qid stringlengths 5 5 | problem stringlengths 41 787 | gold int64 -6,066 674M | rft_solution stringlengths 646 61.1k ⌀ | n_samples_to_correct int64 1 7 ⌀ | n_correct_of_n int64 0 8 | n int64 8 8 |
|---|---|---|---|---|---|---|
Q0001 | In triangle $ABC$, $\sin \angle A = \frac{4}{5}$ and $\angle A < 90^\circ$. Let $D$ be a point outside triangle $ABC$ such that $\angle BAD = \angle DAC$ and $\angle BDC = 90^\circ$. Suppose that $AD = 1$ and that $\frac{BD}{CD} = \frac{3}{2}$. If $AB + AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a, b... | 34 | null | null | 0 | 8 |
Q0002 | Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$. | 113 | We are given a unit square $ABCD$, and points $X$ and $Y$ are chosen independently and uniformly at random on the **perimeter** of the square. We are to compute the **expected area** of triangle $\triangle AXY$, and express it as $\frac{m}{n}$ in lowest terms, then compute $m+n$.
---
### Step 1: Understanding the Set... | 4 | 3 | 8 |
Q0003 | Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b = 0$ also have a common real root. Compute the sum $a + b + c$. | -3 | We are given that:
- $ a, b, c $ are **distinct** numbers.
- The equations:
1. $ x^2 + ax + 1 = 0 $ and $ x^2 + bx + c = 0 $ have a **common real root**.
2. $ x^2 + x + a = 0 $ and $ x^2 + cx + b = 0 $ have a **common real root**.
We are to compute $ a + b + c $.
---
### Step 1: Let the common root of the first... | 1 | 8 | 8 |
Q0004 | There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $1$, the second in box $2$, and so forth up to the seventh card which is placed in box $7$. You then start working back in the other direction, p... | 3 | We are given a pattern of placing 2015 cards into 7 boxes arranged in a row, numbered 1 through 7.
The placement follows a back-and-forth pattern:
- Forward: boxes 1 → 2 → 3 → 4 → 5 → 6 → 7
- Backward: boxes 6 → 5 → 4 → 3 → 2 → 1
- Then forward again: 2 → 3 → 4 → 5 → 6 → 7
- And so on...
Let’s understand the p... | 1 | 6 | 8 |
Q0005 | Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum
\[ \sum_{(a,b,c) \in S} \frac{1}{2^a 3^b 5^c} \]
can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. | 37 | We are given a sum over all triples $(a, b, c)$ of **non-negative integers** such that $a + b + c$ is **even**, and we are to compute:
$$
\sum_{(a,b,c) \in S} \frac{1}{2^a 3^b 5^c}
$$
where $S$ is the set of such triples with $a + b + c$ even.
We want to compute this sum and express it as $\frac{m}{n}$ in lowest ter... | 1 | 8 | 8 |
Q0006 | For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square? | 10 | We are given the expression:
$$
n^4 + 6n^3 + 11n^2 + 3n + 31
$$
and we are to find all integers $ n $ for which this expression is a **perfect square**.
---
### Step 1: Try small integer values of $ n $
Let’s compute the value of the expression for small integers and see if any of them are perfect squares.
#### $... | 1 | 8 | 8 |
Q0007 | Four positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest possible value of $d$ such that the arithmetic mean of the numbers $a, b, c$ is half of the arithmetic mean of the numbers $a, b, c, d$. | 10 | We are given four positive integers $ a, b, c, d $ such that:
$$
a < b < c < d
$$
and the condition that the **arithmetic mean of $ a, b, c $** is **half** the **arithmetic mean of $ a, b, c, d $**.
---
### Step 1: Write the arithmetic means
- Arithmetic mean of $ a, b, c $:
$$
\frac{a + b + c}{3}
$$
- Arit... | 1 | 8 | 8 |
Q0008 | $n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations? | 2 | We are given a problem involving $ n $ coins arranged in a circle. The allowed operation is:
> If two **neighboring** coins are both **heads (H)** or both **tails (T)**, then we can **flip both** of them.
We are to find the number of **distinct arrangements** such that **no arrangement can be transformed into another... | 1 | 8 | 8 |
Q0009 | Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4$. | 509 | We are asked to find the **smallest prime** $ p $ such that:
- $ p^3 \mid 2018! $, but
- $ p^4 \nmid 2018! $
That is, the exponent of $ p $ in the prime factorization of $ 2018! $ is **exactly 3**.
---
### Step 1: Understanding the exponent of a prime in $ n! $
The exponent of a prime $ p $ in $ n! $ is given by *... | 1 | 8 | 8 |
Q0010 | Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$. | 3 | We are given a sphere that is tangent to the $xy$-plane and has its center at a point with positive $z$-coordinate. When this sphere is projected from a point $P = (0, b, a)$ onto the $xy$-plane, the image is the conic section $y = x^2$.
We are to find $a = \frac{p}{q}$ in lowest terms and compute $p + q$.
---
### S... | 1 | 8 | 8 |
Q0012 | Rectangle $HOMF$ has $HO = 11$ and $OM = 5$. Triangle $\triangle ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$, and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$. | 28 | We are given a rectangle $ HOMF $ with:
- $ HO = 11 $
- $ OM = 5 $
Also, triangle $ \triangle ABC $ has:
- Orthocenter $ H $
- Circumcenter $ O $
- $ M $ is the midpoint of $ BC $
- Altitude $ AF $ meets $ BC $ at $ F $
We are to **find the length of $ BC $**.
---
### Step 1: Understand the Geometry
In any trian... | 1 | 7 | 8 |
Q0013 | Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$. | 5 | We are given a cubic equation:
$$
x^3 - (k+1)x^2 + kx + 12 = 0
$$
Let the roots be $ a, b, c $. We are told:
$$
(a-2)^3 + (b-2)^3 + (c-2)^3 = -18
$$
We are to find the value of $ k $.
---
### Step 1: Use identity for sum of cubes
We use the identity:
$$
x^3 + y^3 + z^3 = (x+y+z)^3 - 3(x+y+z)(xy+yz+zx) + 3xyz
$$... | 1 | 8 | 8 |
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