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
value | timestamp stringlengths 32 32 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
24ae0aaaa25fe1c492a30f9541e35bb1f79579990709afd4934f42e367918d5d | In a group of $n$ people, each person exchanges phone numbers with exactly three other people in the group. Let $M$ be the number of ways this phone number exchanging can occur, considering two arrangements different if and only if at least two people who exchange phone numbers under one arrangement do not exchange pho... | We are given a group of $ n $ people, and each person exchanges phone numbers with **exactly three** other people in the group. We are to find the number of **distinct** ways this can happen, considering two arrangements different if **at least one pair** of people who exchange numbers in one arrangement do **not** exc... | Qwen/Qwen3-4B | false | 1 | stop | 2,001 | true | null | [] | 146,470,217 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:37:46.907095+00:00 |
fc84108491c2f70d29b3227fae92ac70f97615d85a4218a87f88b8d4457593e0 | In a circle of radius $1$, a chord is drawn to form a triangle with two radii. The triangle is reflected over one of its radii to create another triangle. If the area of the overlap between the two triangles is $\frac{1}{4}$ of the area of the original triangle, what is the length of the chord? | We are given a **circle of radius 1**, and a **chord** is drawn to form a triangle with **two radii**. This triangle is then **reflected over one of its radii**, forming another triangle. The **area of the overlap** between the two triangles is **1/4** of the area of the original triangle. We are to find the **length o... | Qwen/Qwen3-4B | false | 1 | stop | 3,276 | true | null | [] | 3,434,330,381 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:40:46.442839+00:00 |
b8d922257ae8bfd1e98f8f9c5078f88bf6df0f2f2bdf736d56e3cefbc524991a | Consider a quadrilateral ABCD. The diagonals AC and BD intersect at point O. Given that angle BAC is 30 degrees and angle ABC is 60 degrees, find angle AOB.
[asy]
size(150); defaultpen(linewidth(0.8));
pair A = (0,0), B = (2,0), C = (3,2), D = (1,2), O = intersectionpoint(A--C,B--D);
draw(A--B--C--D--cycle);
draw(A--C... | We are given a quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at point $O$. We are also given:
- $\angle BAC = 30^\circ$
- $\angle ABC = 60^\circ$
We are to find $\angle AOB$, the angle between the diagonals at their intersection point $O$.
---
### Step 1: Use triangle $ABC$
In triangle $ABC$, we a... | Qwen/Qwen3-4B | false | 1 | stop | 1,360 | true | null | [] | 6,095,444 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:45.718532+00:00 |
ffee44931a55befd30d957cff560ed673df529228d56715e6e63788d2f306ff2 | Tetrahedron $ABCD$ has vertices $A(0,0,0)$, $B(1,0,0)$, $C(1,1,0)$, and $D(0,0,1)$. Find the area of the cross-section formed by plane $P$ that contains points $A$ and $B$ and is perpendicular to segment $CD$. | We are given a **tetrahedron** $ABCD$ with vertices:
- $A(0,0,0)$
- $B(1,0,0)$
- $C(1,1,0)$
- $D(0,0,1)$
We are to find the **area of the cross-section** formed by a **plane $P$** that:
- Contains points $A$ and $B$, and
- Is **perpendicular to segment $CD$**.
---
### Step 1: Understand the Geometry
We are given ... | Qwen/Qwen3-4B | false | 2 | stop | 2,866 | true | null | [] | 23,291,411 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:37:32.861897+00:00 |
8475b32370f3e2e06b00671c54726de00cec2952bd14b77b773fd4206c795631 | The roots of the equation $x^2 + bx + c = 0$ are two consecutive integers, and the sum of their squares is equal to the square of the sum of their coefficients. Find the value of $b$. | We are given the quadratic equation:
$$
x^2 + bx + c = 0
$$
with **two consecutive integer roots**. Let's denote the roots as:
$$
r \quad \text{and} \quad r + 1
$$
### Step 1: Use Vieta's formulas
For a quadratic equation $x^2 + bx + c = 0$, Vieta's formulas tell us:
- Sum of the roots: $r + (r + 1) = -b$
- Produ... | Qwen/Qwen3-4B | false | 2 | stop | 5,146 | true | null | [] | 1,494,242,354 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:33:06.528406+00:00 |
72cc8fddd188bd24b6f8af3aba2ad81b9752ed826d7e75e62a913efd90efbee2 | Find the number of polynomials $p(x)$ of degree 4 or less such that $p(x)p(1-x)=p(x^2).$ | We are given the functional equation for a polynomial $ p(x) $ of degree at most 4:
$$
p(x)p(1 - x) = p(x^2)
$$
We are to find the **number of such polynomials**.
---
### Step 1: Understand the degree
Let’s denote the degree of $ p(x) $ as $ d $, where $ 0 \leq d \leq 4 $.
Then:
- $ p(x) $ is degree $ d $
- $ p(... | Qwen/Qwen3-4B | false | 1 | stop | 4,132 | true | null | [] | 1,390,441,722 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:52.964439+00:00 |
7b705c6ed8c7245c667daa9566d3ca2f4a347db43738cba0561d49555e7267c9 | Given a circle with equation $x^2 + y^2 + 2x - 6y + 5 = 0$, find the equation of a circle that is tangent to it at the point $(-1, 3)$ and has a radius twice as large. | We are given the equation of a **circle**:
$$
x^2 + y^2 + 2x - 6y + 5 = 0
$$
We are to find the equation of another **circle** that is **tangent to this circle at the point** $(-1, 3)$, and has a **radius twice as large**.
---
### **Step 1: Find the center and radius of the given circle**
We start by rewriting the... | Qwen/Qwen3-4B | false | 1 | stop | 2,621 | true | null | [] | 3,640,449,378 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:21.264856+00:00 |
5bce9b4cdbe0b6e7bf3503c23f965eae5138667a99d2c36cbf5c898a2dd1f892 | Problem:
Let $P(x)$ be a polynomial with leading coefficient $1$ such that the equation $P(x) = \sin x$ has infinitely many solutions. Find the sum of all values of $x$ for which $P(x) \neq \sin x$.
Note: This problem requires a different approach, as it involves a transcendental function (sine) and a polynomial, and ... | We are given a polynomial $ P(x) $ with **leading coefficient 1** such that the equation $ P(x) = \sin x $ has **infinitely many solutions**. We are to **find the sum of all values of $ x $** for which $ P(x) \neq \sin x $.
---
### Step 1: Understanding the Problem
We are told that the equation $ P(x) = \sin x $ has... | Qwen/Qwen3-4B | false | 2 | stop | 2,741 | true | null | [] | 1,433,730,721 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:34:37.249792+00:00 |
a467d1f73c0c5fc0f712b2f9ee0d7a276497526af007a6ab472f8126ec453d46 | Find the largest positive integer $n$ such that $n^2+2n+1$ is divisible by $3n+1$. | We are given the expression:
$$
n^2 + 2n + 1
$$
and we are told that it is divisible by:
$$
3n + 1
$$
We are to find the **largest positive integer** $ n $ such that:
$$
3n + 1 \mid n^2 + 2n + 1
$$
---
### Step 1: Understand the divisibility condition
We are told that:
$$
3n + 1 \mid n^2 + 2n + 1
$$
This mean... | Qwen/Qwen3-4B | false | 2 | stop | 1,836 | true | null | [] | 3,348,270,174 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:28:39.765921+00:00 |
eb7d8e858785b1a544d89fd550517e958de018b1eb3dbe31ffbfa3447a616d0e | A triangle $ABC$ is given by its vertices $A(-2,1)$, $B(3,-3)$, and $C(7,5)$. A line passing through the centroid of $\triangle ABC$ is such that the projection of $\overrightarrow{AB}$ onto it is $30$ units. Find the equation of this line. | We are given a triangle $ ABC $ with vertices:
- $ A(-2, 1) $
- $ B(3, -3) $
- $ C(7, 5) $
We are told that a line passes through the **centroid** of triangle $ ABC $, and the **projection of vector $ \vec{AB} $** onto this line is **30 units**. We are to find the **equation of this line**.
---
### **Step 1: Find t... | Qwen/Qwen3-4B | false | 3 | stop | 1,921 | true | null | [] | 2,081,518,579 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T20:00:07.160132+00:00 |
5709d808171e6e784f62813e21a0543a84677d7ca16c1614cbe4bc318f923506 | Let $a$, $b$, and $c$ be positive integers such that
\[1 < \frac{a + c}{b + c} < 2.\]
If $\frac{a}{b}$ is an integer, what is its value? | We are given the following:
- $ a, b, c $ are **positive integers**.
- $ 1 < \frac{a + c}{b + c} < 2 $
- $ \frac{a}{b} $ is an **integer**.
- We are to find the **value** of $ \frac{a}{b} $.
---
### Step 1: Understand the inequality
We are given:
$$
1 < \frac{a + c}{b + c} < 2
$$
This means that the **average** o... | Qwen/Qwen3-4B | false | 1 | stop | 989 | true | null | [] | 3,090,660,624 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:45.761621+00:00 |
f2b537ebd272c15a29e802d2f4c826a55c9c972e76a0c2c8e3103540d323dabc | There is no key information about the size of the farm in the text.
On a farm, five grazing areas with only one main entrance and exit are separated by fences. The grazing areas were originally created to feed a specific number of livestock. In spring, new fencing was installed, and the grazing areas were expanded, w... | To solve this problem, we need to understand how the grazing areas' sizes relate to their capacity to hold livestock, and how moving the main entrance and re-fencing affects the grazing areas' effective sizes.
---
### **Step 1: Understand the Problem**
- **Original grazing areas** had sizes: M, L, L, XL, XXL
- **Aft... | Qwen/Qwen3-4B | false | 1 | stop | 2,146 | true | null | [] | 1,116,482,176 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:15.345660+00:00 |
263d6f36c81cc87af080991bd8a69e227a9e5928db628b6f9d742c40e1eea198 | Point $M$ is the midpoint of $AB$ in triangle $ABC$.
The length of the altitude $CH$ of the triangle is 24 cm, and the length of the segment $MH$ is 4 cm.
Find the length of the side $AB$. | We are given the following information about triangle $ ABC $:
- $ M $ is the **midpoint** of $ AB $, so $ AM = MB $.
- $ CH $ is the **altitude** from $ C $ to $ AB $, and its length is $ 24 $ cm.
- The segment $ MH $ is $ 4 $ cm long.
We are to find the **length of side $ AB $**.
---
### Step 1: Understand the Ge... | Qwen/Qwen3-4B | false | 1 | stop | 2,243 | true | null | [] | 3,887,109,607 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:41:12.466810+00:00 |
6e2e59a58c6f7f66030d0f12572568f49fa1822da3127122b67f67ae92768610 | Simplify
\[(1 - \tan B + \cot B)(1 + \csc B - \sec B).\] | We are given the expression:
$$
(1 - \tan B + \cot B)(1 + \csc B - \sec B)
$$
We are to **simplify** this expression.
---
### Step 1: Recall the definitions of the trigonometric functions:
- $\tan B = \frac{\sin B}{\cos B}$
- $\cot B = \frac{\cos B}{\sin B}$
- $\csc B = \frac{1}{\sin B}$
- $\sec B = \frac{1}{\cos ... | Qwen/Qwen3-4B | false | 1 | stop | 2,699 | true | null | [] | 2,863,248,306 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:52.970437+00:00 |
e2516aebeb74b2875f6ea392ecc5be0325d7681a76859afa3bfb44b91f4c08b4 | A plane travels from city A to city B at a constant speed. On the return trip, the plane flies at a constant speed that is 30% faster than the original speed. If the total time for the round trip is 10 hours, and the time spent flying from A to B is 2 hours longer than the time spent flying from B to A, what is the one... | We are given the following:
- The plane travels from **A to B** at a **constant speed**.
- On the **return trip (B to A)**, it flies at a **constant speed 30% faster** than the original speed.
- The **total time for the round trip is 10 hours**.
- The **time from A to B is 2 hours longer** than the time from B to A.
... | Qwen/Qwen3-4B | false | 1 | stop | 4,539 | true | null | [] | 2,676,153,670 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:21.330015+00:00 |
b606316c41e263fc7d8eb6f69b308ae2b0db77181c669e87029994cf5fcb6995 | Find the smallest positive integer $x$ such that $\lfloor x/2\rfloor$ is divisible by 3, $\lfloor x/3\rfloor$ is divisible by 4, $\lfloor x/4\rfloor$ is divisible by 5, $\lfloor x/5\rfloor$ is divisible by 6, $\lfloor x/6\rfloor$ is divisible by 7, and $\lfloor x/7\rfloor$ is divisible by 8. | We are given a set of conditions involving floor functions and divisibility:
We are to find the **smallest positive integer** $ x $ such that:
- $ \left\lfloor \frac{x}{2} \right\rfloor $ is divisible by 3
- $ \left\lfloor \frac{x}{3} \right\rfloor $ is divisible by 4
- $ \left\lfloor \frac{x}{4} \right\rfloor $ ... | Qwen/Qwen3-4B | false | 1 | stop | 2,183 | true | null | [] | 1,936,393,998 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:44:12.351192+00:00 |
79665d01f53a2a8432c9ad6775c3c08a9187409a8d65d818a4bc075e24da7588 | Let $a,$ $b,$ $c$ be distinct complex numbers such that
\[\frac{a^2}{1 - b^2} = \frac{b^2}{1 - c^2} = \frac{c^2}{1 - a^2} = m.\]Find the product of all possible values of $m.$ | We are given that:
$$
\frac{a^2}{1 - b^2} = \frac{b^2}{1 - c^2} = \frac{c^2}{1 - a^2} = m
$$
Let’s denote:
$$
\frac{a^2}{1 - b^2} = m \quad (1) \\
\frac{b^2}{1 - c^2} = m \quad (2) \\
\frac{c^2}{1 - a^2} = m \quad (3)
$$
### Step 1: Express each variable in terms of $ m $
From equation (1):
$$
a^2 = m(1 - b^2) \q... | Qwen/Qwen3-4B | false | 1 | stop | 1,038 | true | null | [] | 3,090,380,554 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:37:46.994926+00:00 |
37d5233bedb454db62d34b72b518b505db332690f26b0e6e5ef468c28ea1b450 | Two circles defined by the equations $(x-a)^2 + (y-b)^2 = r^2$ and $(x-c)^2 + (y-d)^2 = s^2$ intersect at the points $(p,q)$ and $(u,v)$. What is the value of $ac + bd$ if the points of intersection satisfy the relationship $p+q = u+v$? | We are given two circles:
- Circle 1: $(x - a)^2 + (y - b)^2 = r^2$
- Circle 2: $(x - c)^2 + (y - d)^2 = s^2$
These circles intersect at two points: $(p, q)$ and $(u, v)$, and we are told that:
$$
p + q = u + v
$$
We are to find the value of $ac + bd$.
---
### Step 1: Use the fact that both points lie on both cir... | Qwen/Qwen3-4B | false | 3 | stop | 3,337 | true | null | [] | 2,274,422,782 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T20:00:07.164691+00:00 |
cb3b545bacf636daa184e914c49b859187c6e2309d24018b53afcfee751870e1 | What is the value of the digit in the 1000th place in the decimal expansion of $\sqrt{3}+\sqrt{5}$? | We are asked to find the **value of the digit in the 1000th place** in the **decimal expansion of**
$$
\sqrt{3} + \sqrt{5}
$$
---
### Step 1: Understand the Problem
We need to find the **1000th digit after the decimal point** in the decimal expansion of:
$$
\sqrt{3} + \sqrt{5}
$$
This is a **non-repeating, irrati... | Qwen/Qwen3-4B | false | 2 | stop | 779 | true | null | [] | 1,222,546,626 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:37:32.900013+00:00 |
3552be4e53f8619aaab81b4d22365ec3c551c5eb3d54b84e70f7fd8a72244ab1 | If $n$ is the smallest positive integer for which there exist positive integers $a$ and $b$ such that
$$\frac{n}{a} + \frac{n}{b} = \frac{n}{6} + 1,$$
find the remainder when $n$ is divided by $6$. | We are given the equation:
$$
\frac{n}{a} + \frac{n}{b} = \frac{n}{6} + 1
$$
We are to find the **smallest positive integer $ n $** for which there exist **positive integers $ a $ and $ b $** that satisfy this equation. Then, we need to find the **remainder when $ n $ is divided by 6**, i.e., $ n \mod 6 $.
---
### ... | Qwen/Qwen3-4B | false | 1 | stop | 4,545 | true | null | [] | 4,090,008,208 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:41:12.474900+00:00 |
2967dd9427b613b4bccc914cba3678409b4cc0ef975465098371ea34dedd6179 | Quadrilateral $ABCD$ is inscribed in a circle. The measures of arcs $AB$, $BC$, $CD$, and $DA$ are $x^\circ$, $y^\circ$, $z^\circ$, and $w^\circ$, respectively. What is the smallest possible value of $x+y+z+w$ such that quadrilateral $ABCD$ is not a rectangle? | We are given that quadrilateral $ABCD$ is **inscribed in a circle**, meaning it is a **cyclic quadrilateral**. The arcs between the vertices are:
- Arc $AB = x^\circ$
- Arc $BC = y^\circ$
- Arc $CD = z^\circ$
- Arc $DA = w^\circ$
We are asked to find the **smallest possible value of $x + y + z + w$** such that quadri... | Qwen/Qwen3-4B | false | 1 | stop | 713 | true | null | [] | 2,424,282,639 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:53.011250+00:00 |
35f9aff11c4e5ff5a13e46e794995508731d0d12b9b7a65eebbc00d61ce35e3d | A palindrome is a number that reads the same forward and backward. A two-digit number is squared, and then the hundreds digit of the result is moved to the units place, resulting in a palindrome. How many two-digit numbers satisfy this property? | We are given a two-digit number $ N $, which is squared, and then the **hundreds digit** of the resulting square is moved to the **units place**, forming a **palindrome**. We are to find how many such two-digit numbers exist.
---
### Step 1: Understand the Problem
Let $ N $ be a two-digit number, so $ 10 \leq N \leq... | Qwen/Qwen3-4B | false | 3 | stop | 4,355 | true | null | [] | 2,135,510,035 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T20:09:17.982480+00:00 |
ef81cfad1ecd0d80c117671ec3d5952a591eead77eef7145fa6723bb5312a746 | Given the polynomial equation $x^4+ax^3+bx^2+cx+d=0$, determine the value of $b$ such that the equation has four non-real roots, the product of two of the roots is $13+i$, and the sum of the other two roots is $3+4i$, where $i^2=-1$.
New problem:
Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a polynomial with real coeff... | We are given a polynomial:
$$
P(x) = x^4 + ax^3 + bx^2 + cx + d
$$
with **real coefficients**, and it has **four non-real roots**. Since the coefficients are real, the non-real roots must come in **complex conjugate pairs**.
---
### Step 1: Understanding the roots
Let the four roots be:
- $ \alpha, \beta $, and
-... | Qwen/Qwen3-4B | false | 2 | stop | 2,041 | true | null | [] | 1,670,233,035 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:34:37.282620+00:00 |
711993e838ac78c95b169cff150e93b171cc8c1cda84bf7d2cee230acd2e4175 | Consider the polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$ with roots $r_1, r_2, r_3,$ and $r_4$. If $Q(x)$ is a polynomial whose roots are $\frac{1}{r_1r_2}, \frac{1}{r_2r_3}, \frac{1}{r_3r_4},$ and $\frac{1}{r_4r_1}$, and $Q(2) = 5$, find the value of $d$. | We are given a polynomial:
$$
P(x) = x^4 + ax^3 + bx^2 + cx + d
$$
with roots $ r_1, r_2, r_3, r_4 $. Then we define another polynomial $ Q(x) $ whose roots are:
$$
\frac{1}{r_1r_2}, \frac{1}{r_2r_3}, \frac{1}{r_3r_4}, \frac{1}{r_4r_1}
$$
and we are told that:
$$
Q(2) = 5
$$
We are to find the value of $ d $.
--... | Qwen/Qwen3-4B | false | 1 | stop | 4,556 | true | null | [] | 1,816,700,908 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:37:47.074032+00:00 |
f724049bd3d5c25e31db23dffe67c879a5c017277e0968ca491f01f82c2a50a1 | Find all values of $a$ for which the graph of $y = \frac{4x^2 + 3x - 2}{x^2 + ax + 4}$ has exactly two horizontal asymptotes. | We are given the function:
$$
y = \frac{4x^2 + 3x - 2}{x^2 + ax + 4}
$$
We are to find all values of $ a $ such that the graph of this function has **exactly two horizontal asymptotes**.
---
### Step 1: Understand what a horizontal asymptote is
A **horizontal asymptote** of a rational function is a horizontal line... | Qwen/Qwen3-4B | false | 1 | stop | 2,568 | true | null | [] | 2,352,688,282 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:40:46.503673+00:00 |
76162007ecc3023ae88e50b88213c12e275b9801b4fadb649fdcc2ca4a1a8f6a | Let $f(x) = x + \frac{1}{x + \frac{1}{x + \frac{1}{x + ...}}}$. If $f(2) = \frac{7}{2}$, find the value of $g$ such that $f(g) = 4$. | We are given the function:
$$
f(x) = x + \frac{1}{x + \frac{1}{x + \frac{1}{x + \cdots}}}
$$
This is a **continued fraction** that repeats infinitely. Let's denote the continued fraction part as $ y $, so we can write:
$$
f(x) = x + \frac{1}{y}
$$
But notice that the continued fraction is the same as the entire fun... | Qwen/Qwen3-4B | false | 2 | stop | 2,420 | true | null | [] | 714,735,299 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:37:25.426914+00:00 |
cd2bdff207279f4fb0c8ee40078b4e908bcb21a4a3675967680f377b1567924b | The function g(x) is defined as follows:
\[g(x) =
\begin{cases}
sin(x) &\text{if } x<π, \\
cos(x-π) &\text{if } x\geq π.
\end{cases}
\]
If the graph y=g(x) is continuous and has exactly one turning point for x\geq 0, find the value of the turning point. | We are given a piecewise function:
$$
g(x) =
\begin{cases}
\sin(x) & \text{if } x < \pi, \\
\cos(x - \pi) & \text{if } x \geq \pi.
\end{cases}
$$
We are told that the graph of $ y = g(x) $ is **continuous** and has **exactly one turning point for $ x \geq 0 $**.
---
### Step 1: Ensure continuity at $ x = \pi $
For... | Qwen/Qwen3-4B | false | 1 | stop | 1,281 | true | null | [] | 3,963,344,950 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:41:12.529850+00:00 |
2b148893505f82c2d8bcfad92244aaf9e869320c29f7b03c49d3d210b48bec83 | In the complex plane, there exists a complex number $z$ with $|z| = 3$ such that the points $z$, $z^2$, and $z^3$ form an isosceles right triangle. Find the argument of $z$. | We are given a complex number $ z $ such that:
- $ |z| = 3 $, so $ z $ lies on a circle of radius 3 centered at the origin.
- The points $ z $, $ z^2 $, and $ z^3 $ form an **isosceles right triangle**.
- We are to find the **argument** of $ z $, i.e., the angle $ \theta = \arg(z) $.
---
### Step 1: Express $ z $ in... | Qwen/Qwen3-4B | false | 1 | stop | 5,009 | true | null | [] | 1,502,081,489 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:53.020816+00:00 |
af079eea496eea0c1e24a3dc38b23bbb604edf704f6a14e0478fd6286d95c860 | A pyramid has a square base with sides of length 6 feet. The four triangular faces are isosceles and congruent, and the apex of the pyramid is 8 feet from the base. When the pyramid is modified, the lower 2 feet of one of the triangular faces breaks off. Let H be the height in feet of the apex of the pyramid from the b... | We are given a **square pyramid** with:
- **Base side length**: 6 feet
- **Apex height from base**: 8 feet
- **Triangular faces**: Isosceles and congruent (so all edges from apex to base are equal)
Then, the **lower 2 feet** of one of the triangular faces **breaks off**, and we are to find the **new height $ H $** of... | Qwen/Qwen3-4B | false | 1 | stop | 4,121 | true | null | [] | 872,648,668 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:21.384471+00:00 |
3ec1d53ef410d34739236fbecf34138eecd4ac1b26fbb5a345b47a42efef6f58 | Two positive integers $a$ and $b$ have the property that $ab=126$, while the difference of their reciprocals is $1/42$. What is the largest possible value of $a+b$? | We are given:
- Two positive integers $ a $ and $ b $ such that $ ab = 126 $
- The difference of their reciprocals is $ \frac{1}{42} $
We are to find the **largest possible value of** $ a + b $.
---
### Step 1: Use the given conditions
We are given:
$$
ab = 126 \quad \text{and} \quad \frac{1}{a} - \frac{1}{b} = \f... | Qwen/Qwen3-4B | false | 3 | stop | 4,771 | true | null | [] | 4,025,207,813 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:59:14.124044+00:00 |
d77a8b7bbd9675c77c38aba2afaf73e85eb78bbd77a6444249da8f9b0a24e02c | Let $f(n)$ denote the smallest positive integer $k$ for which there exists a positive integer $a$ such that $a^k$ has exactly $n$ positive divisors. Find the maximum value of $f(n)$ for $n<1000$. | We are given a function $ f(n) $, defined as the **smallest positive integer $ k $** such that there exists a **positive integer $ a $** with $ a^k $ having **exactly $ n $ positive divisors**.
We are to **find the maximum value of $ f(n) $ for $ n < 1000 $**.
---
### Step 1: Understanding the number of divisors
Fo... | Qwen/Qwen3-4B | false | 1 | stop | 3,831 | true | null | [] | 2,485,062,682 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:37:47.100502+00:00 |
d9fa4df7bdd20c7b618b89939fadc819338dcbe5bf8242c6e6352432bfc742f6 | Let $a$, $b$, and $c$ be the roots of the equation
$$x^3+px^2+qx+r=0$$
where $p$, $q$, and $r$ are real numbers.
If the roots are such that $a+b$, $b+c$, and $c+a$ are all integers,
determine the value of $p^2-q+2r$. | We are given a cubic polynomial:
$$
x^3 + px^2 + qx + r = 0
$$
with roots $ a, b, c $. We are told that:
- $ a + b $, $ b + c $, and $ c + a $ are all integers.
We are to find the value of:
$$
p^2 - q + 2r
$$
---
### Step 1: Use Vieta's formulas
From Vieta's formulas for a cubic polynomial $ x^3 + px^2 + qx + r... | Qwen/Qwen3-4B | false | 1 | stop | 4,184 | true | null | [] | 408,357,091 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:40:46.509047+00:00 |
331dca1c5a88de69be60e67a10100f6d0307464a40ad31f2b8472a1011bd45af | Given the polynomial $P(x)$ such that $P(1) = 5$ and $\frac{P(x^2)}{P(x+1)} = x^2 - 2x + \frac{2x+1}{x^2+2x+1}$ for $x \in \mathbb{R}$, find $P(4)$. | We are given a polynomial $ P(x) $ such that:
- $ P(1) = 5 $
- $ \frac{P(x^2)}{P(x+1)} = x^2 - 2x + \frac{2x+1}{x^2+2x+1} $ for all $ x \in \mathbb{R} $
We are to **find $ P(4) $**.
---
### Step 1: Simplify the given rational expression
We are given:
$$
\frac{P(x^2)}{P(x+1)} = x^2 - 2x + \frac{2x+1}{x^2+2x+1}
$$
... | Qwen/Qwen3-4B | false | 1 | stop | 4,233 | true | null | [] | 3,103,423,365 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:45.833826+00:00 |
22e499f629c50c88b26d39ad05a8d4a4ffd6ca6f3706de44cedbe83424d6228d | What are the possible values of $c$ for which the equation $x^3 + cx^2 + 3x + 1 = 0$ has exactly one real root? Express your answer in interval notation. | We are given the cubic equation:
$$
x^3 + cx^2 + 3x + 1 = 0
$$
We are to find the values of $ c $ for which the equation has **exactly one real root**.
---
### Step 1: Understand the nature of cubic equations
A **cubic polynomial** has **at least one real root**, and the number of real roots (counting multiplicity... | Qwen/Qwen3-4B | false | 1 | stop | 658 | true | null | [] | 2,932,241,448 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:15.395313+00:00 |
9d0865695f2387b4b777d69ab49bd7ac5dacf1a88eaf31492c6648ed23a35a10 | Find all values of $b$ for which the system of equations
\begin{align*}
2x^2 + 3y^2 &= 7,\\
x^2 + by^2 &= 2,
\end{align*}has at least one solution where both $x$ and $y$ are integers. Enter all possible values of $b,$ separated by commas. | We are given the system of equations:
$$
\begin{align*}
2x^2 + 3y^2 &= 7, \\
x^2 + by^2 &= 2.
\end{align*}
$$
We are to find **all values of $ b $** such that the system has **at least one solution where both $ x $ and $ y $ are integers**.
---
### Step 1: Understand the constraints
We are looking for integer valu... | Qwen/Qwen3-4B | false | 1 | stop | 735 | true | null | [] | 1,571,371,406 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:41:12.533495+00:00 |
c8dd47eefa1fcc4986e85689dc56cd554c1485cb761ea56265e6313486ca8f4f | Find the value of $n$ for which
\[\frac{1}{2} \cdot 2 + \frac{1}{3} \cdot 3^2 + \frac{1}{4} \cdot 4^3 + \dots + \frac{1}{n} \cdot n^{n-1} = \frac{n^n}{2}.\] | We are given the equation:
$$
\frac{1}{2} \cdot 2 + \frac{1}{3} \cdot 3^2 + \frac{1}{4} \cdot 4^3 + \dots + \frac{1}{n} \cdot n^{n-1} = \frac{n^n}{2}
$$
We are to find the value of $ n $ that satisfies this equation.
---
### Step 1: Understand the general term
The left-hand side (LHS) is a sum of terms of the form... | Qwen/Qwen3-4B | false | 1 | stop | 4,005 | true | null | [] | 3,847,117,185 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:53.074129+00:00 |
50b84c8c51b260702213ecbe28dccac404d95d2020d3dbce94a6643c86739c8b | After a tennis tournament, each player shook hands once with every player on every team (except themselves). Afterwards, a tournament organizer came down and only shook hands with each player from their own team. There were a total of 225 handshakes. What is the fewest number of handshakes the organizer could have part... | We are told:
- After a tennis tournament, **each player shook hands once with every other player on every team**, **except themselves**.
- Then, a **tournament organizer** came down and **shook hands with each player from their own team**.
- The **total number of handshakes** is **225**.
- We are to find the **fewest ... | Qwen/Qwen3-4B | false | 3 | stop | 2,760 | true | null | [] | 2,510,292,489 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T20:09:18.009209+00:00 |
3a33fda57416068a45ca843527ad6817f278821accefc39a2f32a58b56ed72b6 | Find all solutions to the equation $x^2 + 7 = 2 \sqrt{x^3 + 2}$. | We are given the equation:
$$
x^2 + 7 = 2 \sqrt{x^3 + 2}
$$
Our goal is to **find all real solutions** to this equation.
---
### **Step 1: Understand the structure of the equation**
This is a **transcendental equation** (contains both polynomial and radical terms). We'll try to isolate the radical and then square ... | Qwen/Qwen3-4B | false | 1 | stop | 8,087 | true | null | [] | 3,615,235,295 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:44:12.497978+00:00 |
a7af07348b1dc9aa87a40a5ba3cecb6023bc97191299ab0342eb433ba4a335f7 | Consider the polynomial $P(x) = x^4 + 2x^3 + ax + b$ with $a$ and $b$ being integers. The equation $P(x) = 0$ has only real roots, and the difference between the largest and smallest roots is a rational number. Determine $a+b$. | We are given a polynomial:
$$
P(x) = x^4 + 2x^3 + ax + b
$$
with integer coefficients $ a $ and $ b $, and it is given that **all roots are real** and that the **difference between the largest and smallest root is a rational number**. We are to **find $ a + b $**.
---
### Step 1: Understand the structure of the pol... | Qwen/Qwen3-4B | false | 2 | stop | 1,269 | true | null | [] | 2,060,592,819 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:28:39.817704+00:00 |
6f7e7b0cd2d2a8eb95a8a828729093774387feef5840fe2d18122ad343e23bef | A ball is thrown from a height of 40 meters towards the ground. At the same time, a boy standing on the ground throws a ball upwards towards the falling ball. The initial speed of the falling ball is 15 meters per second, and the initial speed of the ball thrown by the boy is 20 meters per second. At what height do the... | We are given the following:
- A **ball is thrown downward** from a height of **40 meters** with an **initial speed of 15 m/s**.
- A **boy on the ground throws a ball upward** with an **initial speed of 20 m/s**.
- The **angle between the two balls' directions is 45°**.
- We are to find the **height above the ground** ... | Qwen/Qwen3-4B | false | 1 | stop | 4,655 | true | null | [] | 2,849,573,025 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:40:46.557836+00:00 |
8b1c061c5268565cc49feba0b6b3da09bd40a125c763d65efd78ee12b00be85f | Let $x$ be a three-digit positive integer such that its square, $x^2$, is divisible by $200$, and the quotient of the integer division of $x^2$ by $200$ is $x-2$. What is the value of $x$? | We are given a three-digit positive integer $ x $, such that:
1. $ x^2 $ is divisible by 200.
2. The quotient of the integer division of $ x^2 $ by 200 is $ x - 2 $.
That is:
$$
\frac{x^2}{200} = x - 2
$$
---
### Step 1: Multiply both sides by 200
$$
x^2 = 200(x - 2)
$$
$$
x^2 = 200x - 400
$$
$$
x^2 - 200x + 40... | Qwen/Qwen3-4B | false | 1 | stop | 576 | true | null | [] | 2,536,717,175 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:45.836803+00:00 |
c449968b01f864e1e6c2bb6005c170179cfcb294ca1ca5a5b479f62c4c12256b | Consider three linear functions $f(x),g(x),$ and $h(x)$ such that the maximum of these functions, denoted by $j(x)$, satisfies $j(-3) = 5$, $j(0) = 2$, and $j(3) = 5$. What is the least possible value of $j(x)$ for $-3 \leq x \leq 3$? | We are given three linear functions $ f(x), g(x), h(x) $, and their **maximum** is denoted by $ j(x) $. We are told that:
- $ j(-3) = 5 $
- $ j(0) = 2 $
- $ j(3) = 5 $
We are to find the **least possible value** of $ j(x) $ for $ -3 \leq x \leq 3 $.
---
### Step 1: Understand the meaning of $ j(x) = \max(f(x), g(x)... | Qwen/Qwen3-4B | false | 1 | stop | 2,472 | true | null | [] | 254,423,576 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:15.415437+00:00 |
07c94cc5da1a0453a0551ead07f2124fed83a572aad2ea787cfbffd59ef8babd | A square pyramid is formed by stacking $N$ identical wooden blocks, each with dimensions 1 inch x 1 inch x 2 inches. When the pyramid is viewed from the side, exactly $105$ of the blocks are completely hidden from view. Find the smallest possible value of $N$ such that the height of the pyramid is equal to its base len... | We are given a square pyramid made of **$ N $** identical wooden blocks, each with dimensions $ 1 \times 1 \times 2 $ inches. When viewed from the side, **105 blocks are completely hidden**, and we are to **find the smallest possible value of $ N $** such that the **height of the pyramid is equal to its base length**.
... | Qwen/Qwen3-4B | false | 1 | stop | 2,109 | true | null | [] | 3,438,500,201 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:41:12.575392+00:00 |
e11a92f7914e28a33ed05c927a45c55f0fa18aaa77094186c6c84e8a20cd19df | In the Cartesian plane, there exists a constant $k$ such that for any line $L$ passing through the origin $(0,0)$, the sum of the reciprocals of the distances from the points where $L$ intersects the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ to the origin is equal to $k$. Find the constant $k$. | We are given an ellipse:
$$
\frac{x^2}{4} + \frac{y^2}{9} = 1
$$
and a line $ L $ passing through the origin. We are told that for **any** such line $ L $, the **sum of the reciprocals of the distances** from the points where $ L $ intersects the ellipse to the origin is a **constant** $ k $. We are to find the value... | Qwen/Qwen3-4B | false | 1 | stop | 4,562 | true | null | [] | 2,859,425,873 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:21.424249+00:00 |
8409cdf17cd6f82697174d721b9d2647acf1f55a40a7d3b34e0d30e77f022926 | Let $f(x)$ be a quadratic function such that the roots of the equation $f(x) = 0$ are $r$ and $s$. Let $f^{-1}(x)$ be the inverse of $f(x)$ and let the roots of the equation $f^{-1}(x) = 0$ be $u$ and $v$. Suppose that the equation $x^2 - tx + 4 = 0$ has roots $r + u$ and $s + v$. Determine $t$. | We are given:
- $ f(x) $ is a **quadratic function** with **roots** $ r $ and $ s $, so we can write:
$$
f(x) = a(x - r)(x - s)
$$
where $ a \ne 0 $.
- $ f^{-1}(x) $ is the **inverse** of $ f(x) $, and its **roots** are $ u $ and $ v $. So:
$$
f^{-1}(x) = 0 \Rightarrow f(u) = x, \quad f(v) = x
$$
But ... | Qwen/Qwen3-4B | false | 2 | stop | 2,664 | true | null | [] | 2,843,917,725 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:34:37.313513+00:00 |
f56b43bcb66ca42eb0d78172f0d43ca58ca09d407af4b0e67e4c6ee63ee36931 | Let p(x) be a monic polynomial such that p(1) = 2, p(2) = 5, p(3) = 10, p(4) = 17 and p(5) = 26. Determine the number of distinct prime factors of the product p(1)p(2)...p(10). | We are given that $ p(x) $ is a **monic polynomial** (i.e., the leading coefficient is 1), and we are told:
$$
p(1) = 2,\quad p(2) = 5,\quad p(3) = 10,\quad p(4) = 17,\quad p(5) = 26
$$
We are to **find the number of distinct prime factors of the product**:
$$
p(1)p(2)p(3)p(4)p(5)p(6)p(7)p(8)p(9)p(10)
$$
---
### S... | Qwen/Qwen3-4B | false | 1 | stop | 1,473 | true | null | [] | 1,082,697,151 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:37:47.250134+00:00 |
bb46c486453517ebf023dab07968774a1125f66c7a27266771e58423b738e089 | Solve for all real values of $x$ that satisfy $\frac{1}{\sqrt{x^2 + 1}} + \frac{2}{\sqrt{x^2 + 4}} \ge \frac{3}{2}.$ (Give your answer in interval notation.) | We are given the inequality:
$$
\frac{1}{\sqrt{x^2 + 1}} + \frac{2}{\sqrt{x^2 + 4}} \ge \frac{3}{2}
$$
We are to find all **real values of $ x $** that satisfy this inequality, and express the solution in **interval notation**.
---
### Step 1: Let’s define a substitution to simplify the expression.
Let:
$$
y = x^... | Qwen/Qwen3-4B | false | 1 | stop | 2,814 | true | null | [] | 3,752,051,559 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:40:46.562723+00:00 |
e51a6795a9c53a20c2bca365f485f71cbc9aed0f52b7e2632d2755629938accd | Problem:
A square pyramid with a base of side length $s$ is inscribed in a cube with edge length $2s$. The pyramid is cut by a plane that is parallel to its base and passes through the midpoints of three adjacent edges of the cube. What is the ratio of the volume of the top part of the pyramid to the volume of the bott... | We are given a square pyramid inscribed in a cube of edge length $ 2s $. The pyramid is cut by a plane that is **parallel to its base** and passes through the **midpoints of three adjacent edges** of the cube. We are to find the **ratio of the volume of the top part of the pyramid to the volume of the bottom part**.
-... | Qwen/Qwen3-4B | false | 2 | stop | 1,058 | true | null | [] | 1,877,581,958 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T19:37:25.441765+00:00 |
f60b568c367df81383c89774f2b13d46af6de400875b8b1057a67de9f1cf04bf | A light beam is reflected off the sides of a regular tetrahedron, starting from one of its vertices. The beam reflects off each face exactly once before reaching another vertex. The sides of the tetrahedron have a length of 10 units. If the beam travels a total distance of $p\sqrt{q}$ units, where $p$ and $q$ are integ... | We are given a **regular tetrahedron** with side length 10 units. A **light beam** starts at one vertex, reflects off each of the **four faces** (each face is a triangle), and ends at another vertex. The beam reflects **once per face**, so it must reflect **three times** (since there are four faces, and one face is the... | Qwen/Qwen3-4B | false | 1 | stop | 1,536 | true | null | [] | 1,075,449,044 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:15.419580+00:00 |
702b7ce2814cc8f68b415a2a10419a2ea914b9e396827b9e21c546bcc29a663d | A circular necklace consists of 8 white beads and 8 black beads, with each bead having a distinct shape (e.g., circle, square, triangle, etc.). The beads are arranged such that no two adjacent beads have the same shape. How many distinguishable necklaces can be formed, considering rotations and reflections as equivalen... | We are given a **circular necklace** with:
- **8 white beads**
- **8 black beads**
- Each bead has a **distinct shape** (e.g., circle, square, triangle, etc.)
- **No two adjacent beads have the same shape**
- We are to count **distinguishable necklaces**, considering **rotations and reflections** as equivalent.
---
... | Qwen/Qwen3-4B | false | 1 | stop | 1,037 | true | null | [] | 469,643,948 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:41:12.578364+00:00 |
06a2a90319c7970cb91a27ab281ce4e5db0724850f67102a2571810c37a25cf3 | The sides of a rectangle have lengths $x$ and $y$.
If $x$ is increased by 15%, and $y$ is decreased by 3%, then the area of the rectangle is reduced by $24 cm^2$.
If $x$ is decreased by 10%, and $y$ is increased by 1%, then the area of the rectangle is increased by $21 cm^2$.
Find the original area of the rectangle. | We are given a rectangle with side lengths $ x $ and $ y $, and we are told:
1. If $ x $ is increased by 15% and $ y $ is decreased by 3%, the **area is reduced by 24 cm²**.
2. If $ x $ is decreased by 10% and $ y $ is increased by 1%, the **area is increased by 21 cm²**.
We are to **find the original area** of the r... | Qwen/Qwen3-4B | false | 1 | stop | 2,421 | true | null | [] | 2,696,806,407 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:53.106327+00:00 |
7c9b3d96052ba2d65f2e4a79726d60b5d332a5b2f6f8be233b0f13c80e14a812 | Determine the value of the infinite series $(2^{1/3})(3^{1/9})(4^{1/27})(5^{1/81}) \dotsm.$ Enter your answer in the form "$\sqrt[a]{b}$", which stands for $\sqrt[a]{b}.$ | We are given the infinite product:
$$
P = 2^{1/3} \cdot 3^{1/9} \cdot 4^{1/27} \cdot 5^{1/81} \cdots
$$
We are to determine the value of this infinite product and express it in the form $\sqrt[a]{b}$.
---
### Step 1: Understand the general term
Observe the pattern:
- The base of each term is an integer: $2, 3, 4,... | Qwen/Qwen3-4B | false | 1 | stop | 1,700 | true | null | [] | 2,214,945,155 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:42:21.426967+00:00 |
e945fdd5bc30f797054e48d573eb65586dfb70829f2f8a8696742956c52fb8c2 | In a triangle with side lengths $3x$, $5x$, and $7x$, the angle opposite the longest side is twice the measure of the angle opposite the shortest side. Determine the value of $x$. | We are given a triangle with side lengths:
- $ a = 3x $
- $ b = 5x $
- $ c = 7x $
We are also told:
- The **angle opposite the longest side** is **twice** the measure of the **angle opposite the shortest side**.
---
### Step 1: Label the triangle
Let’s denote:
- $ A $: angle opposite side $ a = 3x $ (shortest si... | Qwen/Qwen3-4B | false | 1 | stop | 1,594 | true | null | [] | 2,845,906,399 | 1cfa9a7208912126459214e8b04321603b3df60c | 2026-10-11T18:44:12.592996+00:00 |
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