problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? | 45 | 1. **Identify the games where the team lost by one run**: The problem states that the team lost by one run in exactly five games. Since they cannot score twice as many runs as their opponents and still lose by one run, these games must be the ones where they scored odd numbers of runs. Therefore, the games where they s... | 0.5 | 6,254.5625 | 4,389.875 | 8,119.25 |
Let $f(x) = x^3 - 9x^2 + 27x - 25$ and let $g(f(x)) = 3x + 4$. What is the sum of all possible values of $g(7)$? | 39 | 0.375 | 7,379.25 | 6,314.833333 | 8,017.9 | |
Solve the equation \( x^{[x]} = \frac{9}{2} \) for real numbers \( x \), where \( [x] \) represents the greatest integer less than or equal to \( x \). | \frac{3\sqrt{2}}{2} | 0 | 6,193.1875 | -1 | 6,193.1875 | |
The denominator of the fraction $15 \cdot 18$ in simplest form is 30. Find the sum of all such positive rational numbers less than 10. | 400 | 0.125 | 7,331.625 | 5,620.5 | 7,576.071429 | |
A batch of fragile goods totaling $10$ items is transported to a certain place by two trucks, A and B. Truck A carries $2$ first-class items and $2$ second-class items, while truck B carries $4$ first-class items and $2$ second-class items. Upon arrival at the destination, it was found that trucks A and B each broke on... | \frac{29}{48} | 0.0625 | 7,982.125 | 5,532 | 8,145.466667 | |
Let \( P_{1} \) and \( P_{2} \) be any two different points on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\), and let \( P \) be a variable point on the circle with diameter \( P_{1} P_{2} \). Find the maximum area of the circle with radius \( OP \). | 13 \pi | 0 | 8,192 | -1 | 8,192 | |
Given that the price savings of buying the computer at store A is $15 more than buying it at store B, and store A offers a 15% discount followed by a $90 rebate, while store B offers a 25% discount and no rebate, calculate the sticker price of the computer. | 750 | 0.5 | 2,143.375 | 2,605.75 | 1,681 | |
Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. Point $X$ is chosen on edge $A_{1} D_{1}$, and point $Y$ is chosen on edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects ray $D A$ at point $Z$. Find $D Z$. | 20 | 0.3125 | 7,539.4375 | 6,103.8 | 8,192 | |
Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points, and destroys the apple if he drops the bomb either on the point containin... | \frac{1}{2} | Let the points be $0, \ldots, 7(\bmod 8)$, and view Alice's reveal as revealing the three possible locations of the apple. If Alice always picks $0,2,4$ and puts the apple randomly at 0 or 4 , by symmetry Bob cannot achieve more than $\frac{1}{2}$. Here's a proof that $\frac{1}{2}$ is always possible. Among the three r... | 0 | 7,862.5625 | -1 | 7,862.5625 |
How many distinct lines pass through the point $(0, 2016)$ and intersect the parabola $y = x^2$ at two lattice points? (A lattice point is a point whose coordinates are integers.) | 36 | 0.1875 | 7,894.0625 | 6,802 | 8,146.076923 | |
The vertices and midpoints of the sides of a regular decagon (thus a total of 20 points marked) are noted.
How many triangles can be formed with vertices at the marked points? | 1130 | 0.0625 | 7,612.875 | 5,318 | 7,765.866667 | |
Calculate the arc lengths of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\
y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t
\end{array}\right. \\
& \frac{\pi}{2} \leq t \leq \frac{2 \pi}{3}
\end{aligned}
$$ | \sqrt{2} - 1 | 0.9375 | 4,731.75 | 4,501.066667 | 8,192 | |
Wanda is trying to locate the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(8,-1)$, and $C$ is at $(5,4)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at $P = (4,2)$, and computes the sum of ... | 8 | 1 | 2,104.0625 | 2,104.0625 | -1 | |
In the 2017 Shanghai college entrance examination reform plan, it is required that each candidate must choose 3 subjects from 6 subjects including Physics, Chemistry, Biology, Politics, History, and Geography to take the level examination. Xiaoming decided to choose at most one subject from Biology, Politics, and Histo... | 10 | 0.5625 | 4,672.375 | 2,793.444444 | 7,088.142857 | |
$S$ is a subset of the set $\{1, 2, \cdots, 2023\}$, such that the sum of the squares of any two elements is not a multiple of 9. What is the maximum value of $|S|$? (Here, $|S|$ represents the number of elements in $S$.) | 1350 | 0.125 | 7,745.9375 | 7,321.5 | 7,806.571429 | |
There is a \(4 \times 4\) square. Its cells are called neighboring if they share a common side. All cells are painted in two colors: red and blue. It turns out that each red cell has more red neighbors than blue ones, and each blue cell has an equal number of red and blue neighbors. It is known that cells of both colo... | 12 | 0 | 8,192 | -1 | 8,192 | |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | 71 | 1 | 2,008.8125 | 2,008.8125 | -1 | |
Determine how many solutions the following equation has:
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-50)}{(x-2^2)(x-4^2)(x-6^2)\dotsm(x-24^2)} = 0
\] | 47 | 0.8125 | 4,712.75 | 3,909.846154 | 8,192 | |
Find real numbers \( x, y, z \) greater than 1 that satisfy the equation
\[ x + y + z + \frac{3}{x - 1} + \frac{3}{y - 1} + \frac{3}{z - 1} = 2(\sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}). \] | \frac{3 + \sqrt{13}}{2} | 0 | 7,061.9375 | -1 | 7,061.9375 | |
The function $y=f(x)$ is an even function with the smallest positive period of $4$, and when $x \in [-2,0]$, $f(x)=2x+1$. If there exist $x\_1$, $x\_2$, $…x\_n$ satisfying $0 \leqslant x\_1 < x\_2 < … < x\_n$, and $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_1)|+…+|f(x\_{n-1}-f(x\_n))|=2016$, then the minimum value of $n+x\_n$ is ... | 1513 | 0.0625 | 7,717.3125 | 8,192 | 7,685.666667 | |
In the complex plane, $z,$ $z^2,$ $z^3$ form, in some order, three of the vertices of a non-degenerate square. Enter all possible areas of the square, separated by commas. | \frac{5}{8}, 2, 10 | 0 | 8,192 | -1 | 8,192 | |
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and a positive geometric sequence $\{b_{n}\}$ with the sum of the first $n$ terms as $T_{n}$, where $a_{1}=2$, $b_{1}=1$, and $b_{3}=3+a_{2}$. <br/>$(1)$ If $b_{2}=-2a_{4}$, find the general formula for the sequence $\{b_{n}\}$; <b... | 18 | 0.6875 | 5,838.9375 | 4,769.363636 | 8,192 | |
Given that one root of the equation $x^{2}+mx+3=0$ is $1$, find the other root and the value of $m$. | -4 | 0.6875 | 1,320.5625 | 1,621.181818 | 659.2 | |
Compute $\begin{pmatrix} 2 & 0 \\ 5 & -3 \end{pmatrix} \begin{pmatrix} 8 & -2 \\ 1 & 1 \end{pmatrix}.$ | \begin{pmatrix} 16 & -4 \\ 37 & -13 \end{pmatrix} | 1 | 2,011.625 | 2,011.625 | -1 | |
Expand the product $(9x+2)(4x^2+3)$. | 36x^3+8x^2+27x+6 | 1 | 1,609.375 | 1,609.375 | -1 | |
Compute the product of $0.\overline{123}$ and $9$, and write your result as a fraction in simplified form. | \frac{41}{37} | 0.875 | 4,453.125 | 3,919 | 8,192 | |
Compute $\sin 240^\circ$. | -\frac{\sqrt{3}}{2} | 0 | 2,187.125 | -1 | 2,187.125 | |
Given line segments $OA$, $OB$, $OC$ are pairwise perpendicular, with $OA=1$, $OB=1$, $OC=2$. If the projections of line segments $OA$, $OB$, $OC$ on line $OP$ have equal lengths, then the length of these projections is $\_\_\_\_\_\_.$ | \frac{2}{3} | 0.9375 | 3,483.75 | 3,169.866667 | 8,192 | |
Star lists the whole numbers $1$ through $50$ once. Emilio copies Star's numbers, but he replaces each occurrence of the digit $2$ by the digit $1$ and each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum and Emilio's sum. | 210 | 0 | 8,020.875 | -1 | 8,020.875 | |
You have a number of gold coins that you were going to distribute equally among your 11 best friends. However, after dividing your gold coins into 11 equal piles, you realize that if you give away all your gold coins, 2 people will receive an extra gold coin. You have less than 100 gold coins. What is the largest numbe... | 90 | 1 | 4,295.9375 | 4,295.9375 | -1 | |
Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | \frac{31}{2} | 0 | 6,552.875 | -1 | 6,552.875 | |
Let $x$ be a positive real number. Find the maximum possible value of $$\frac{x^{2}+2-\sqrt{x^{4}+4}}{x}$$ | 2 \sqrt{2}-2 | Rationalizing the numerator, we get $$\begin{aligned} \frac{x^{2}+2-\sqrt{x^{4}+4}}{x} \cdot \frac{x^{2}+2+\sqrt{x^{4}+4}}{x^{2}+2+\sqrt{x^{4}+4}} & =\frac{\left(x^{2}+2\right)^{2}-\left(x^{4}+4\right)}{x\left(x^{2}+2+\sqrt{x^{4}+4}\right)} \\ & =\frac{4 x^{2}}{x\left(x^{2}+2+\sqrt{x^{4}+4}\right)} \\ & =\frac{4}{\frac... | 0.6875 | 6,449.6875 | 6,029.545455 | 7,374 |
One cube has each of its faces covered by one face of an identical cube, making a solid as shown. The volume of the solid is \(875 \ \text{cm}^3\). What, in \(\text{cm}^2\), is the surface area of the solid?
A) 750
B) 800
C) 875
D) 900
E) 1050 | 750 | 0 | 7,756.5625 | -1 | 7,756.5625 | |
Let \[g(x) =
\begin{cases}
2x - 4 &\text{if } x < 0, \\
5 - 3x &\text{if } x \geq 0.
\end{cases}\]
Find $g(-2)$ and $g(3)$. | -4 | 1 | 1,591 | 1,591 | -1 | |
Given $\tan\alpha= \dfrac {1}{3}$, find the values of the following expressions:
1. $\dfrac {\sin \alpha+\cos\alpha}{5\cos\alpha-\sin\alpha}$
2. $\dfrac {1}{2\sin\alpha\cdot \cos\alpha+\cos ^{2}\alpha}$. | \dfrac {2}{3} | 1 | 3,341.4375 | 3,341.4375 | -1 | |
In the Cartesian coordinate plane \(xOy\), the circle \(\Omega\) and the parabola \(\Gamma: y^2 = 4x\) have exactly one common point, and the circle \(\Omega\) is tangent to the \(x\)-axis at the focus \(F\) of the parabola \(\Gamma\). Find the radius of the circle \(\Omega\). | \frac{4 \sqrt{3}}{9} | 0 | 7,732.3125 | -1 | 7,732.3125 | |
Given $$\sin\left(\alpha+ \frac {\pi}{3}\right)=- \frac {4}{5}$$, and $$- \frac {\pi}{2}<\alpha<0$$, find the value of $\cos\alpha$. | \frac {3-4 \sqrt {3}}{10} | 0 | 8,192 | -1 | 8,192 | |
What is the greatest common divisor of $1729$ and $1768$? | 13 | 1 | 1,977.125 | 1,977.125 | -1 | |
Let $T$ denote the sum of all three-digit positive integers where each digit is different and none of the digits are 5. Calculate the remainder when $T$ is divided by $1000$. | 840 | 0 | 7,097.0625 | -1 | 7,097.0625 | |
Teacher Shi distributed cards with the numbers 1, 2, 3, and 4 written on them to four people: Jia, Yi, Bing, and Ding. Then the following conversation occurred:
Jia said to Yi: "The number on your card is 4."
Yi said to Bing: "The number on your card is 3."
Bing said to Ding: "The number on your card is 2."
Ding said ... | 2341 | 0 | 7,873.4375 | -1 | 7,873.4375 | |
Given the parabola $y=ax^{2}+bx+c$ ($a\neq 0$) with its axis of symmetry to the left of the $y$-axis, where $a$, $b$, $c \in \{-3,-2,-1,0,1,2,3\}$, let the random variable $X$ be the value of "$|a-b|$". Then, the expected value $EX$ is \_\_\_\_\_\_. | \dfrac {8}{9} | 0.375 | 7,103.75 | 6,168.333333 | 7,665 | |
Given that our number system has a base of eight, determine the fifteenth number in the sequence. | 17 | 0.3125 | 5,666.1875 | 5,126 | 5,911.727273 | |
Given a regular triangular pyramid \( S A B C \). Point \( S \) is the apex of the pyramid, \( AB = 1 \), \( AS = 2 \), \( BM \) is the median of triangle \( ABC \), and \( AD \) is the angle bisector of triangle \( SAB \). Find the length of segment \( DM \). | \frac{\sqrt{31}}{6} | 0 | 6,209.125 | -1 | 6,209.125 | |
Let $a, b, c$ be integers. Define $f(x)=a x^{2}+b x+c$. Suppose there exist pairwise distinct integers $u, v, w$ such that $f(u)=0, f(v)=0$, and $f(w)=2$. Find the maximum possible value of the discriminant $b^{2}-4 a c$ of $f$. | 16 | By the factor theorem, $f(x)=a(x-u)(x-v)$, so the constraints essentially boil down to $2=f(w)=a(w-u)(w-v)$. We want to maximize the discriminant $b^{2}-4 a c=a^{2}\left[(u+v)^{2}-4 u v\right]=a^{2}(u-v)^{2}=a^{2}[(w-v)-(w-u)]^{2}$. Clearly $a \mid 2$. If $a>0$, then $(w-u)(w-v)=2 / a>0$ means the difference $|u-v|$ is... | 0.3125 | 7,396.8125 | 6,334.4 | 7,879.727273 |
In the sequence $\{a_n\}$, $a_n+a_{n+1}+a_{n+2}=(\sqrt{2})^{n}$. Find the sum of the first $9$ terms of the sequence $\{a_n\}$ (express the answer as a numerical value). | 4+9\sqrt{2} | 0.1875 | 7,769.375 | 6,227 | 8,125.307692 | |
A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? | 15 | To find the minimum number of small bottles necessary to completely fill a large bottle, we need to determine how many times $35$ milliliters (the capacity of one small bottle) goes into $500$ milliliters (the capacity of one large bottle).
1. **Calculate the number of small bottles needed:**
We perform the divisio... | 0.9375 | 3,077.25 | 2,736.266667 | 8,192 |
Given a prism \(ABC-A'B'C'\) with a base that is an equilateral triangle with side length 2, the lateral edge \(AA'\) forms a 45-degree angle with the edges \(AB\) and \(AC\) of the base. Point \(A'\) is equidistant from the planes \(ABC\) and \(BB'C'C\). Find \(A'A = \_\_\_\_\_ \). | \sqrt{6} | 0 | 7,842 | -1 | 7,842 | |
Find all pairs of integer solutions $(n, m)$ to $2^{3^{n}}=3^{2^{m}}-1$. | (0,0) \text{ and } (1,1) | We find all solutions of $2^{x}=3^{y}-1$ for positive integers $x$ and $y$. If $x=1$, we obtain the solution $x=1, y=1$, which corresponds to $(n, m)=(0,0)$ in the original problem. If $x>1$, consider the equation modulo 4. The left hand side is 0, and the right hand side is $(-1)^{y}-1$, so $y$ is even. Thus we can wr... | 0 | 7,644.4375 | -1 | 7,644.4375 |
Place several small circles with a radius of 1 inside a large circle with a radius of 11, so that each small circle is tangentially inscribed in the large circle and these small circles do not overlap. What is the maximum number of small circles that can be placed? | 31 | 0.625 | 6,819.5625 | 5,996.1 | 8,192 | |
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $1... | 25 | 1. **Understanding the Problem**: Bernardo writes the base-5 and base-6 representations of a three-digit number $N$ on a blackboard. LeRoy, treating these numbers as base-10, adds them to get $S$. We need to find how many such $N$ exist such that the last two digits of $S$ are the same as those of $2N$.
2. **Represent... | 0 | 8,192 | -1 | 8,192 |
With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in the passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the revote was $\frac{12}{11}$ of the number voting against it originally. How m... | 60 | Let $x$ be the number of members who voted for the bill the first time, and $y$ be the number of members who voted against it the first time. Since the total number of members voting is $400$, we have:
\[ x + y = 400 \]
The bill was defeated the first time, so $y > x$. Let the margin by which the bill was defeated be ... | 0.875 | 3,912.6875 | 3,301.357143 | 8,192 |
Given that \( f(x) \) is an odd function defined on \( \mathbf{R} \), and for any \( x \in \mathbf{R} \), the following holds:
$$
f(2+x) + f(2-x) = 0.
$$
When \( x \in [-1, 0) \), it is given that
$$
f(x) = \log_{2}(1-x).
$$
Find \( f(1) + f(2) + \cdots + f(2021) \). | -1 | 0.5 | 7,434.4375 | 6,676.875 | 8,192 | |
In the rectangular coordinate system $(xOy)$, there is an ellipse $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity $e=\frac{\sqrt{2}}{2}$. Also, point $P(2,1)$ is on the ellipse $(C)$.
1. Find the equation of the ellipse $(C)$.
2. If points $A$ and $B$ are both on the ellipse $(C)$,... | \frac{3 \sqrt{2}}{2} | 0 | 7,316.5 | -1 | 7,316.5 | |
What is the volume, in cubic inches, of a rectangular box, whose faces have areas of $24$ square inches, $16$ square inches and $6$ square inches? | 48 | 1 | 2,116.6875 | 2,116.6875 | -1 | |
A number which when divided by $10$ leaves a remainder of $9$, when divided by $9$ leaves a remainder of $8$, by $8$ leaves a remainder of $7$, etc., down to where, when divided by $2$, it leaves a remainder of $1$, is: | 2519 | 1. **Understanding the Problem:**
The problem states that a number $n$ has specific remainders when divided by integers from $10$ down to $2$. Specifically, when $n$ is divided by any integer $k$ (where $10 \geq k \geq 2$), it leaves a remainder of $k-1$.
2. **Formulating the Equations:**
This can be expressed a... | 0.9375 | 2,979.8125 | 2,632.333333 | 8,192 |
Let $x$ be a positive integer, and define the integers $n=x^2+2x+17$ and $d=2x+5$. When dividing $n$ by $d$, the quotient is $x$, and the remainder is $7$. Find $x$. | 2 | 0.9375 | 2,543.75 | 2,167.2 | 8,192 | |
The updated stem-and-leaf plot shows the duration of rides for each of the $21$ top-rated roller coasters. Each entry in the plot represents ride time, where, for example, $3 \ 05$ means $3$ minutes, $5$ seconds. Convert this time to seconds to find the median of the data set.
\begin{tabular}{c|cccccc}
0&28&28&50&55&&... | 163 | 0.3125 | 4,864.5 | 2,213.4 | 6,069.545455 | |
Given the sequence $\{a_n\}$, $a_1=1$, $a_2=2$, and $a_{n+2}-a_{n}=1+(-1)^{n}$ $(n\in\mathbb{N}_{+})$, calculate the value of $S_{100}$. | 2600 | 0.625 | 6,075.5625 | 4,805.7 | 8,192 | |
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? | 207 | 1. **Identify the constraints on the digits of prime numbers**:
- Prime numbers greater than 2 are odd, so they cannot end in an even digit (0, 2, 4, 6, 8).
- Additionally, a prime number cannot end in 5 unless it is 5 itself, because any other number ending in 5 is divisible by 5.
2. **Determine the digits tha... | 0 | 8,192 | -1 | 8,192 |
How many irreducible fractions with numerator 2015 exist that are less than \( \frac{1}{2015} \) and greater than \( \frac{1}{2016} \)? | 1440 | 0.0625 | 8,055.1875 | 7,433 | 8,096.666667 | |
Let $ABC$ be a triangle with circumcenter $O$, incenter $I, \angle B=45^{\circ}$, and $OI \parallel BC$. Find $\cos \angle C$. | 1-\frac{\sqrt{2}}{2} | Let $M$ be the midpoint of $BC$, and $D$ the foot of the perpendicular of $I$ with $BC$. Because $OI \parallel BC$, we have $OM=ID$. Since $\angle BOC=2 \angle A$, the length of $OM$ is $OA \cos \angle BOM=OA \cos A=R \cos A$, and the length of $ID$ is $r$, where $R$ and $r$ are the circumradius and inradius of $\trian... | 0 | 8,150.5625 | -1 | 8,150.5625 |
For how many positive integers $n$ does $1+2+\cdots+n$ evenly divide $6n$? | 5 | 1 | 2,381.8125 | 2,381.8125 | -1 | |
Solve for $r$: \[\frac{r-45}{2} = \frac{3-2r}{5}.\] | \frac{77}{3} | 0.9375 | 2,570 | 2,634 | 1,610 | |
Gavrila found that the front tires of the car last for 21,000 km, and the rear tires last for 28,000 km. Therefore, he decided to swap them at some point so that the car would travel the maximum possible distance. Find this maximum distance (in km). | 24000 | 0.0625 | 7,514.25 | 3,875 | 7,756.866667 | |
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the side... | 2(w+h)^2 | 1. **Understanding the Problem Setup**: We have a square sheet of wrapping paper and a box with a square base of side $w$ and height $h$. The box is placed such that its base vertices lie on the midlines of the wrapping paper. The wrapping paper is folded up to meet at a point $A$ at the center of the top of the box.
... | 0 | 8,192 | -1 | 8,192 |
In the diagram, four circles with centers $P$, $Q$, $R$, and $S$ each have a radius of 2. These circles are tangent to one another and to the sides of $\triangle ABC$ as shown. The circles centered at $P$ and $Q$ are tangent to side $AB$, the circle at $R$ is tangent to side $BC$, and the circle at $S$ is tangent to si... | 36 | 0 | 8,192 | -1 | 8,192 | |
Determine the coefficient of $x^{8}$ in the expansion of \\((x^{3}+ \frac{1}{2 \sqrt {x}})^{5}\\). | \frac{5}{2} | 1 | 3,134 | 3,134 | -1 | |
The expression $x^2 + 15x + 54$ can be written as $(x + a)(x + b),$ and the expression $x^2 - 17x + 72$ written as $(x - b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$? | 23 | 0.4375 | 3,631.3125 | 4,061.714286 | 3,296.555556 | |
Given a $4\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the first row being $3$ and the fourth term being $21$, and the first term of the fourth row being $15$ and the fourth term being $45$. | \frac{43}{3} | 0 | 7,808.0625 | -1 | 7,808.0625 | |
If \(x + \frac{1}{y} = 3\) and \(y + \frac{1}{z} = 3\), what is the value of the product \(xyz\)? | -1 | 0 | 7,936.125 | -1 | 7,936.125 | |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? | 2\sqrt{10} | 1 | 2,838.375 | 2,838.375 | -1 | |
If
\begin{align*}
a + b + c &= 1, \\
a^2 + b^2 + c^2 &= 2, \\
a^3 + b^3 + c^3 &= 3,
\end{align*}find $a^4 + b^4 + c^4.$ | \frac{25}{6} | 0.8125 | 4,831.9375 | 4,056.538462 | 8,192 | |
Given triangle $ABC$ . Let $A_1B_1$ , $A_2B_2$ , $ ...$ , $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor... | 29985 | 0 | 7,917.6875 | -1 | 7,917.6875 | |
Let $M$ be a point on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, with $F_1$ and $F_2$ as its foci. If $\angle F_1MF_2 = \frac{\pi}{6}$, calculate the area of $\triangle MF_1F_2$. | 16(2 - \sqrt{3}) | 0.1875 | 7,757.6875 | 6,298.333333 | 8,094.461538 | |
Find the equation of the directrix of the parabola $y = \frac{x^2 - 6x + 5}{12}.$ | y = -\frac{10}{3} | 1 | 2,942.6875 | 2,942.6875 | -1 | |
In a $k \times k$ chessboard, a set $S$ of 25 cells that are in a $5 \times 5$ square is chosen uniformly at random. The probability that there are more black squares than white squares in $S$ is $48 \%$. Find $k$. | 9 | We know that there must be fewer black squares than white squares, and $k$ must be odd. Additionally, we know that there are $k-4$ ways to pick the left column of the $5 \times 5$ square so that the right column can fit within the $k \times k$ grid, and $k-4$ ways to pick the top row by similar logic. Therefore, there ... | 0 | 8,192 | -1 | 8,192 |
Given the letters in the word $SUCCESS$, determine the number of distinguishable rearrangements where all the vowels are at the end. | 20 | 0.375 | 5,137.9375 | 4,681.5 | 5,411.8 | |
Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. What is the number of square units in the area of the fourth square?
Note that the diagram is not provided, but imagine it similarly to the reference whe... | 10 | 0.125 | 6,921.1875 | 5,481.5 | 7,126.857143 | |
How many natural numbers between 200 and 400 are divisible by 8? | 24 | 0.5625 | 5,248.75 | 4,142.666667 | 6,670.857143 | |
Determine the area of the circle described by the graph of the equation
\[r = 4 \cos \theta - 3 \sin \theta.\] | \frac{25\pi}{4} | 0.125 | 2,528.25 | 2,746 | 2,497.142857 | |
The function \[f(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x} & \quad \text{ if } x \ge 5 \end{aligned} \right.\] has an inverse $f^{-1}.$ Find the value of $f^{-1}(0) + f^{-1}(1) + \dots + f^{-1}(9).$ | 291 | 0.125 | 6,040.375 | 5,593 | 6,104.285714 | |
In $\triangle ABC$ , $AB = 40$ , $BC = 60$ , and $CA = 50$ . The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$ . Find $BP$ .
*Proposed by Eugene Chen* | 40 | 0.3125 | 7,456.0625 | 6,610.2 | 7,840.545455 | |
At Jefferson Summer Camp, $60\%$ of the children play soccer, $30\%$ of the children swim, and $40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer? | 51\% | Let's denote the total number of children at the camp as $N$. We are given the following percentages:
- $60\%$ of the children play soccer, which translates to $0.6N$ children.
- $30\%$ of the children swim, which translates to $0.3N$ children.
- $40\%$ of the soccer players also swim. Since $0.6N$ children play soccer... | 1 | 3,091.1875 | 3,091.1875 | -1 |
What is the area of the shaded pinwheel shown in the $5 \times 5$ grid?
[asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } [/asy] | 6 | To find the area of the shaded pinwheel in the $5 \times 5$ grid, we can use Pick's Theorem. However, the theorem requires all vertices of the polygons to be lattice points (points with integer coordinates), and the center of the pinwheel is not a lattice point. To address this, we scale the figure by a factor of 2, ma... | 0.375 | 7,700.5 | 6,881.333333 | 8,192 |
Find the number of ordered pairs of integers $(a, b) \in\{1,2, \ldots, 35\}^{2}$ (not necessarily distinct) such that $a x+b$ is a "quadratic residue modulo $x^{2}+1$ and 35 ", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following equivalent conditions holds: - there exist po... | 225 | By the Chinese remainder theorem, we want the product of the answers modulo 5 and modulo 7 (i.e. when 35 is replaced by 5 and 7, respectively). First we do the modulo 7 case. Since $x^{2}+1$ is irreducible modulo 7 (or more conceptually, in $\mathbb{F}_{7}[x]$ ), exactly half of the nonzero residues modulo $x^{2}+1$ an... | 0.0625 | 7,859.4375 | 4,976 | 8,051.666667 |
$$\frac {4}{5} + 9 \frac {4}{5} + 99 \frac {4}{5} + 999 \frac {4}{5} + 9999 \frac {4}{5} + 1 = \_\_\_\_\_\_.$$ | 11111 | 0.5625 | 2,032.8125 | 2,988.555556 | 804 | |
Roger has exactly one of each of the first 22 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. The graph below shows the number of states that joined the union in each decade. What fraction of Roger's 22 coins represents states that joined the union during... | \frac{6}{11} | 0 | 8,192 | -1 | 8,192 | |
Given that \( f(x-1)=|x|-|x-2| \) and \( f(f(m))=f(2002)-\frac{7}{2} \), find the value of the real number \( m \). | -\frac{3}{8} | 1 | 3,704.0625 | 3,704.0625 | -1 | |
Let $x, y, z$ be real numbers such that:
\begin{align*}
y+z & = 16, \\
z+x & = 18, \\
x+y & = 20.
\end{align*}
Find $\sqrt{xyz(x+y+z)}$. | 9\sqrt{77} | 0 | 6,351.625 | -1 | 6,351.625 | |
Given $\sin \alpha - \cos \alpha = \frac{\sqrt{10}}{5}$, $\alpha \in (\pi, 2\pi)$,
$(1)$ Find the value of $\sin \alpha + \cos \alpha$;
$(2)$ Find the value of $\tan \alpha - \frac{1}{\tan \alpha}$. | -\frac{8}{3} | 0.8125 | 4,676.75 | 4,262.692308 | 6,471 | |
Given the function $f(x) = \sin x \cos x - \sqrt{3} \cos (x+\pi) \cos x, x \in \mathbb{R}$.
(Ⅰ) Find the minimal positive period of $f(x)$;
(Ⅱ) If the graph of the function $y = f(x)$ is translated by $\overrightarrow{b}=\left( \frac{\pi}{4}, \frac{\sqrt{3}}{2} \right)$ to obtain the graph of the function $y = g(x)$, f... | \frac{3\sqrt{3}}{2} | 0 | 5,307.5625 | -1 | 5,307.5625 | |
I randomly pick an integer $p$ between $1$ and $20$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 6p - 3q = 3$? Express your answer as a common fraction. | \frac{3}{20} | 0.0625 | 5,197.8125 | 4,599 | 5,237.733333 | |
Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2 &=& r^2+s^2 &=& 2500, \\
pr &=& qs &=& 1200.
\end{array}
\]
If \( T = p + q + r + s \), compute the value of \( \lfloor T \rfloor \). | 120 | 0 | 5,747.25 | -1 | 5,747.25 | |
Given that the number of parcels received by a person in the months from January to May are $1$, $3$, $2$, $2$, $2$ respectively, find the variance ($s^{2}=$ ___) of these $5$ numbers. | \frac{2}{5} | 0.375 | 2,893.8125 | 2,851.5 | 2,919.2 | |
A square with sides 8 inches is shown. If $Q$ is a point such that the segments $\overline{QA}$, $\overline{QB}$, $\overline{QC}$ are equal in length, and segment $\overline{QC}$ is perpendicular to segment $\overline{HD}$, find the area, in square inches, of triangle $AQB$. [asy]
pair A, B, C, D, H, Q;
A = (0,0); B= (... | 12 | 0.1875 | 5,878.3125 | 4,566 | 6,181.153846 | |
Below is the graph of $y = a \csc bx$ for some positive constants $a$ and $b.$ Find $a.$
[asy]import TrigMacros;
size(500);
real g(real x)
{
return 2*csc(x/3);
}
draw(graph(g,-6*pi + 0.01, -3*pi - 0.01),red);
draw(graph(g,-3*pi + 0.01,-0.01),red);
draw(graph(g,0.01,3*pi - 0.01),red);
draw(graph(g,3*pi + 0.01,6*pi... | 2 | 0.875 | 5,010.6875 | 4,556.214286 | 8,192 | |
Given the sets $M={x|m\leqslant x\leqslant m+ \frac {3}{4}}$ and $N={x|n- \frac {1}{3}\leqslant x\leqslant n}$, both of which are subsets of ${x|0\leqslant x\leqslant 1}$, what is the minimum "length" of the set $M\cap N$? (Note: The "length" of a set ${x|a\leqslant x\leqslant b}$ is defined as $b-a$.) | \frac{1}{12} | 0.4375 | 7,350.0625 | 6,448.142857 | 8,051.555556 | |
A polynomial $g(x)=x^4+px^3+qx^2+rx+s$ has real coefficients, and it satisfies $g(3i)=g(3+i)=0$. | 49 | 0.0625 | 4,323.125 | 5,647 | 4,234.866667 | |
Kelvin the frog lives in a pond with an infinite number of lily pads, numbered $0,1,2,3$, and so forth. Kelvin starts on lily pad 0 and jumps from pad to pad in the following manner: when on lily pad $i$, he will jump to lily pad $(i+k)$ with probability $\frac{1}{2^{k}}$ for $k>0$. What is the probability that Kelvin ... | \frac{1}{2} | Suppose we combine all of the lily pads with numbers greater than 2019 into one lily pad labeled $\infty$. Also, let Kelvin stop once he reaches one of these lily pads. Now at every leap, Kelvin has an equal chance of landing on 2019 as landing on $\infty$. Furthermore, Kelvin is guaranteed to reach 2019 or $\infty$ wi... | 0 | 8,192 | -1 | 8,192 |
Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$ . A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers... | 106 | 0 | 8,192 | -1 | 8,192 |
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