d828ea137278f3b6c46b0c5 | Let $a,$ $b,$ $x,$ and $y$ be real numbers such that
$$a\cos x+b\cos y=1,$$$$a\cos 2x+b\cos 2y=2,$$$$a\cos 3x+b\cos 3y=-4,$$$$a\cos 4x+b\cos 4y=-8,$$find $a\cos 5x+b\cos 5y.$ | We are given the following system of equations involving trigonometric functions and real numbers:
$$
\begin{aligned}
(1) &\quad a\cos x + b\cos y = 1 \\
(2) &\quad a\cos 2x + b\cos 2y = 2 \\
(3) &\quad a\cos 3x + b\cos 3y = -4 \\
(4) &\quad a\cos 4x + b\cos 4y = -8 \\
\end{aligned}
$$
We are to find:
$$
a\cos 5x + ... | Qwen/Qwen3-4B | false | 1 | stop | 1,037 | true | null | [] | 745,356,962 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.590485+00:00 |
d95e9e43235d0a71fe162e29c3f0e02030d4e31b45bc76eea9760b8fe4e250a4 | A polynomial is called "$t$-special" if its coefficients are integers and all its roots are complex numbers of the form $r(t+i)$, where $r$ and $t$ are real numbers. For each integer $n>1$, let $T_n$ denote the set of all monic polynomials of degree $n$ that are $t$-special for some real number $t$. Find the largest po... | We are given the definition of a **$t$-special** polynomial and asked to find the **largest possible value of $n$** such that the set $T_n$ of all monic, degree-$n$ $t$-special polynomials is **finite**.
---
### Step 1: Understanding the Definition
A polynomial is **$t$-special** if:
- Its **coefficients are intege... | Qwen/Qwen3-4B | false | 1 | stop | 2,743 | true | null | [] | 1,123,850,420 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.948918+00:00 |
deb624954fa12225b4113d372f9d19edee7826a73b4f95d31101a43a9af4ca98 | Rafael scores an average of 75% on his first n tests and an average of 80% on his next n tests. If his overall average on all 2n tests is 77%, what is the smallest positive integer value of n? | We are given:
- Rafael scores an **average of 75%** on his **first n tests**.
- He scores an **average of 80%** on his **next n tests**.
- His **overall average** on all $ 2n $ tests is **77%**.
- We are to find the **smallest positive integer value of n**.
---
### Step 1: Express total scores
Let’s denote:
- Tota... | Qwen/Qwen3-4B | false | 1 | stop | 1,985 | true | null | [] | 997,216,008 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.079084+00:00 |
5e5fe47885aa1a89889e813c06eb9094b2b9a270db28da35e9d9eef76fa94dba | In trapezoid $PQRS$, $\angle{PQS}\cong\angle{SRQ}$ and $\angle{PSQ}\cong\angle{QSR}$, $PQ = 12$, $SQ = 8$, and $RS = 9$. If the diagonals $PR$ and $QS$ intersect at point $T$, find the ratio of the areas of triangles $\triangle PQT$ and $\triangle RST$. | We are given a trapezoid $ PQRS $ with the following information:
- $ \angle PQS \cong \angle SRQ $
- $ \angle PSQ \cong \angle QSR $
- $ PQ = 12 $
- $ SQ = 8 $
- $ RS = 9 $
- Diagonals $ PR $ and $ QS $ intersect at point $ T $
We are to find the **ratio of the areas** of triangles $ \triangle PQT $ and $ \triangle ... | Qwen/Qwen3-4B | false | 1 | stop | 1,290 | true | null | [] | 1,131,302,489 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.708911+00:00 |
3760e1250cc357a1d473d75df0fbe75a14c4860bcdfaf64afb700cd93db85d80 | Eight students are standing in a row, and each one is wearing a shirt of a different color. The colors are yellow, blue, green, purple, orange, pink, red, and black. For some reason, all students wearing a shirt of a primary color (red, blue, and yellow) refuse to stand next to anyone wearing a shirt of a secondary col... | We are given:
- 8 students, each wearing a **different** shirt color: yellow, blue, green, purple, orange, pink, red, and black.
- **Primary colors**: red, blue, yellow.
- **Secondary colors**: green, purple, orange.
- **Neutral colors**: pink, black (these students don't care about color restrictions).
- **Rule**: St... | Qwen/Qwen3-4B | false | 1 | stop | 1,525 | true | null | [] | 488,009,531 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.709033+00:00 |
ce7294066b982e4e159ce0a3d6f8d06ba44ec37d8d4135de63feda2f8f117566 | In a regular octagon $ABCDEFGH$, the midpoints of the sides are joined to form a square $EFGH$. If the area of the octagon is $480$, what is the area of square $EFGH$? | We are given:
- A **regular octagon** $ABCDEFGH$.
- The **midpoints of its sides** are joined to form a **square** $EFGH$.
- The **area of the octagon** is $480$.
- We are to find the **area of square $EFGH$**.
---
### Step 1: Understand the Geometry
A **regular octagon** has 8 equal sides and 8 equal angles. The m... | Qwen/Qwen3-4B | false | 1 | stop | 1,514 | true | null | [] | 1,013,724,257 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:00.278993+00:00 |
4dea1ccb07e1215b30d52f76eee4a13136fcc54ef23cbb7cd26f3366fbcb8b5a | In quadrilateral $ABCD$, $AB = 6$, $BC = 8$, $CD = 5$, and $DA = 7$. If the diagonals $AC$ and $BD$ intersect at a point $O$ such that $\angle AOB = 60^\circ$, find the length of $AO$. | We are given a quadrilateral $ABCD$ with side lengths:
- $AB = 6$
- $BC = 8$
- $CD = 5$
- $DA = 7$
The diagonals $AC$ and $BD$ intersect at point $O$, and $\angle AOB = 60^\circ$. We are to **find the length of $AO$**.
---
### Step 1: Use the Law of Cosines in triangle $AOB$
Let’s denote:
- $AO = x$
- $BO = y$
- ... | Qwen/Qwen3-4B | false | 1 | stop | 1,964 | true | null | [] | 4,033,302,475 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:52.164106+00:00 |
aa77eeeb1c6a2200c19955faf3581736c9b064d4f295e242fa56beee008f13e7 | A spherical shell with an outer radius of 6 units and an inner radius of 4 units is given. A bug starts at a point on the outer surface of the shell and crawls along the surface to a point on the inner surface. The bug's path is constrained to lie on a plane that passes through the center of the sphere. Find the minimu... | We are given a **spherical shell** with:
- **Outer radius** = 6 units
- **Inner radius** = 4 units
A **bug starts on the outer surface** and **ends on the inner surface**, and its **path is constrained to lie on a plane that passes through the center of the sphere**.
We are to find the **minimum distance the bug... | Qwen/Qwen3-4B | false | 1 | stop | 631 | true | null | [] | 2,858,021,804 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.880908+00:00 |
f6c29f15c167ab9dd8798d9317f83bd5017417ef05f6212fe7288da9775154e0 | In an isosceles triangle, the vertices are $(0,0)$, $(10,0)$ and $(5, \sqrt{50})$. If a line passing through the points $(0,0)$ and $(5,0)$ cuts the triangle into two smaller triangles of equal area, what is the equation of that line? | We are given an **isosceles triangle** with vertices at:
- $ A = (0, 0) $
- $ B = (10, 0) $
- $ C = (5, \sqrt{50}) $
We are told that a line passes through the points $ (0, 0) $ and $ (5, 0) $, and **cuts the triangle into two smaller triangles of equal area**. We are to find the **equation of that line**.
---
### ... | Qwen/Qwen3-4B | false | 1 | stop | 2,123 | true | null | [] | 846,604,393 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.660341+00:00 |
7991b0317ae5cf29c69dd679e0f2c357e0cf190ac2a1c27a2e91cb3e693a373a | Problem:
Given two ellipses
$$\frac{(x-2)^2}{25}+\frac{(y-1)^2}{16}=1$$
and
$$\frac{(x+3)^2}{20}+\frac{(y-4)^2}{12}=1,$$
find the largest positive value of $b$ for which the line $y=bx$ is tangent to both ellipses and the point of tangency with each ellipse lies inside the other ellipse. | We are given two ellipses:
$$
\frac{(x-2)^2}{25}+\frac{(y-1)^2}{16}=1 \tag{1}
$$
$$
\frac{(x+3)^2}{20}+\frac{(y-4)^2}{12}=1 \tag{2}
$$
We are to find the **largest positive value of $ b $** for which the line $ y = bx $ is **tangent** to **both** ellipses, and the **point of tangency with each ellipse lies inside the... | Qwen/Qwen3-4B | false | 1 | stop | 2,764 | true | null | [] | 1,130,111,127 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.953584+00:00 |
925b7f9be519ec86f3b0e056005b56287e5225ee335b708e51888a9798c2bb3b | Consider a base-$b$ number system, where $b$ is a positive integer greater than 1. In this system, the digits 0 through $b-1$ are used to represent numbers. Let $S_b(n)$ denote the sum of the digits of the base-$b$ representation of the positive integer $n$. Find the smallest base $b$ such that $S_b(1001)+S_b(2002)+S_b... | We are given the following:
- $ S_b(n) $ is the **sum of the digits** of the number $ n $ in **base $ b $**.
- We are to find the **smallest base $ b > 1 $** such that:
$$
S_b(1001) + S_b(2002) + S_b(3003) = S_b(4004) + S_b(5005) + S_b(6006)
$$
---
### Step 1: Understand what the numbers 1001, 2002, 3003, etc., rep... | Qwen/Qwen3-4B | false | 2 | stop | 2,808 | true | null | [] | 3,370,980,940 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:17.059104+00:00 |
59a664c1f62adf8750ceb9a5d09d992dc102edbe90730f5f1f7768f4dedf5da6 | A circular sector has an angle of $\theta$ radians and a radius of $\theta+2$ units. A circle has a radius of $\theta-3$ units. If the area of the circular sector is three times the area of the circle, what is the value of $\theta$? | We are given:
- A **circular sector** with angle $\theta$ radians and radius $r = \theta + 2$ units.
- A **circle** with radius $R = \theta - 3$ units.
- The **area of the sector** is **three times** the **area of the circle**.
We are to find the value of $\theta$.
---
### Step 1: Write the formula for the area of ... | Qwen/Qwen3-4B | false | 1 | stop | 3,574 | true | null | [] | 3,524,343,716 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.759804+00:00 |
f6441f249c315b1350896c893c634b3786a1a3a236bac72d7016f837e1165f40 | The equation
$$(x - \sqrt[3]{19})(x - \sqrt[3]{41})(x - \sqrt[3]{97}) = \frac{1}{7}$$
has three distinct solutions $p,$ $q,$ and $r.$ Calculate the value of $p^3 + q^3 + r^3.$ | We are given the equation:
$$
(x - \sqrt[3]{19})(x - \sqrt[3]{41})(x - \sqrt[3]{97}) = \frac{1}{7}
$$
We are told that this equation has **three distinct solutions** $ p, q, r $, and we are to compute:
$$
p^3 + q^3 + r^3
$$
---
### Step 1: Let’s define the cubic polynomial
Let’s denote:
$$
f(x) = (x - a)(x - b)(... | Qwen/Qwen3-4B | false | 1 | stop | 2,011 | true | null | [] | 1,177,243,616 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.752160+00:00 |
afece126f23888588afdd7bb4c25778ac3a2edee890661e2966111107435fe71 | In the triangle $ABC$, $\angle B = 90^{\circ}$, $AC = 3\sqrt{2}$, and the bisector of angle $A$ intersects side $BC$ at point $D$. If a circle with center $D$ is tangent to side $AB$ at point $E$, find the length of segment $CE$. | We are given a right triangle $ ABC $ with $ \angle B = 90^\circ $, $ AC = 3\sqrt{2} $, and the angle bisector of $ \angle A $ intersects side $ BC $ at point $ D $. A circle centered at $ D $ is tangent to side $ AB $ at point $ E $. We are to find the length of segment $ CE $.
---
### Step 1: Understand the triangl... | Qwen/Qwen3-4B | false | 1 | stop | 3,571 | true | null | [] | 3,391,824,457 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:00.289913+00:00 |
7842deda1341f7fe166135117e8b8385841c28bf83c3823a5aeb816e24ec9b0c | Consider a list of positive integers where each number has digits that alternate between even and odd numbers, starting from an even digit. The list is in increasing order, and the first number is 204. What is the seventh number in the list? | We are given a list of **positive integers** where:
- Each number has **digits that alternate between even and odd**, starting with an **even digit**.
- The list is in **increasing order**.
- The **first number** is **204**.
- We are to find the **seventh number** in the list.
---
### Step 1: Understand the pattern
... | Qwen/Qwen3-4B | false | 1 | stop | 1,476 | true | null | [] | 3,464,159,185 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:52.264726+00:00 |
2fe67c2c98d105855f99e558ee232b1093d820638cdd6ed2d80c544cae68eb3d | The graphs of the equations $y^2 = x^3 - x + 1$ and $x^2 + y^2 = r^2$ intersect in exactly two points for some real number $r$. Find the largest possible value of $r$. | We are given two equations:
1. $ y^2 = x^3 - x + 1 $ (a cubic curve)
2. $ x^2 + y^2 = r^2 $ (a circle with radius $ r $)
We are told that these graphs **intersect in exactly two points** for **some real number $ r $**, and we are to **find the largest possible value of $ r $** for which this is true.
---
### Step... | Qwen/Qwen3-4B | false | 1 | stop | 1,327 | true | null | [] | 2,649,216,102 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.740507+00:00 |
bab1baf6e3a50fd17ff51e9a59405dd34dee936a0647d8f1f8bc937e8ea71e48 | A security camera is placed at a random location along a 1-kilometer stretch of a highway. A car travels along this highway, and its speed is also random, uniformly distributed between 60 and 120 km/h. What is the probability that the car will be captured by the camera, assuming that the camera can capture the car if i... | We are given the following:
- A **security camera** is placed **uniformly at random** along a **1-kilometer** stretch of highway.
- A **car** is traveling along the same highway.
- The **car's speed** is **uniformly distributed** between **60 km/h and 120 km/h**.
- The **camera can capture the car** if it is **within ... | Qwen/Qwen3-4B | false | 1 | stop | 4,118 | true | null | [] | 3,321,672,660 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.992627+00:00 |
48b7eaee3bd40b5fcb7e61c8fe2a8670d1393bc4022057800a550751fc37cc81 | Determine all real values of $x$ such that $x^3-x^2-6x+2$ is a perfect square. | We are given the polynomial:
$$
f(x) = x^3 - x^2 - 6x + 2
$$
We are to **find all real values of $ x $** such that $ f(x) $ is a **perfect square**.
---
### Step 1: Understand the Problem
We are looking for **real numbers $ x $** such that:
$$
f(x) = x^3 - x^2 - 6x + 2 = y^2
$$
for some real number $ y $.
So, w... | Qwen/Qwen3-4B | false | 3 | stop | 3,627 | true | null | [] | 792,501,932 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:10:33.994371+00:00 |
f866b374b2505ece310478d8dd3a10f6ac6b5136733556842232f74ad74f525e | Let $S_n$ be the set of strings of $A$'s and $B$'s such that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length, and let $c_n$ be the number of strings in $S_n$ of length $n$. Define $a_n$ to be the number of strings in $S_n$ that end in $A$, and define $b_n$ to be the num... | We are asked to find the number of sequences of length 14 consisting of the letters $ A $, $ B $, and $ C $, such that:
- Every **run of $ A $'s** has **even** length,
- Every **run of $ B $'s** has **odd** length,
- Every **run of $ C $'s** has **length divisible by 3**.
Let’s define the problem more formally.
---
... | Qwen/Qwen3-4B | false | 1 | stop | 2,440 | true | null | [] | 1,907,926,108 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.776710+00:00 |
ee4710fe028351d70d07b28d61adc9245aba91e568bef373a2b18f0b5c322f03 | Let $C$ be the circle with equation $(x - 2)^2 + (y - 1)^2 = 4.$ Find the set of all points $(a,b)$ that are the center of a circle that passes through the point $(3,3)$ and is tangent to $C$ at some point. Enter your answer in the form
\[Pa^2 + Qb^2 + Ra + Sb + T = 0,\]where all the coefficients are integers, $P$ is... | We are given the circle $ C $ with equation:
$$
(x - 2)^2 + (y - 1)^2 = 4
$$
This is a circle with **center** $ (2, 1) $ and **radius** $ \sqrt{4} = 2 $.
We are to find all points $ (a, b) $ that are the **center** of a **circle** that:
- Passes through the point $ (3, 3) $
- Is **tangent** to the circle $ C $
---... | Qwen/Qwen3-4B | false | 1 | stop | 2,172 | true | null | [] | 2,009,374,953 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.755195+00:00 |
212b127e53578eeb3ba37f33eb2f1ecaa21b14fed66566ac8b3e090717b85f6c | Determine the largest positive integer $m$ such that there exist positive integers $a, b, c, d$ so that \[
m^3 = a^3 + b^3 + c^3 + d^3 + ab(a+b) + bc(b+c) + cd(c+d) + da(d+a) - 24
\] | We are given the equation:
$$
m^3 = a^3 + b^3 + c^3 + d^3 + ab(a + b) + bc(b + c) + cd(c + d) + da(d + a) - 24
$$
We are to find the **largest positive integer** $ m $ such that there exist **positive integers** $ a, b, c, d $ satisfying this equation.
---
### Step 1: Understand the structure of the expression
Let... | Qwen/Qwen3-4B | false | 3 | stop | 2,877 | true | null | [] | 3,572,616,438 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:16:10.245819+00:00 |
53d31d7bb983113735b6f27a6239fa9469be1f1b6982016f175b0fbd29d92dff | In a right-angled triangle, the length of the hypotenuse is $\sqrt{29}$ units, and one of the legs has a length of 5 units. The triangle is reflected across the hypotenuse, creating a new triangle. The vertex of the reflected triangle that was not on the hypotenuse now lies on the $y$-axis. What is the $y$-coordinate o... | We are given:
- A **right-angled triangle** with **hypotenuse** $ \sqrt{29} $ units.
- One of the **legs** is **5 units**.
- The triangle is **reflected across the hypotenuse**, and the **vertex not on the hypotenuse** now lies on the **y-axis**.
- We are to find the **y-coordinate** of this vertex.
---
### Step 1: ... | Qwen/Qwen3-4B | false | 1 | stop | 2,761 | true | null | [] | 2,796,651,965 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:52.385891+00:00 |
6f965cf5d294face08ff9b331ab5914e47dde2ec538d0090c4eab9542531bfed | A cyclist travels uphill at an average speed of 10 km/h for 30 minutes, then downhill at an average speed of 30 km/h for 20 minutes, and finally on flat terrain at an average speed of 20 km/h for 40 minutes. If the total distance traveled is 30 km, what is the average gradient of the uphill section, assuming the downhi... | We are given the following:
- **Uphill**:
- Speed: 10 km/h
- Time: 30 minutes = 0.5 hours
- **Downhill**:
- Speed: 30 km/h
- Time: 20 minutes = 1/3 hour
- **Flat**:
- Speed: 20 km/h
- Time: 40 minutes = 2/3 hour
- **Total distance**: 30 km
We are to find the **average gradient of the uphill section**, ass... | Qwen/Qwen3-4B | false | 1 | stop | 3,793 | true | null | [] | 396,851,803 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:14.961662+00:00 |
6c408d37555bce28f84eb5b562f95bac7f9a15bc8f0452d8f2881dee759055d5 | A sequence of consecutive integers starts with a positive number $a$. The sum of the first $n$ terms of the sequence is $1234$. What is the value of $a$ if the sum of the squares of the first $n$ terms is $399099$? | We are given:
- A sequence of **consecutive integers** starting with a positive number $ a $.
- The **sum of the first $ n $ terms** is $ 1234 $.
- The **sum of the squares of the first $ n $ terms** is $ 399099 $.
We are to find the value of $ a $.
---
### Step 1: Express the sequence
Since it's a sequence of **c... | Qwen/Qwen3-4B | false | 2 | stop | 4,420 | true | null | [] | 3,870,301,337 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:10:16.455788+00:00 |
9f6fc71a6e289bcc6da12865ffd4c131260bb6455715ee857069ccb8c9307f02 | Find all solutions to the equation $x^3 + x^2 - 4x - 4 = \sqrt{x + 1}$ and determine the sum of their squares.
This problem requires a different approach than the original problem, as it involves solving a cubic equation with a radical term. The student will need to use algebraic manipulations and possibly numerical m... | We are given the equation:
$$
x^3 + x^2 - 4x - 4 = \sqrt{x + 1}
$$
Our goal is to **find all real solutions** to this equation and then compute the **sum of the squares of the solutions**.
---
### Step 1: Understand the domain of the function
The right-hand side is a square root: $\sqrt{x + 1}$, which is defined o... | Qwen/Qwen3-4B | false | 1 | stop | 2,879 | true | null | [] | 1,754,105,443 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:09.997397+00:00 |
4985f105fe78a8d43791b8eba6c911dfc9d0a3902f78f3974a01c4af75d828a8 | Find the number of integer solutions to the equation $x^3 + y^3 + z^3 = 30$. | We are given the equation:
$$
x^3 + y^3 + z^3 = 30
$$
and are asked to find the number of **integer solutions** $(x, y, z)$ that satisfy this equation.
---
### Step 1: Understand the nature of the equation
This is a **sum of cubes** equation with three variables. We are looking for all **integer** values of $x, y,... | Qwen/Qwen3-4B | false | 2 | stop | 6,475 | true | null | [] | 3,286,917,222 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:17.178126+00:00 |
91b9ad025ef56b64ac35e75e312537cc72fab2e7d5728bae1629deb05f8b3cec | Let $a_n = \frac{n!}{2^n}$ for each positive integer $n$. Find the largest value of $n$ for which $a_n$ is not an integer. | We are given the sequence:
$$
a_n = \frac{n!}{2^n}
$$
We are to find the **largest value of $ n $** for which $ a_n $ is **not an integer**.
---
### Step 1: Understand the expression
We are looking at the expression:
$$
a_n = \frac{n!}{2^n}
$$
We want to find the **largest** $ n $ such that $ a_n $ is **not an i... | Qwen/Qwen3-4B | false | 1 | stop | 6,816 | true | null | [] | 2,658,217,757 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.858087+00:00 |
75a5b4e88355fb9a6cd6784ffe5b16ce05817405229ad761a8e0b786aae3c7ab | Find all values of $k$ such that the equation $(x^2-kx+5)^2-13x^2-4kx=0$ is satisfied by infinitely many values of $x$. | We are given the equation:
$$
(x^2 - kx + 5)^2 - 13x^2 - 4kx = 0
$$
We are to find all values of $ k $ such that this equation is **satisfied by infinitely many values of $ x $**.
---
### Step 1: Expand the expression
Let’s expand the left-hand side:
$$
(x^2 - kx + 5)^2 - 13x^2 - 4kx = 0
$$
First, expand $ (x^2 ... | Qwen/Qwen3-4B | false | 1 | stop | 1,329 | true | null | [] | 2,320,717,367 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.793813+00:00 |
825d598c15e26811186409ff65eba8716828b5f129431e795788b93625168fd3 | Let $a$, $b$, and $c$ be constants such that the function $f(x) = \log_a (\log_b (\log_c x))$ has the domain $x \in (10, \infty)$ and the range $f(x) \in (1, \infty)$. Find the value of $c$. | We are given the function:
$$
f(x) = \log_a (\log_b (\log_c x))
$$
and we are told that the **domain** is $ x \in (10, \infty) $, and the **range** is $ f(x) \in (1, \infty) $. We are to find the value of $ c $.
---
### Step 1: Understand the domain
The function $ f(x) = \log_a (\log_b (\log_c x)) $ is defined onl... | Qwen/Qwen3-4B | false | 1 | stop | 1,664 | true | null | [] | 4,237,010,396 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:00.400160+00:00 |
64943a5b58645a3829215672ea73d1db49b5f7653dacd848412e63befb0af5f3 | The point $(2,7)$ is rotated $90^\circ$ clockwise about the point $(3,5)$, and then reflected across a line passing through the point $(6,1)$. If the final image of the point $(2,7)$ is $(4,3)$, find the equation of the line of reflection. | We are given that the point $ (2,7) $ is:
1. Rotated $ 90^\circ $ **clockwise** about the point $ (3,5) $,
2. Then reflected over a line that passes through the point $ (6,1) $,
3. The final image is $ (4,3) $.
We are to **find the equation of the line of reflection**.
---
### Step 1: Rotate the point $ (2,7) $ 90°... | Qwen/Qwen3-4B | false | 1 | stop | 856 | true | null | [] | 871,664,588 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:52.456423+00:00 |
ceca4a5e19dedc708a7cbb84e84d93ff7f4e7ff7c1bdc487a731d6b7de6854d5 | Let $a$, $b$, and $c$ be positive integers such that $\text{lcm}(ab,c)=72$ and $a+b+c=36$. What is the smallest possible value of $\gcd(a,b,c)$? | We are given:
- $ a, b, c $ are **positive integers**
- $ \text{lcm}(ab, c) = 72 $
- $ a + b + c = 36 $
- We are to find the **smallest possible value of** $ \gcd(a, b, c) $
---
### Step 1: Understand the LCM condition
We are told:
$$
\text{lcm}(ab, c) = 72
$$
This tells us that $ ab $ and $ c $ must be **factors**... | Qwen/Qwen3-4B | false | 1 | stop | 1,237 | true | null | [] | 2,622,380,170 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:15.009020+00:00 |
143b64f216dab73e5ee83afa3e46c454098702e8586493322cf55d4925369168 | Consider a sequence $(b_n)$ defined by $b_0 = 1$ and $b_{n+1} = b_n + \frac{1}{2}\arctan{\frac{1}{b_n}}$ for $n \geq 0$. Find the value of $b_{20}$. | We are given a sequence defined by:
- $ b_0 = 1 $
- $ b_{n+1} = b_n + \frac{1}{2} \arctan\left( \frac{1}{b_n} \right) $
We are to find the value of $ b_{20} $.
---
### Step 1: Understand the recurrence
We have a recursive definition:
$$
b_{n+1} = b_n + \frac{1}{2} \arctan\left( \frac{1}{b_n} \right)
$$
This is a... | Qwen/Qwen3-4B | false | 1 | stop | 1,194 | true | null | [] | 1,167,485,557 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.823724+00:00 |
07acd1fc7511886322f5c53913414003c21832811f3becaca367f08a5c51e93b | In pentagon $ABCDE$, angles $A$ and $B$ are adjacent and measure $45^{\circ}$ and $135^{\circ}$ respectively. When drawn, the angle bisectors of angles $A$ and $B$ intersect at point $F$. If line segment $CF$ bisects angle $C$, what is the measure of obtuse angle $AFC$? | We are given a pentagon $ABCDE$ with the following information:
- Angles $A$ and $B$ are adjacent and measure $45^\circ$ and $135^\circ$ respectively.
- The **angle bisectors** of angles $A$ and $B$ intersect at point $F$.
- Line segment $CF$ **bisects** angle $C$.
- We are to find the **measure of the obtuse angle $A... | Qwen/Qwen3-4B | false | 1 | stop | 1,623 | true | null | [] | 3,218,560,147 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:10.022090+00:00 |
0746e58baecd5ceee63fdfdb569ccb6ce1c89c408b2cd5bc4d7c794e518b5278 | In the diagram, both polygons are regular. What is the value, in degrees, of the sum of the measures of angles $MNO$ and $MNP$?
[asy]
draw(8dir(0)--8dir(45)--8dir(90)--8dir(135)--8dir(180)--8dir(225)--8dir(270)--8dir(315)--cycle,linewidth(2));
draw(8dir(135)--8dir(180)--8dir(180)+(0,-8)--8dir(135)+(0,-8)--8dir(135)--c... | We are given a diagram with two regular polygons. The goal is to find the **sum of the measures of angles $ \angle MNO $ and $ \angle MNP $**.
---
### Step 1: Understand the Diagram
From the Asymptote code and the labels:
- **M** is at **8dir(135)**, which is **west** of the center.
- **N** is at **8dir(180)**, whi... | Qwen/Qwen3-4B | false | 1 | stop | 2,073 | true | null | [] | 3,659,942,948 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.339496+00:00 |
e89d5ec3eb284454b882b7102c3d53cb08cd84b7497d554749167a76cdf96923 | In triangle $ABC$, points $P$, $Q$, $R$ are the circumcenters of triangles $BCD$, $ACD$, $ABD$, respectively. Find $\frac{[PQR]}{[ABC]}$. | We are given a triangle $ ABC $, and three points $ P, Q, R $, which are the **circumcenters** of triangles $ BCD $, $ ACD $, and $ ABD $, respectively. We are to find the ratio $ \frac{[PQR]}{[ABC]} $, where $ [PQR] $ is the area of triangle $ PQR $, and $ [ABC] $ is the area of triangle $ ABC $.
---
### Step 1: Und... | Qwen/Qwen3-4B | false | 1 | stop | 578 | true | null | [] | 1,938,679,709 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.873400+00:00 |
3c2a24150338ac8168451e7c84db000b5b436325769d34993d9d0878ee59d6cc | Consider the functions $h(x) = x^2 - 4x + 2$ and $k(x) = x^4 + x^3 - 1$. Find the value of $x$ such that $h(k(x)) = k(h(x))$. | We are given two functions:
- $ h(x) = x^2 - 4x + 2 $
- $ k(x) = x^4 + x^3 - 1 $
We are to find the value of $ x $ such that:
$$
h(k(x)) = k(h(x))
$$
---
### Step 1: Understand the equation
We are to solve:
$$
h(k(x)) = k(h(x))
$$
This means that we must compute both sides of the equation and find the value(s) ... | Qwen/Qwen3-4B | false | 1 | stop | 1,428 | true | null | [] | 2,405,642,075 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.869344+00:00 |
1a2c21b4fab76a79a5e600e164e1d0ae2aa87e502087138b0ff5873879cd4823 | A three-digit integer $ABC$ is equal to the sum of the two-digit integers $AB$ and $BC$, where $A$, $B$, and $C$ represent distinct digits from 1 to 9. What is the largest possible value of the three-digit integer $ABC$? | We are given a three-digit integer $ ABC $, where:
- $ A, B, C $ are **distinct digits from 1 to 9** (i.e., no zero allowed),
- $ ABC $ is equal to the **sum of the two-digit numbers** $ AB $ and $ BC $,
- We are to find the **largest possible value** of $ ABC $.
---
### Step 1: Express the numbers in terms of digit... | Qwen/Qwen3-4B | false | 3 | stop | 10,548 | true | null | [] | 3,300,052,707 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:16:27.848096+00:00 |
8908bd89eaf7d28bfba0b36259265e48a0781f52f212bdcb63853015fcd42340 | Let $g(x)$ be a function that satisfies the equation
\[xg(x^2) + g(x) = 2x\]for all $x.$ Given that $g(2) = 5,$ find $g(8)$. | We are given a functional equation:
$$
xg(x^2) + g(x) = 2x \quad \text{for all } x,
$$
and we are told that $ g(2) = 5 $. We are to find $ g(8) $.
---
### Step 1: Understand the functional equation
We are given:
$$
xg(x^2) + g(x) = 2x \tag{1}
$$
This is true for all $ x $, so we can plug in specific values of $ x... | Qwen/Qwen3-4B | false | 3 | stop | 989 | true | null | [] | 171,302,350 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:15:44.133025+00:00 |
470cd52e8af72c1ef144dd8292f787cfbe1c37491a4acdc4e1683312d6bcf5ea | A regular tetrahedron with edge length $6$ cm is truncated by cutting off a smaller regular tetrahedron from one of its vertices. The edge length of the smaller tetrahedron is $1$ cm. What is the maximum number of unit cubes with edge length $1$ cm that can be packed inside the resulting frustum (the remaining part of ... | We are given a **regular tetrahedron** with **edge length 6 cm**, and we **cut off a smaller regular tetrahedron** from one of its vertices. The smaller tetrahedron has **edge length 1 cm**. The remaining shape is called a **frustum** of the tetrahedron.
We are to find the **maximum number of unit cubes (1 cm edge len... | Qwen/Qwen3-4B | false | 1 | stop | 1,729 | true | null | [] | 1,410,801,535 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:08:48.837300+00:00 |
573f484b4135c39b91b2623fbc94fce7d7fedcd93286b863b0caba07a524138a | Let $S$ be a set of positive integers with the following property: for every integer $x$ in $S$, there exists a unique positive integer $n$ such that the arithmetic mean of $x$ and the next $n$ consecutive integers is a perfect square. What is the smallest integer that can be in $S$? | We are given a set $ S $ of **positive integers** with the following property:
> For **every** integer $ x \in S $, there exists a **unique** positive integer $ n $ such that the **arithmetic mean** of $ x $ and the next $ n $ consecutive integers is a **perfect square**.
We are to find the **smallest integer** that ... | Qwen/Qwen3-4B | false | 1 | stop | 1,196 | true | null | [] | 747,786,449 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:10.024341+00:00 |
b5d0ce3ea02b3aaf491b97a376b08bcb631f4279739e283f8cf5fd895e2628c5 | In isosceles triangle $ABC$ ($AB=BC$), $P$ is a point on $AC$ such that $\angle APB=90^{\circ}$. $M$ and $N$ are the midpoints of $AB$ and $BC$, respectively, and $PQ$ is perpendicular to $MN$. Given that $BM=10$, $MP=12$, and $PQ=9$, find the length of $AM$ and the area of $\triangle APB$. The area of $\triangle APB$ ... | We are given an **isosceles triangle** $ ABC $ with $ AB = BC $, so $ \angle B = 90^\circ $ (since it's isosceles with equal sides $ AB = BC $, and the base is $ AC $), making triangle $ ABC $ a **right isosceles triangle**.
---
### Step 1: Understand the Geometry
- Triangle $ ABC $ is **right isosceles** with $ AB ... | Qwen/Qwen3-4B | false | 1 | stop | 4,295 | true | null | [] | 2,241,042,535 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:07:03.385306+00:00 |
4abb81219ebbaa85885e83ed8ad185dcb0a9b2428cc3ff8871d17b0658541c5f | A rhombus-shaped region is formed by two congruent triangles that are externally tangent to a smaller circle, and internally tangent to a larger circle. The radius of the smaller circle is 3, and the radius of the larger circle is 6. What is the area of the region outside the smaller circle and inside the larger circle... | We are given a **rhombus-shaped region** formed by **two congruent triangles** that are **externally tangent to a smaller circle** (radius = 3) and **internally tangent to a larger circle** (radius = 6). We are to find the area **outside the smaller circle**, **inside the larger circle**, but **not in the rhombus-shape... | Qwen/Qwen3-4B | false | 1 | stop | 2,425 | true | null | [] | 1,763,816,238 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:02.914511+00:00 |
4d5b7237b6d1b910c4c424779c747182fd64c7bc6d3406587414f2b23680629f | A circle is divided into 4 sectors, and each sector is further divided into smaller equal arcs by a total of 24 equally spaced radii. The sum of the lengths of these 24 radii is equal to the circumference of the circle. What is the area of the circle? | We are given the following:
- A **circle** is divided into **4 sectors**.
- Each sector is further divided into **smaller equal arcs** by a **total of 24 equally spaced radii**.
- The **sum of the lengths of these 24 radii is equal to the circumference of the circle**.
- We are to find the **area of the circle**.
---... | Qwen/Qwen3-4B | false | 1 | stop | 1,208 | true | null | [] | 1,463,735,927 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:06:05.910576+00:00 |
4a2dbbb20912cde845206d2ada90faf4f6890a3b30821583f089774de317785c | In the land of Azura, there are three magical artifacts: a Sword, a Shield, and an Elixir. Each artifact has a specific weight, and when combined in pairs, they create a new, heavier artifact. The weights of the artifacts are represented by distinct, non-zero digits less than 7 in base 7. The following equation holds t... | We are given a cryptarithmetic puzzle in **base 7**:
$$
\begin{array}{c@{}c@{}c@{}c}
& S & H & E_7 \\
& \times & S & H_7 \\
\cline{2-4}
& E & L & I_7 \\
\end{array}
$$
We are to find the **weights** of the **Sword (S)**, **Shield (H)**, and **Elixir (E)**, which are **distinct, non-zero digits less than 7** (i.e.,... | Qwen/Qwen3-4B | false | 1 | stop | 1,167 | true | null | [] | 673,230,312 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:09:00.587460+00:00 |
1815e1533cfb0ab6b494e82fe65ee5a64e7cf8e2c820e07ea604b9afe5128ec2 | Find all 7-digit multiples of 33 of the form $4a6,\!39b$ where $a$ and $b$ are digits. What is the minimum value of $a$? | We are given a 7-digit number of the form:
$$
4a6,\!39b
$$
This is a number with 7 digits: the digits are:
- First digit: 4
- Second digit: $ a $ (unknown digit)
- Third digit: 6
- Fourth digit: 3
- Fifth digit: 9
- Sixth digit: $ b $ (unknown digit)
- Seventh digit: ?
Wait — actually, the number is written as:
$$... | Qwen/Qwen3-4B | false | 2 | stop | 3,513 | true | null | [] | 1,340,730,541 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:45.883855+00:00 |
0ada092195e98d3e2e3df13f482097313456f5de2d4c9db0f527242e768e694f | Problem:
A player is presented with a set of 16 envelopes, each containing a different integer from 1 to 16. The player's goal is to find the envelope with the largest integer by opening a subset of the envelopes one at a time, without being able to see the integer inside until it is opened. However, after opening an e... | We are given:
- 16 envelopes, each containing a **distinct integer** from **1 to 16**.
- The player can **open** envelopes one at a time.
- After opening an envelope, the player **can compare** the revealed number with the numbers in the **remaining envelopes**, and **eliminate some** of them (i.e., rule out envelopes... | Qwen/Qwen3-4B | false | 1 | stop | 1,129 | true | null | [] | 1,773,119,897 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:26:20.043851+00:00 |
389325164d5a253eb495cc01b8a9edfc916b9dc3afb4ec45fe9f6f9b6cf4863f | The graph of the parabola $y^2 = 4x$ and the circle $x^2 + y^2 = 6x + 2y + 5$ intersect at two points. Find the distance between these two points. | We are given two curves:
1. **Parabola**: $ y^2 = 4x $
2. **Circle**: $ x^2 + y^2 = 6x + 2y + 5 $
We are to find the **distance between the two points of intersection** of these curves.
---
### Step 1: Simplify the Circle Equation
Start with the circle:
$$
x^2 + y^2 = 6x + 2y + 5
$$
Bring all terms to one side:
... | Qwen/Qwen3-4B | false | 1 | stop | 7,068 | true | null | [] | 1,044,107,319 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:26:06.082430+00:00 |
14710bdc4e7bc26bf6fda4bc3113070e5678566851f84c42c5140f1f79ad1d69 | Find the integer that is closest to $1000\sum_{n=3}^{10000}\frac1{n^2-4}$.
Here's another problem inspired by this one:
Let $S$ be the sum of all the terms of the form $\frac{1}{n^2-4}$, where $n$ ranges from 3 to 10000. Find the value of $S$ modulo 1000.
This problem requires a different approach than the original ... | We are given:
$$
S = \sum_{n=3}^{10000} \frac{1}{n^2 - 4}
$$
We are to **find $ S \mod 1000 $**.
---
### Step 1: Simplify the general term
We start by simplifying the expression:
$$
\frac{1}{n^2 - 4} = \frac{1}{(n - 2)(n + 2)}
$$
This is a rational function that can be decomposed using **partial fractions**.
Le... | Qwen/Qwen3-4B | false | 1 | stop | 3,419 | true | null | [] | 1,392,441,297 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:26:53.581924+00:00 |
060489f93baed91178ececd9c7cc7fa3b58a94b02b07eb92bdef1a5e21629a32 | A polynomial $P(x)$ of degree 3 with coefficients in $\mathbb{Q}$ satisfies
$$P(x) + \frac{P(1)}{(x-1)^2} + \frac{P(2)}{(x-2)^2} = 0$$
for all $x \in \mathbb{R}$.
Find $P(3)$. | We are given a polynomial $ P(x) $ of degree 3 with rational coefficients, and it satisfies the identity:
$$
P(x) + \frac{P(1)}{(x-1)^2} + \frac{P(2)}{(x-2)^2} = 0
$$
for all $ x \in \mathbb{R} $. We are to find $ P(3) $.
---
### Step 1: Understand the equation
We are told that the expression:
$$
P(x) + \frac{P(1... | Qwen/Qwen3-4B | false | 3 | stop | 1,450 | true | null | [] | 1,738,752,150 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:35:49.063303+00:00 |
7da9fdbb7f9e3c229ad9eb2699ed580205bf11d710b938d629db75556f193047 | Let $A,B,C$ be angles of a triangle, where angle $C$ is acute, and \begin{align*}
\sin^2 A + \sin^2 B + 2 \cos A \cos B \sin C &= \frac{5}{6} \text{ and} \\
\sin^2 B + \sin^2 C + 2 \cos B \cos C \sin A &= \frac{7}{12}.
\end{align*}There are positive integers $p$, $q$, $r$, and $s$ for which \[ \sin^2 C + \sin^2 A + 2 \... | We are given three angles $ A, B, C $ of a triangle, with $ C $ being acute. The problem gives us two trigonometric equations involving sines and cosines of the angles, and we are to find the value of a third similar expression:
$$
\sin^2 C + \sin^2 A + 2 \cos C \cos A \sin B = \frac{p+q\sqrt{r}}{s}
$$
with the condi... | Qwen/Qwen3-4B | false | 1 | stop | 5,919 | true | null | [] | 3,985,434,691 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:25:58.041507+00:00 |
7f8af226f140883483446315b8ded01a231c9c9d8ac3a366b485d207fce8e396 | A sphere of radius 8 units contains a right cone with a base radius of 6 units and a height of 7 units. The volume, in cubic units, of the region inside the sphere and outside the cone is $V\pi$. Find $V$ as a common fraction. | We are given:
- A **sphere** of radius **8 units**.
- A **right cone** with:
- **Base radius** = 6 units
- **Height** = 7 units
- The cone is **inside** the sphere.
- We are to find the volume of the **region inside the sphere but outside the cone**, which is given as $ V\pi $, and find the value of $ V $ as a **c... | Qwen/Qwen3-4B | false | 1 | stop | 477 | true | null | [] | 4,558,587 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:26:38.779228+00:00 |
964c0b9bb12ad259797f9e0db75ad4664cc0b55083b7c26cd35dcb53b01800ac | Find all the solutions to
\[\sqrt[5]{x^2 + 1} - \sqrt{x + 2} = 2.\] | We are given the equation:
$$
\sqrt[5]{x^2 + 1} - \sqrt{x + 2} = 2
$$
Our goal is to find **all real solutions** to this equation.
---
### Step 1: Understand the domain
We need to ensure that all expressions under the radicals are defined and real.
- $\sqrt{x + 2}$ is defined when $x + 2 \geq 0 \Rightarrow x \geq... | Qwen/Qwen3-4B | false | 1 | stop | 2,679 | true | null | [] | 2,010,090,833 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:26:09.452142+00:00 |
6011fe063746adb8cd73ffc275cbce75af4b1252e7ba853c48c9c8bedfe55395 | Let ABCDEFGH be a cube. Points M and N are chosen on segments GH and GE respectively, such that $\overline{\text{GM}}=\frac{1}{3}\cdot\overline{\text{GH}}$ and $\overline{\text{GN}}=\frac{1}{4}\cdot\overline{\text{GE}}.$
Find the ratio of the volume of pyramid GMBCN to the volume of cube ABCDEFGH.
Express your answer ... | We are given a cube $ ABCDEFGH $, and points $ M $ and $ N $ are chosen on edges $ GH $ and $ GE $, respectively, such that:
- $ \overline{GM} = \frac{1}{3} \cdot \overline{GH} $
- $ \overline{GN} = \frac{1}{4} \cdot \overline{GE} $
We are to find the **ratio of the volume of pyramid $ GMBCN $** to the **volume of th... | Qwen/Qwen3-4B | false | 1 | stop | 1,647 | true | null | [] | 2,299,024,197 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T16:26:20.590391+00:00 |
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