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 |
|---|---|---|---|---|---|---|
What is the median of the following list of numbers that includes integers from $1$ to $2020$, their squares, and their cubes? \[1, 2, 3, \ldots, 2020, 1^2, 2^2, \ldots, 2020^2, 1^3, 2^3, \ldots, 2020^3\]
A) $2040200$
B) $2040201$
C) $2040202$
D) $2040203$
E) $2040204$ | 2040201 | 0 | 8,192 | -1 | 8,192 | |
What is the value of $\log_{10}{4}+2\log_{10}{5}+3\log_{10}{2}+6\log_{10}{5}+\log_{10}{8}$? | 8 | 0.875 | 3,991.1875 | 3,391.071429 | 8,192 | |
A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression? | 1 | 1 | 2,395.6875 | 2,395.6875 | -1 | |
In right triangle $ABC$, shown below, $\cos{C}=\frac{9\sqrt{130}}{130}$. Find $AC$.
[asy]
draw((0,0)--(7,0)--(0,9)--cycle,black+linewidth(1));
draw(rightanglemark((7,0),(0,0),(0,9),20),black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(7,0),E);
label("$C$",(0,9),W);
label("$\sqrt{130}$",(7,0)--(0,9),NE);
[/asy] | 9 | 0.6875 | 5,482.3125 | 4,331.636364 | 8,013.8 | |
It is known that there are a total of $n$ students in the first grade of Shuren High School, with $550$ male students. They are divided into layers based on gender, and $\frac{n}{10}$ students are selected to participate in a wetland conservation knowledge competition. It is given that there are $10$ more male students... | 1000 | 0.0625 | 7,431.1875 | 1,684 | 7,814.333333 | |
The numbers assigned to 100 athletes range from 1 to 100. If each athlete writes down the largest odd factor of their number on a blackboard, what is the sum of all the numbers written by the athletes? | 3344 | 0 | 8,192 | -1 | 8,192 | |
Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + XY^2 = 1143$. Find $XY^2$.
| 717 | 0 | 8,192 | -1 | 8,192 | |
How many three-digit positive integers \( x \) are there with the property that \( x \) and \( 2x \) have only even digits? (One such number is \( x=420 \), since \( 2x=840 \) and each of \( x \) and \( 2x \) has only even digits.) | 18 | 0 | 8,192 | -1 | 8,192 | |
Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$ . If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ? | 121 | 0.25 | 7,913 | 7,076 | 8,192 | |
The decimal representation of \(\dfrac{1}{25^{10}}\) consists of a string of zeros after the decimal point, followed by non-zero digits. Find the number of zeros in that initial string of zeros after the decimal point. | 20 | 0 | 7,481.5 | -1 | 7,481.5 | |
The ratio of $x+2$ to $2x+2$ is equal to the ratio of $4x+3$ to $7x+3$. What is the product of all real x which satisfy this statement? | 0 | 1 | 2,738.25 | 2,738.25 | -1 | |
Jia, Yi, Bing, Ding, and Wu sit around a circular table to play cards. Jia has a fixed seat. If Yi and Ding cannot sit next to each other, how many different seating arrangements are possible? | 12 | 0.25 | 7,520.125 | 6,064.75 | 8,005.25 | |
In cube \( ABCD A_{1} B_{1} C_{1} D_{1} \), with an edge length of 6, points \( M \) and \( N \) are the midpoints of edges \( AB \) and \( B_{1} C_{1} \) respectively. Point \( K \) is located on edge \( DC \) such that \( D K = 2 K C \). Find:
a) The distance from point \( N \) to line \( AK \);
b) The distance bet... | \frac{66}{\sqrt{173}} | 0 | 6,860.4375 | -1 | 6,860.4375 | |
What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $15(3x-4) \equiv 30 \pmod{10}$? | 240 | 0.875 | 3,507.375 | 2,838.142857 | 8,192 | |
The solution of $\sqrt{5x-1}+\sqrt{x-1}=2$ is: | $x=1$ | 1. **Start by squaring both sides of the equation**:
\[
\left(\sqrt{5x-1} + \sqrt{x-1}\right)^2 = 2^2
\]
Expanding the left side using the identity $(a+b)^2 = a^2 + 2ab + b^2$, we get:
\[
\sqrt{5x-1}^2 + 2\sqrt{5x-1}\sqrt{x-1} + \sqrt{x-1}^2 = 4
\]
Simplifying the squares of the square roots, we... | 0 | 3,795.375 | -1 | 3,795.375 |
Given a circular park 20 feet in diameter with a straight walking path 5 feet wide passing through its center, calculate the area of the park covered by grass after the path is laid. | 100\pi - 100 | 0.125 | 4,486.8125 | 4,020 | 4,553.5 | |
Let $d_1$, $d_2$, $d_3$, $d_4$, $e_1$, $e_2$, $e_3$, and $e_4$ be real numbers such that for every real number $x$, we have
\[
x^8 - 2x^7 + 2x^6 - 2x^5 + 2x^4 - 2x^3 + 2x^2 - 2x + 1 = (x^2 + d_1 x + e_1)(x^2 + d_2 x + e_2)(x^2 + d_3 x + e_3)(x^2 + d_4 x + e_4).
\]
Compute $d_1 e_1 + d_2 e_2 + d_3 e_3 + d_4 e_4$. | -2 | 0.375 | 7,619.5 | 6,815.666667 | 8,101.8 | |
Given that $\tan \alpha$ and $\frac{1}{\tan \alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $\pi < \alpha < \frac{3}{2}\pi$.
$(1)$ Find the value of $\cos \alpha + \sin \alpha$.
$(2)$ Find the value of $\frac{\sin (\alpha-3\pi)+\cos (\pi-\alpha)+\sin ( \frac{3\pi}{2}-\alpha)-2\cos ( \fra... | \frac{1}{2} | 0.8125 | 5,369.875 | 5,166.615385 | 6,250.666667 | |
Express $0.4\overline5$ as a common fraction. | \frac{41}{90} | 1 | 2,243.0625 | 2,243.0625 | -1 | |
Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q... | 71 | Let $Q(x) = x^2 + ax + 2$ and $R(x) = x^2 + bx + c$. We can write the following: \[P + Q = 2x^2 + (a - 3)x - 5\] \[P + R = 2x^2 + (b - 3)x + (c - 7)\] \[Q + R = 2x^2 + (a + b)x + (c + 2)\] Let the common root of $P+Q,P+R$ be $r$; $P+R,Q+R$ be $s$; and $P+Q,Q+R$ be $t$. We then have that the roots of $P+Q$ are $r,t$, th... | 0.3125 | 7,439.4375 | 5,783.8 | 8,192 |
Given that the mean of the data set \\(k\_{1}\\), \\(k\_{2}\\), \\(…\\), \\(k\_{8}\\) is \\(4\\), and the variance is \\(2\\), find the mean and variance of \\(3k\_{1}+2\\), \\(3k\_{2}+2\\), \\(…\\), \\(3k\_{8}+2\\). | 18 | 1 | 1,937.875 | 1,937.875 | -1 | |
An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. The number 12 is an abundant number since $1 + 2 + 3 + 4 + 6 > 12$. What is the smallest abundant number that is not a multiple of 6? | 20 | 0.8125 | 4,470.375 | 3,611.538462 | 8,192 | |
Given that $\alpha$ is an angle in the second quadrant, and $P(x, 4)$ is a point on its terminal side, with $\cos\alpha= \frac {1}{5}x$, then $x= \_\_\_\_\_\_$, and $\tan\alpha= \_\_\_\_\_\_$. | -\frac {4}{3} | 1 | 1,724.5 | 1,724.5 | -1 | |
Let $a=x^3-3x^2$, then the coefficient of the $x^2$ term in the expansion of $(a-x)^6$ is $\boxed{-192}$. | -192 | 0.75 | 8,192 | 8,192 | 8,192 | |
Given vectors $\overrightarrow{a}=(\sqrt{2}\cos \omega x,1)$ and $\overrightarrow{b}=(2\sin (\omega x+ \frac{\pi}{4}),-1)$ where $\frac{1}{4}\leqslant \omega\leqslant \frac{3}{2}$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$, and the graph of $f(x)$ has an axis of symmetry at $x= \frac{5\pi}{8}$... | \frac{2\sqrt{10}+2}{9} | 0 | 6,973.125 | -1 | 6,973.125 | |
Let $n$ be the smallest positive integer such that the remainder of $3n+45$ , when divided by $1060$ , is $16$ . Find the remainder of $18n+17$ upon division by $1920$ . | 1043 | 0.875 | 5,313 | 4,901.714286 | 8,192 | |
Given that $(a+1)x - 1 - \ln x \leqslant 0$ holds for any $x \in [\frac{1}{2}, 2]$, find the maximum value of $a$. | 1 - 2\ln 2 | 0.875 | 4,543.8125 | 4,197.5 | 6,968 | |
Let the function be $$f(x)=\sin(2\omega x+ \frac {\pi}{3})+ \frac { \sqrt {3}}{2}+a(\omega＞0)$$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $$\frac {\pi}{6}$$.
(1) Find the value of $\omega$;
(2) If the minimum value of $f(x)$ in the interval $$[- \frac {... | \frac { \sqrt {3}+1}{2} | 0 | 7,285.4375 | -1 | 7,285.4375 | |
Two distinct positive integers \( x \) and \( y \) are factors of 48. If \( x \cdot y \) is not a factor of 48, what is the smallest possible value of \( x \cdot y \)? | 32 | 0.125 | 7,597.5625 | 7,054.5 | 7,675.142857 | |
At 7:00, five sheep, designated as A, B, C, D, and E, have distances to Wolf Castle forming an arithmetic sequence with a common difference of 20 meters. At 8:00, these same five sheep have distances to Wolf Castle forming another arithmetic sequence, but with a common difference of 30 meters, and their order has chang... | 140 | 0 | 8,114.625 | -1 | 8,114.625 | |
Let set $M=\{x|-1\leq x\leq 5\}$, and set $N=\{x|x-k\leq 0\}$.
1. If $M\cap N$ has only one element, find the value of $k$.
2. If $k=2$, find $M\cap N$ and $M\cup N$. | -1 | 0 | 2,506 | -1 | 2,506 | |
If the system of equations $$\begin{aligned} & |x+y|=99 \\ & |x-y|=c \end{aligned}$$ has exactly two real solutions $(x, y)$, find the value of $c$. | 0 | If $c<0$, there are no solutions. If $c>0$ then we have four possible systems of linear equations given by $x+y= \pm 99, x-y= \pm c$, giving four solutions $(x, y)$. So we must have $c=0$, and then we do get two solutions ( $x=y$, so they must both equal $\pm 99 / 2$ ). | 0.5 | 7,505.4375 | 6,818.875 | 8,192 |
Denis has cards with numbers from 1 to 50. How many ways are there to choose two cards such that the difference of the numbers on the cards is 11, and their product is divisible by 5?
The order of the selected cards does not matter: for example, selecting cards with numbers 5 and 16, as well as selecting cards with nu... | 15 | 0.75 | 6,218.3125 | 5,713.916667 | 7,731.5 | |
Find \(\cos \frac{\alpha - \beta}{2}\), given \(\sin \alpha + \sin \beta = -\frac{27}{65}\), \(\tan \frac{\alpha + \beta}{2} = \frac{7}{9}\), \(\frac{5}{2} \pi < \alpha < 3 \pi\) and \(-\frac{\pi}{2} < \beta < 0\). | \frac{27}{7 \sqrt{130}} | 0 | 6,558.5 | -1 | 6,558.5 | |
Given two numbers selected randomly from the interval $[0,1]$, find the probability that the sum of these two numbers is less than $\frac{8}{5}$. | \frac{23}{25} | 0.5 | 6,884.25 | 5,576.5 | 8,192 | |
In a chess tournament, students from the 9th and 10th grades participated. There were 10 times more 10th graders than 9th graders. | 10 | 0.375 | 5,074.9375 | 5,760.833333 | 4,663.4 | |
Given points $A\left(\begin{matrix} \cos \alpha , & \sin \alpha \end{matrix}\right)$ and $B\left(\begin{matrix} \cos \beta , & \sin \beta \end{matrix}\right)$, where $\alpha$ and $\beta$ are acute angles, and the distance between $A$ and $B$ is $\frac{\sqrt{10}}{5}$.
$(1)$ Find the value of $\cos (\alpha -\beta)$; $(2... | \frac{24}{25} | 0.9375 | 5,704.5 | 5,538.666667 | 8,192 | |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}+1, n \text{ is odd}}\\{{a}_{n}+2, n \text{ is even}}\end{array}\right.$
$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$.
$(2)$ Find the sum of the first $20$ te... | 300 | 1 | 4,852.8125 | 4,852.8125 | -1 | |
If $x + 2y= 4$ and $xy = -8$, what is the value of $x^2 + 4y^2$? | 48 | 1 | 2,452.4375 | 2,452.4375 | -1 | |
Given a quadratic function \( f(x) = x^2 + ax + b \), determine the minimum value of \( F = \max_{|x| \leq 1} \mid f(x) \), where \( a \) and \( b \) are real numbers. | \frac{1}{2} | 0.3125 | 7,709.125 | 6,646.8 | 8,192 | |
Define $[a,b,c]$ to mean $\frac {a+b}c$, where $c \neq 0$. What is the value of $\left[[60,30,90],[2,1,3],[10,5,15]\right]?$ | 2 | 1. **Understanding the operation $[a,b,c]$:**
The operation $[a,b,c]$ is defined as $\frac{a+b}{c}$, where $c \neq 0$.
2. **Simplifying each component:**
- For $[60,30,90]$, we calculate:
\[
[60,30,90] = \frac{60+30}{90} = \frac{90}{90} = 1
\]
- For $[2,1,3]$, we calculate:
\[
[2,... | 1 | 2,441.1875 | 2,441.1875 | -1 |
Let \( A \) and \( B \) be the endpoints of a semicircular arc of radius \( 3 \). The arc is divided into five congruent arcs by four equally spaced points \( C_1, C_2, C_3, C_4 \). All chords of the form \( \overline{AC_i} \) or \( \overline{BC_i} \) are drawn. Find the product of the lengths of these eight chords. | 32805 | 0.25 | 7,746.0625 | 6,408.25 | 8,192 | |
Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$ ?
*Proposed by Giacomo Rizzo* | 444 | 0.25 | 7,958.25 | 7,257 | 8,192 | |
What is the greatest four-digit number that is one more than a multiple of 7 and five more than a multiple of 8? | 9997 | 0.5625 | 5,555.1875 | 4,353 | 7,100.857143 | |
Observe the following equations:
1=1
1-4=-（1+2）=-3
1-4+9=1+2+3=6
1-4+9-16=-（1+2+3+4）=-10
Then, the 5th equation is
The value of the 20th equation is
These equations reflect a certain pattern among integers. Let $n$ represent a positive integer, try to express the pattern you discovered using an eq... | -210 | 0.625 | 3,915.125 | 3,155 | 5,182 | |
An employee receives an average of two requests per hour. Assuming a simple flow of requests, what is the probability of receiving four requests in four hours? | 0.0572 | 0 | 3,998.6875 | -1 | 3,998.6875 | |
Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a $60 \%$ chance of winning each point, what is the probability that he will win the game? | 9 / 13 | Consider the situation after two points. Daniel has a $9 / 25$ chance of winning, Scott, $4 / 25$, and there is a $12 / 25$ chance that the players will be tied. In the latter case, we revert to the original situation. In particular, after every two points, either the game returns to the original situation, or one play... | 0.5625 | 6,525.8125 | 5,438 | 7,924.428571 |
What is the largest number, all of whose digits are 3 or 2, and whose digits add up to $11$? | 32222 | 0.6875 | 6,039.0625 | 5,139.727273 | 8,017.6 | |
Find the sum of the squares of the solutions to
\[\left| x^2 - x + \frac{1}{2023} \right| = \frac{1}{2023}.\] | \frac{4042}{2023} | 0.5625 | 7,450.1875 | 7,017.222222 | 8,006.857143 | |
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that
$$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$. | f(x) = x + 1 |
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function that satisfies the equation:
\[
f(x) + f(yf(x) + f(y)) = f(x + 2f(y)) + xy
\]
for all \( x, y \in \mathbb{R} \).
We need to find all such functions \( f \).
### Step 1: Check for Simple Solutions
Assume \( f(x) = x + c \) for some constant \( c \). Substitute into... | 0 | 8,192 | -1 | 8,192 |
Given 5 different letters from the word "equation", find the total number of different arrangements that contain "qu" where "qu" are consecutive and in the same order. | 480 | 0.5 | 6,564.4375 | 5,450.5 | 7,678.375 | |
If each side of a regular hexagon consists of 6 toothpicks, and there are 6 sides, calculate the total number of toothpicks used to build the hexagonal grid. | 36 | 0.4375 | 5,611.375 | 5,557.714286 | 5,653.111111 | |
Eight distinct points, $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$, $P_7$, and $P_8$, are evenly spaced around a circle. If four points are chosen at random from these eight points to form two chords, what is the probability that the chord formed by the first two points chosen intersects the chord formed by the last two p... | \frac{1}{3} | 0.375 | 7,595.625 | 6,663.666667 | 8,154.8 | |
Four chess players - Ivanov, Petrov, Vasiliev, and Kuznetsov - played a round-robin tournament (each played one game against each of the others). A victory awards 1 point, a draw awards 0.5 points to each player. It was found that the player in first place scored 3 points, and the player in last place scored 0.5 points... | 36 | 0 | 8,192 | -1 | 8,192 | |
With all angles measured in degrees, consider the product $\prod_{k=1}^{22} \sec^2(4k)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | 24 | 0 | 6,956.625 | -1 | 6,956.625 | |
Given points $A$, $B$, and $C$ are on the curve $y=\sqrt{x}$ $(x \geqslant 0)$, with x-coordinates $1$, $m$, and $4$ $(1 < m < 4)$, find the value of $m$ that maximizes the area of $\triangle ABC$. | \frac{9}{4} | 1 | 4,655.9375 | 4,655.9375 | -1 | |
A cylindrical hole of 6 cm in length is drilled through a sphere, with the axis of the cylinder passing through the center of the sphere. What is the remaining volume? (Note: The volume of a spherical cap is $\pi h^{2}(R-h / 3)$, where $R$ is the radius of the sphere and $h$ is the height of the cap.) | 36 \pi | 0.1875 | 7,569 | 6,640.333333 | 7,783.307692 | |
If $\frac{1}{9}$ of 60 is 5, what is $\frac{1}{20}$ of 80? | 6 | In base 15, 6. | 0 | 3,895.875 | -1 | 3,895.875 |
At the beginning of the school year, Jinshi Middle School has 17 classrooms. Teacher Dong Helong takes 17 keys to open the doors, knowing that each key can only open one door, but not knowing which key matches which door. What is the maximum number of attempts he needs to make to open all 17 locked doors? | 136 | 0.1875 | 5,874.125 | 4,288.666667 | 6,240 | |
Given \(\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+}\) and \(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=\pi\), find the minimum value of \(\left(2 \sin ^{2} \theta_{1}+\frac{1}{\sin ^{2} \theta_{1}}\right)\left(2 \sin ^{2} \theta_{2}+\frac{1}{\sin ^{2} \theta_{2}}\right)\left(2 \sin ^{2} \theta_{3... | 81 | 0.1875 | 7,884.9375 | 7,396.666667 | 7,997.615385 | |
Given that $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively, and $c\cos A=5$, $a\sin C=4$.
(1) Find the length of side $c$;
(2) If the area of $\triangle ABC$, $S=16$, find the perimeter of $\triangle ABC$. | 13+ \sqrt {41} | 0 | 5,710.4375 | -1 | 5,710.4375 | |
What is the least integer greater than $\sqrt{300}$? | 18 | 0.9375 | 3,745.3125 | 3,448.866667 | 8,192 | |
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with 3 buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines a... | 9 | Clearly we do not want to reset the system at any time. After pressing the red button $r$ times, the yellow button $y$ times, and the green button $g$ times, there will be $3+r-2 y$ armed mines and $3+y-2 g$ closed doors, so we want the values of $r, y$, and $g$ that make both of these quantities 0 while minimizing $r+... | 0 | 8,047.5 | -1 | 8,047.5 |
The sum of all integers between 50 and 350 which end in 1 is | 5880 | 1. **Identify the sequence**: The problem asks for the sum of all integers between 50 and 350 that end in 1. These integers are $51, 61, 71, \ldots, 341$.
2. **Determine the sequence type**: The sequence is an arithmetic sequence where each term increases by 10.
3. **Find the first term ($a$) and common difference ($... | 1 | 3,108.75 | 3,108.75 | -1 |
Four foreign guests visit a school and need to be accompanied by two security personnel. Six people enter the school gate in sequence. For safety reasons, the two security personnel must be at the beginning and the end. If the guests A and B must be together, calculate the total number of sequences for the six people e... | 24 | 0.6875 | 6,282.625 | 5,957.636364 | 6,997.6 | |
The year 2009 has a unique property: by rearranging the digits of the number 2009, it is impossible to form a smaller four-digit number (numbers do not start with zero). In which future year will this property first repeat again? | 2022 | 0.0625 | 7,740.0625 | 7,290 | 7,770.066667 | |
Find the largest negative integer $x$ which satisfies the congruence $34x+6\equiv 2\pmod {20}$. | -6 | 1 | 3,189.3125 | 3,189.3125 | -1 | |
Given that $F\_1$ and $F\_2$ are two foci of the hyperbola $x^2-y^2=1$, and $P$ is a point on the hyperbola such that $\angle F\_1PF\_2=60^{\circ}$, determine the area of $\triangle F\_1PF\_2$. | \sqrt{3} | 0.8125 | 5,866.4375 | 5,329.769231 | 8,192 | |
Evaluate the expression $\log_{y^8}{x^2}\cdot\log_{x^7}{y^3}\cdot\log_{y^5}{x^4}\cdot\log_{x^4}{y^5}\cdot\log_{y^3}{x^7}$ and express it as $b\log_y{x}$ for some constant $b$. | \frac{1}{4} | 0.375 | 6,533.5625 | 5,247.833333 | 7,305 | |
How many square units are in the area of the triangle whose vertices are the $x$ and $y$ intercepts of the curve $y = (x-3)^2 (x+2)$? | 45 | 1 | 3,415.125 | 3,415.125 | -1 | |
Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$ . | 13 | 0 | 8,192 | -1 | 8,192 | |
In an equilateral triangle \(ABC\), a point \(P\) is chosen such that \(AP = 10\), \(BP = 8\), and \(CP = 6\). Find the area of this triangle. | 36 + 25\sqrt{3} | 0.5 | 7,367.1875 | 6,542.375 | 8,192 | |
An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as a center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle. | 2 | 0.4375 | 6,949.5 | 5,352 | 8,192 | |
At 7:00 AM, Xiaoming leaves his house and heads to school at a speed of 52 meters per minute. When he reaches the school, the hour and minute hands on his watch are positioned symmetrically around the number 7 on the clock. It's known that Xiaoming walked for less than an hour. How far is the distance between Xiaoming'... | 1680 | 0.125 | 7,125.875 | 5,027.5 | 7,425.642857 | |
Given the vertices of a rectangle are $A(0,0)$, $B(2,0)$, $C(2,1)$, and $D(0,1)$. A particle starts from the midpoint $P_{0}$ of $AB$ and moves in a direction forming an angle $\theta$ with $AB$, reaching a point $P_{1}$ on $BC$. The particle then sequentially reflects to points $P_{2}$ on $CD$, $P_{3}$ on $DA$, and $P... | \frac{1}{2} | 0.125 | 7,966.25 | 7,796.5 | 7,990.5 | |
Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be sequences of integers such that $a_1 = 20$ , $b_1 = 15$ , and for $n \ge 1$ , \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2, b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$ . Determine the number of positive int... | 525825 | 0.25 | 6,896.3125 | 4,989.5 | 7,531.916667 | |
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? | \frac{63}{64} | To solve this problem, we will use complementary counting to determine the probability that the product of the numbers rolled on the six dice is not divisible by $4$. We then subtract this probability from $1$ to find the probability that the product is divisible by $4$.
#### Step 1: Calculate the probability that the... | 0 | 5,852.9375 | -1 | 5,852.9375 |
Find the greatest common divisor of $8!$ and $(6!)^2.$ | 7200 | 0 | 3,376 | -1 | 3,376 | |
Determine the value of $x^2 + y^2$ if $x - y = 25$ and $xy = 36$. Additionally, find what $x+y$ equals. | \sqrt{769} | 0.5625 | 3,586.5625 | 3,349.666667 | 3,891.142857 | |
A calculator has a squaring key $\boxed{x^2}$ which replaces the current number displayed with its square. For example, if the display is $\boxed{000003}$ and the $\boxed{x^2}$ key is depressed, then the display becomes $\boxed{000009}$. If the display reads $\boxed{000002}$, how many times must you depress the $\box... | 4 | We start with the number $2$ displayed on the calculator. Each time we press the $\boxed{x^2}$ key, the displayed number is replaced by its square. We need to determine how many times we must press the key until the displayed number exceeds $500$.
1. **Initial Display**: $2$
2. **First Press**: $2^2 = 4$
3. **Second P... | 0.9375 | 4,042.625 | 3,766 | 8,192 |
$12 \cos ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}+\cos ^{4} \frac{5 \pi}{8}+\cos ^{4} \frac{7 \pi}{8}=$ | \frac{3}{2} | 0.0625 | 7,606.9375 | 6,359 | 7,690.133333 | |
Let \( z \) be a complex number that satisfies
\[ |z - 2| + |z - 7i| = 10. \]
Find the minimum value of \( |z| \). | 1.4 | 0 | 8,192 | -1 | 8,192 | |
If $\odot$ and $\nabla$ represent different positive integers less than 20, and $\odot \times \odot \times \odot = \nabla$, what is the value of $\nabla \times \nabla$? | 64 | If $\odot = 2$, then $\nabla = \odot \times \odot \times \odot = 2 \times 2 \times 2 = 8$, which is possible. Thus, $\odot = 2$ and so $\nabla = 8$. This means that $\nabla \times \nabla = 8 \times 8 = 64$. | 0.875 | 1,011.0625 | 1,032.071429 | 864 |
Given an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ whose right focus is $F$, and a circle with center $F$: $x^2 + y^2 - 4x - 32 = 0$ that is tangent to the right directrix of the ellipse, find the eccentricity of the ellipse. | \frac{1}{2} | 0.4375 | 6,074.875 | 3,914.285714 | 7,755.333333 | |
Which of the following expressions is not equivalent to $3x + 6$? | \frac{1}{3}(3x) + \frac{2}{3}(9) | We look at each of the five choices: (A) $3(x + 2) = 3x + 6$ (B) $\frac{-9x - 18}{-3} = \frac{-9x}{-3} + \frac{-18}{-3} = 3x + 6$ (C) $\frac{1}{3}(3x) + \frac{2}{3}(9) = x + 6$ (D) $\frac{1}{3}(9x + 18) = 3x + 6$ (E) $3x - 2(-3) = 3x + (-2)(-3) = 3x + 6$ The expression that is not equivalent to $3x + 6$ is the expressi... | 0 | 5,929.75 | -1 | 5,929.75 |
From a barrel, 4 liters of wine are drawn, and this is replaced with 4 liters of water. From the resulting mixture, 4 liters are drawn again and replaced with 4 liters of water. This operation is repeated a total of three times, and the final result is that there are 2.5 liters more water than wine. How many liters of ... | 16 | 0.625 | 6,274.3125 | 5,123.7 | 8,192 | |
Find the largest integer $n$ satisfying the following conditions:
(i) $n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $2n + 79$ is a perfect square. | 181 | Let us generate numbers $1$ to $1000$ for the second condition, for squares. We know for $N$ to be integer, the squares must be odd. So we generate $N = 1, 21, 45, 73, 105, 141, 181, 381, 441, 721, 801$. $N$ cannot exceed $1000$ since it is AIME problem. Now take the first criterion, let $a$ be the smaller consecutive ... | 0.125 | 7,991.6875 | 6,589.5 | 8,192 |
Evaluate $27^{-\frac{1}{3}} + 32^{-\frac{2}{5}}$. Express your answer as a common fraction. | \frac{7}{12} | 1 | 1,599.5625 | 1,599.5625 | -1 | |
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.1$ cm, $8.2$ cm and $9.7$ cm. What is the area of the square in square centimeters? | 36 | 1 | 1,337.0625 | 1,337.0625 | -1 | |
In 1860, someone deposited 100,000 florins at 5% interest with the goal of building and maintaining an orphanage for 100 orphans from the accumulated amount. When can the orphanage be built and opened if the construction and furnishing costs are 100,000 florins, the yearly personnel cost is 3,960 florins, and the maint... | 1896 | 0 | 8,192 | -1 | 8,192 | |
The sequence $1, 4, 5, 16, 17, 20, 21 \cdots$ consists of all those positive integers which are powers of 4 or sums of distinct powers of 4. Find the $150^{\mbox{th}}$ term of this sequence. | 16660 | 0.6875 | 6,313.5625 | 5,509.090909 | 8,083.4 | |
A right triangular prism \( ABC-A_{1}B_{1}C_{1} \) has 9 edges of equal length. Point \( P \) is the midpoint of \( CC_{1} \). The dihedral angle \( B-A_{1}P-B_{1} \) is \( \alpha \). What is \( \sin \alpha \)? | \frac{\sqrt{10}}{4} | 0 | 7,530.1875 | -1 | 7,530.1875 | |
Point $P_{}$ is located inside triangle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$ | 463 | Let $\angle{PAB} = \angle{PBC} = \angle{PCA} = x.$ Then, using Law of Cosines on the three triangles containing vertex $P,$ we have \begin{align*} b^2 &= a^2 + 169 - 26a \cos x \\ c^2 &= b^2 + 196 - 28b \cos x \\ a^2 &= c^2 + 225 - 30c \cos x. \end{align*} Add the three equations up and rearrange to obtain \[(13a + 14b... | 0.75 | 5,844.375 | 5,466.583333 | 6,977.75 |
A Saxon silver penny, from the reign of Ethelbert II in the eighth century, was sold in 2014 for £78000. A design on the coin depicts a circle surrounded by four equal arcs, each a quarter of a circle. The width of the design is 2 cm. What is the radius of the small circle, in centimetres?
A) \(\frac{1}{2}\)
B) \(2 -... | 2 - \sqrt{2} | 0 | 7,292.625 | -1 | 7,292.625 | |
The cells of a $5 \times 5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through 9 cells on his path, including the start and end cells. Compute the number of colorings for wh... | 1680 | Let $c_{i, j}$ denote the cell in the $i$-th row from the bottom and the $j$-th column from the left, so Sam starts at $c_{1,1}$ and is traveling to $c_{5,5}$. The key observation (from, say, trying small cases) is that Claim. For $1 \leq i, j<5$, the cells $c_{i+1, j}$ and $c_{i, j+1}$ must be the same color. Proof. C... | 0.0625 | 8,132.5 | 7,400 | 8,181.333333 |
An integer $x$ is chosen so that $3x+1$ is an even integer. Which of the following must be an odd integer? (A) $x+3$ (B) $x-3$ (C) $2x$ (D) $7x+4$ (E) $5x+3$ | 7x+4 | Solution 1: If $x=1$, then $3x+1=4$, which is an even integer. In this case, the five given choices are (A) $x+3=4$, (B) $x-3=-2$, (C) $2x=2$, (D) $7x+4=11$, (E) $5x+3=8$. Of these, the only odd integer is (D). Therefore, since $x=1$ satisfies the initial criteria, then (D) must be the correct answer as the result must... | 0.0625 | 1,119.8125 | 798 | 1,141.266667 |
A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
| 438 | 0.1875 | 8,097.5 | 7,688 | 8,192 | |
In quadrilateral $ABCD,\ BC=8,\ CD=12,\ AD=10,$ and $m\angle A= m\angle B = 60^\circ.$ Given that $AB = p + \sqrt{q},$ where $p$ and $q$ are positive integers, find $p+q.$
| 150 | 0.0625 | 7,659 | 4,239 | 7,887 | |
Given the complex number z = $$\frac{a^2i}{2-i} + \frac{1-2ai}{5}$$ (where a ∈ R, i is the imaginary unit), find the value(s) of a if z is a purely imaginary number. | -1 | 0.5 | 5,445.25 | 4,754.375 | 6,136.125 | |
What is the largest four-digit negative integer congruent to $1 \pmod{17}$? | -1002 | 0.3125 | 6,347.75 | 3,667.6 | 7,566 |
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