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 |
|---|---|---|---|---|---|---|
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $... | 13 | Let $a$ be the number of $2$-point shots, $b$ be the number of $3$-point shots, and $x$ be the number of free throws. We need to find $x$.
Given:
1. The points from two-point shots equal the points from three-point shots, so $2a = 3b$.
2. The number of free throws is one more than the number of two-point shots, so $x ... | 1 | 1,878.9375 | 1,878.9375 | -1 |
Find $\lfloor |-4.2| \rfloor + |\lfloor -4.2 \rfloor|$. | 9 | 1 | 1,379.4375 | 1,379.4375 | -1 | |
Point \(C\) divides diameter \(AB\) in the ratio \(AC:BC = 2:1\). A point \(P\) is selected on the circle. Determine the possible values that the ratio \(\tan \angle PAC: \tan \angle APC\) can take. Specify the smallest such value. | 1/2 | 0.375 | 7,939.3125 | 7,518.166667 | 8,192 | |
If $x_{k+1} = x_k + \frac{1}{2}$ for $k=1, 2, \dots, n-1$ and $x_1=1$, find $x_1 + x_2 + \dots + x_n$. | \frac{n^2+3n}{4} | 1. **Identify the sequence type**: Given the recurrence relation $x_{k+1} = x_k + \frac12$, we observe that each term increases by $\frac12$ from the previous term. This indicates that the sequence is an arithmetic sequence.
2. **Determine the first term and common difference**: The first term $x_1$ is given as 1. The... | 0.0625 | 3,154.75 | 2,632 | 3,189.6 |
Let $r$, $s$, and $t$ be solutions of the equation $x^3-5x^2+6x=9$.
Compute $\frac{rs}t + \frac{st}r + \frac{tr}s$. | -6 | 0.5625 | 6,676.625 | 5,498 | 8,192 | |
Given points $A=(4,10)$ and $B=(10,8)$ lie on circle $\omega$ in the plane, and the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis, find the area of $\omega$. | \frac{100\pi}{9} | 0.0625 | 6,191.9375 | 4,527 | 6,302.933333 | |
The kindergarten received flashcards for reading: some say "MA", and the others say "NYA". Each child took three cards to form words. It turned out that 20 children can form the word "MAMA" from their cards, 30 children can form the word "NYANYA", and 40 children can form the word "MANYA". How many children have all th... | 10 | 0.0625 | 8,021.625 | 8,191 | 8,010.333333 | |
Suppose $a, b, c, d$, and $e$ are objects that we can multiply together, but the multiplication doesn't necessarily satisfy the associative law, i.e. ( $x y) z$ does not necessarily equal $x(y z)$. How many different ways are there to interpret the product abcde? | 14 | $C($ number of letters -1$)=C(4)=14$. | 0.9375 | 3,671.25 | 3,369.866667 | 8,192 |
Two companies, A and B, each donated 60,000 yuan to the "See a Just Cause Foundation". It is known that the average donation per person in company B is 40 yuan more than that in company A, and the number of people in company A is 20% more than that in company B. What are the respective number of people in companies A a... | 250 | 0.8125 | 2,430.3125 | 2,116 | 3,792.333333 | |
Let $ABCD$ be a square with each side of 4 units. Construct equilateral triangles $ABE$, $BCF$, $CDG$, and $DAH$ inscribed on each side of the square, inside the square. Let $E, F, G, H$ be the centers, respectively, of these equilateral triangles. What is the ratio of the area of square $EFGH$ to the area of square $A... | \frac{2}{3} | 0 | 8,192 | -1 | 8,192 | |
In Phoenix, AZ, the temperature was given by the quadratic equation $-t^2 + 14t + 40$, where $t$ is the number of hours after noon. What is the largest $t$ value when the temperature was exactly 77 degrees? | 11 | 0 | 3,042.5 | -1 | 3,042.5 | |
Nine copies of a certain pamphlet cost less than $10.00 while ten copies of the same pamphlet (at the same price) cost more than $11.00. How much does one copy of this pamphlet cost? | $1.11 | 1. **Define the variable**: Let $p$ be the cost of one pamphlet in dollars.
2. **Translate the conditions into inequalities**:
- From the first condition, nine copies of the pamphlet cost less than $10.00. This can be expressed as:
\[
9p < 10
\]
- From the second condition, ten copies of the pamph... | 0 | 5,690.0625 | -1 | 5,690.0625 |
From the set $S={1,2,3,...,100}$, three numbers are randomly selected and arranged in ascending order. Find the probability that $50$ is exactly in the middle. | \frac{1}{66} | 0.9375 | 5,213.5 | 5,014.933333 | 8,192 | |
Three circles with radii 2, 3, and 3 touch each other. What is the area of the triangle formed by joining the centers of these circles? | 12 | 0.9375 | 3,381.1875 | 3,060.466667 | 8,192 | |
What are the last two digits in the sum of the factorials of the first 15 positive integers? | 13 | 1 | 4,191.1875 | 4,191.1875 | -1 | |
Let $f(x)=16x+3$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f((2x)^{-1})$. | 51 | 1 | 3,151.9375 | 3,151.9375 | -1 | |
For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$.
[i]Romania[/i] | 7 |
To find the greatest common divisor (GCD) of the sequence \( A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2} \) for each nonnegative integer \( n \), we begin by examining the expression for \( A_n \):
\[
A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}.
\]
Our goal is to determine \( \gcd(A_0, A_1, \ldots, A_{1999}) \).
### Step 1: Check Di... | 0.6875 | 6,726.375 | 6,060.181818 | 8,192 |
<u>Set 4</u> **p10.** Eve has nine letter tiles: three $C$ ’s, three $M$ ’s, and three $W$ ’s. If she arranges them in a random order, what is the probability that the string “ $CMWMC$ ” appears somewhere in the arrangement?**p11.** Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$ 4 $ per p... | \frac{1}{28} | 0.0625 | 7,676.6875 | 8,192 | 7,642.333333 | |
In a survey, $82.5\%$ of respondents believed that mice were dangerous. Of these, $52.4\%$ incorrectly thought that mice commonly caused electrical fires. Given that these 27 respondents were mistaken, determine the total number of people surveyed. | 63 | 0.1875 | 8,080.0625 | 7,595 | 8,192 | |
Using the digits $0$, $1$, $2$, $3$, $4$, $5$, how many different five-digit even numbers greater than $20000$ can be formed without repetition? | 240 | 0.1875 | 7,369.0625 | 6,254.333333 | 7,626.307692 | |
When $\frac{1}{909}$ is expressed as a decimal, what is the sum of the first 30 digits after the decimal point? | 14 | 0.125 | 7,080.6875 | 5,018 | 7,375.357143 | |
In the Cartesian coordinate system $xOy$, it is known that the line $l_1$ is defined by the parametric equations $\begin{cases}x=t\cos \alpha\\y=t\sin \alpha\end{cases}$ (where $t$ is the parameter), and the line $l_2$ by $\begin{cases}x=t\cos(\alpha + \frac{\pi}{4})\\y=t\sin(\alpha + \frac{\pi}{4})\end{cases}$ (where ... | 2\sqrt{2} | 0.5625 | 7,077 | 6,209.777778 | 8,192 | |
Given the function $y=\cos (2x-\frac{\pi }{4})$, determine the horizontal translation of the graph of the function $y=\sin 2x$. | \frac{\pi }{8} | 0.75 | 5,420.1875 | 4,821.75 | 7,215.5 | |
The square of a three-digit number ends with three identical digits different from zero. Write the smallest such three-digit number. | 462 | 0 | 8,192 | -1 | 8,192 | |
If the remainders of 2017, 1029, and 725 divided by $\mathrm{d}$ are all $\mathrm{r}$, what is the maximum value of $\mathrm{d} - \mathrm{r}$? | 35 | 0.9375 | 3,647.9375 | 3,355.133333 | 8,040 | |
A stationery store sells a certain type of pen bag for $18$ yuan each. Xiao Hua went to buy this pen bag. When checking out, the clerk said, "If you buy one more, you can get a 10% discount, which is $36 cheaper than now." Xiao Hua said, "Then I'll buy one more, thank you." According to the conversation between the two... | 486 | 0.375 | 1,378.1875 | 1,449.666667 | 1,335.3 | |
Convert $813_9$ to base 3. | 220110_3 | 0.9375 | 5,041.5 | 4,831.466667 | 8,192 | |
Given a regular 14-gon, where each vertex is connected to every other vertex by a segment, three distinct segments are chosen at random. What is the probability that the lengths of these three segments refer to a triangle with positive area?
A) $\frac{73}{91}$
B) $\frac{74}{91}$
C) $\frac{75}{91}$
D) $\frac{76}{91}$
E)... | \frac{77}{91} | 0 | 8,192 | -1 | 8,192 | |
The graph of the function $f(x)=3 \sqrt {2}\cos (x+\varphi)+\sin x$, where $x\in \mathbb{R}$ and $\varphi\in\left(- \frac {\pi}{2}, \frac {\pi}{2}\right)$, passes through the point $\left( \frac {\pi}{2},4\right)$. Find the minimum value of $f(x)$. | -5 | 0.75 | 5,553.75 | 4,937.166667 | 7,403.5 | |
In a chorus performance, there are 6 female singers (including 1 lead singer) and 2 male singers arranged in two rows.
(1) If there are 4 people per row, how many different arrangements are possible?
(2) If the lead singer stands in the front row and the male singers stand in the back row, with again 4 people per row, ... | 5760 | 0.125 | 7,179.875 | 5,401.5 | 7,433.928571 | |
Compute the sum of the roots of the equation \[x\sqrt{x} - 6x + 7\sqrt{x} - 1 = 0,\]given that all of the roots are real and nonnegative. | 22 | 0.75 | 5,722 | 4,898.666667 | 8,192 | |
Solve the following problems:<br/>$(1)$ Given: $2^{m}=32$, $3^{n}=81$, find the value of $5^{m-n}$;<br/>$(2)$ Given: $3x+2y+1=3$, find the value of $27^{x}\cdot 9^{y}\cdot 3$. | 27 | 0.875 | 2,088.3125 | 2,133.285714 | 1,773.5 | |
In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are on sides $YZ$, $XZ$, and $XY$ respectively. Given that $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$, and that $\frac{XP}{PX'} + \frac{YP}{PY'} + \frac{ZP}{PZ'} = 100$, find $\frac{XP}{PX'} \cdot \frac{YP}{PY'} \cdot \frac{ZP}{PZ'}$. | 98 | 0 | 7,098 | -1 | 7,098 | |
Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$. We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed ... | (2n-1)!! |
Consider an integer \( n > 0 \) and a balance with \( n \) weights of weights \( 2^0, 2^1, \ldots, 2^{n-1} \). Our task is to place each of these weights on the balance, one by one, so that the right pan is never heavier than the left pan. We aim to determine the number of ways to achieve this.
### Understanding the ... | 0 | 8,192 | -1 | 8,192 |
When the expression $3(x^2 - 3x + 3) - 8(x^3 - 2x^2 + 4x - 1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms? | 2395 | 1 | 2,696 | 2,696 | -1 | |
Two of the roots of
\[ax^3 + (a + 2b) x^2 + (b - 3a) x + (8 - a) = 0\]are $-2$ and 3. Find the third root. | \frac{4}{3} | 0.75 | 5,688.125 | 4,853.5 | 8,192 | |
The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1)... | 2 | We analyze each of the four statements to determine which transformations will map the figure onto itself.
1. **Some rotation around a point on line $\ell$:**
- Consider a rotation of $180^\circ$ around a point that is exactly halfway between an up-facing square and a down-facing square on line $\ell$. This point i... | 0 | 7,625.125 | -1 | 7,625.125 |
The same amount of steel used to create six solid steel balls, each with a radius of 2 inches, is used to create one larger steel ball. What is the radius of the larger ball? | 4\sqrt[3]{2} | 0 | 2,887.5625 | -1 | 2,887.5625 | |
The average of four different positive whole numbers is $4.$ If the difference between the largest and smallest of these numbers is as large as possible, what is the average of the other two numbers? | 2\frac{1}{2} | 0 | 4,280.3125 | -1 | 4,280.3125 | |
Let $\Gamma$ be the region formed by the points $(x, y)$ that satisfy
$$
\left\{\begin{array}{l}
x \geqslant 0, \\
y \geqslant 0, \\
x+y+[x]+[y] \leqslant 5
\end{array}\right.
$$
where $[x]$ represents the greatest integer less than or equal to the real number $x$. Find the area of the region $\Gamma$. | 9/2 | 0.4375 | 7,577.6875 | 7,372.285714 | 7,737.444444 | |
How many different four-digit numbers, divisible by 4, can be made from the digits 1, 2, 3, and 4,
a) if each digit can be used only once?
b) if each digit can be used multiple times? | 64 | 0.5625 | 5,160.5625 | 5,277 | 5,010.857143 | |
Given a harmonic progression with the first three terms 3, 4, 6, find the value of $S_4$. | 25 | 0.8125 | 3,273.75 | 3,399.230769 | 2,730 | |
An author writes a book containing 60,000 words in 100 hours. For the first 20% of the time, the author writes with 50% increased productivity. How many words per hour did the author write on average? | 600 | 0.375 | 2,164.125 | 4,030 | 1,044.6 | |
Seven points are marked on the circumference of a circle. How many different chords can be drawn by connecting two of these seven points? | 21 | 1 | 1,529.9375 | 1,529.9375 | -1 | |
Two circles of radius 3 are centered at $(0,3)$ and at $(3,0)$. What is the area of the intersection of the interiors of these two circles? Express your answer in terms of $\pi$ in its simplest form. | \frac{9\pi - 18}{2} | 0 | 5,046.0625 | -1 | 5,046.0625 | |
For some positive integers $a$ and $b$, the product \[\log_a(a+1) \cdot \log_{a+1} (a+2) \dotsm \log_{b-2} (b-1) \cdot\log_{b-1} b\]contains exactly $1000$ terms, and its value is $3.$ Compute $a+b.$ | 1010 | 0 | 8,192 | -1 | 8,192 | |
The rapid growth of the Wuhan Economic and Technological Development Zone's economy cannot be separated from the support of the industrial industry. In 2022, the total industrial output value of the entire area reached 3462.23 billion yuan, ranking first in the city. Express the number 3462.23 in scientific notation as... | 3.46223 \times 10^{3} | 0.5625 | 363.1875 | 347.888889 | 382.857143 | |
A rectangular prism measures 10-inches by 20-inches by 10-inches. What is the length, in inches, of the diagonal connecting point A and point B? Express your answer in simplest radical form. [asy]
unitsize(0.75cm);
defaultpen(linewidth(0.7pt)+fontsize(10pt));
dotfactor=4;
draw((0,1)--(1,1)--(1,0)--(0,0)--(0,1)--(1,2)-... | 10\sqrt{6} | 1 | 1,800.875 | 1,800.875 | -1 | |
Given $\tan (\theta-\pi)=2$, then $\sin ^{2}\theta+\sin \theta\cos \theta-2\cos ^{2}\theta=$ \_\_\_\_\_\_ . | \frac {4}{5} | 0.9375 | 4,679.125 | 4,444.933333 | 8,192 | |
How many ways can you mark 8 squares of an $8 \times 8$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.) | 21600 | In the top row, you can mark any of the 6 squares that is not a corner. In the bottom row, you can then mark any of the 5 squares that is not a corner and not in the same column as the square just marked. Then, in the second row, you have 6 choices for a square not in the same column as either of the two squares alread... | 0.0625 | 7,652.75 | 7,411 | 7,668.866667 |
If a triangle has two sides of lengths 5 and 7 units, then how many different integer lengths can the third side be? | 9 | 1 | 1,760.3125 | 1,760.3125 | -1 | |
Simplify and find the value: $4(a^{2}b+ab^{2})-3(a^{2}b-1)+2ab^{2}-6$, where $a=1$, $b=-4$. | 89 | 0.875 | 1,431.3125 | 1,538.285714 | 682.5 | |
Find the maximum and minimum values of the function $f(x)=2\cos^2x+3\sin x$ on the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. $\boxed{\text{Fill in the blank}}$ | -3 | 0.9375 | 4,481.875 | 4,234.533333 | 8,192 | |
Triangle $ABC$ has vertices with coordinates $A(2,3),$ $B(7,8),$ and $C(-4,6)$. The triangle is reflected about line $L$. The image points are $A'(2,-5),$ $B'(7,-10),$ and $C'(-4,-8)$. What is the equation of line $L$? | y = -1 | 0.9375 | 2,204.4375 | 1,805.266667 | 8,192 | |
Given that the sum of the binomial coefficients in the expansion of {(5x-1/√x)^n} is 64, determine the constant term in its expansion. | 375 | 0.4375 | 2,365.5625 | 2,642.285714 | 2,150.333333 | |
Two balls, one blue and one orange, are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin $k$ is $3^{-k}$ for $k = 1, 2, 3,...$. What is the probability that the blue ball is tossed into a higher-numbered bin than the orange ball?
A) $... | \frac{7}{16} | 0 | 8,072.5 | -1 | 8,072.5 | |
The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P,$ and its radius is $21$. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.
| 345 | 0.8125 | 5,938 | 5,417.846154 | 8,192 | |
If $\sin\theta + \cos\theta = \frac{2\sqrt{2}-1}{3}$ ($0 < \theta < \pi$), then $\tan\theta = \_\_\_\_\_\_$. | -2\sqrt{2} | 0.8125 | 5,697.25 | 5,432.461538 | 6,844.666667 | |
For a natural number \( N \), if at least seven out of the nine natural numbers from 1 to 9 are factors of \( N \), \( N \) is called a "seven-star number." What is the smallest "seven-star number" greater than 2000? | 2016 | 0.0625 | 8,178.625 | 7,978 | 8,192 | |
An investor placed \$12,000 in a one-year fixed deposit that yielded a simple annual interest rate of 8%. After one year, the total amount was reinvested in another one-year deposit. At the end of the second year, the total amount was \$13,500. If the annual interest rate of the second deposit is \( s\% \), what is \( ... | 4.17\% | 0 | 4,596.75 | -1 | 4,596.75 | |
The equations of the asymptotes of a hyperbola are $y = 2x+5$ and $y = -2x+1.$ Given that the hyperbola passes through the point $(0, 7),$ the standard form for the equation of the hyperbola is \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1,\]where $a,$ $b$, $h,$ and $k$ are constants with $a, b > 0.$ Find $a + h.$ | 2\sqrt{3}-1 | 1 | 3,453.625 | 3,453.625 | -1 | |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, that is, $\lfloor x \rfloor$ is the largest integer not exceeding $x$. Calculate the value of $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \lfloor \log_{2}4 \rfloor + \ldots + \lfloor \log... | 8204 | 0.4375 | 6,331.6875 | 5,216 | 7,199.444444 | |
The cube below has sides of length 5 feet. If a cylindrical section of radius 1 foot is removed from the solid at an angle of $45^\circ$ to the top face, what is the total remaining volume of the cube? Express your answer in cubic feet in terms of $\pi$. | 125 - 5\sqrt{2}\pi | 0 | 7,673.9375 | -1 | 7,673.9375 | |
In a triangle $ABC$ , the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$ . Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$ . Given that $BE=3,BA=4$ , find the integer nearest to $BC^2$ . | 29 | 0.125 | 7,750.6875 | 6,260 | 7,963.642857 | |
Calculate the definite integral:
$$
\int_{2 \operatorname{arctg} \frac{1}{3}}^{2 \operatorname{arctg} \frac{1}{2}} \frac{d x}{\sin x(1-\sin x)}
$$ | \ln 3 - \ln 2 + 1 | 0 | 6,753.5 | -1 | 6,753.5 | |
For how many pairs of consecutive integers in $\{1000,1001,1002,\ldots,2000\}$ is no carrying required when the two integers are added?
| 156 | 0 | 8,192 | -1 | 8,192 | |
Three of the four vertices of a rectangle are \((1, 7)\), \((14, 7)\), and \((14, -4)\). What is the area of the intersection of this rectangular region and the region inside the graph of the equation \((x - 1)^2 + (y + 4)^2 = 16\)? | 4\pi | 0.6875 | 6,886.4375 | 6,293 | 8,192 | |
Dragoons take up \(1 \times 1\) squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the \(x\)-coordinates of their centers and the difference in the \(y\)-coordinates of their centers are both... | 168 | 0.0625 | 7,968.25 | 8,192 | 7,953.333333 | |
A factory produces a certain product with an annual fixed cost of $250 million. When producing $x$ million units, an additional cost of $C(x)$ million is required. When the annual production volume is less than 80 million units, $C(x)= \frac {1}{3}x^{2}+10x$; when the annual production volume is not less than 80 millio... | 1000 | 0.875 | 4,754.3125 | 4,456.285714 | 6,840.5 | |
In the final stage of a professional bowling tournament, the competition between the top five players is conducted as follows: the fifth and fourth place players compete first, the loser gets the 5th place prize; the winner competes with the third place player, the loser gets the 4th place prize; the winner competes wi... | 16 | 0.1875 | 7,835.9375 | 6,293 | 8,192 | |
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what... | \frac{1}{21,000} | To solve this problem, we need to calculate the total number of possible license plates and then determine the probability of the specific license plate "AMC8".
1. **Calculate the total number of possible license plates:**
- **First symbol (vowel):** There are 5 vowels (A, E, I, O, U). Thus, there are 5 choices for... | 0 | 2,404.3125 | -1 | 2,404.3125 |
Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\angle POQ$ and $\angle POR$ are $a$ and $b$ respectively, and $\angle... | 41 | Noting that $\angle OQP$ and $\angle ORP$ are right angles, we realize that we can draw a semicircle with diameter $\overline{OP}$ and points $Q$ and $R$ on the semicircle. Since the radius of the semicircle is $100$, if $\overline{QR} \leq 100$, then $\overarc{QR}$ must be less than or equal to $60^{\circ}$.
This sim... | 0.25 | 7,494.3125 | 6,651.75 | 7,775.166667 |
Suppose you have an equilateral triangle divided into 9 smaller equilateral triangles with the bottom side horizontal. Starting from the top corner labeled \( A \), you must walk to the bottom right corner labeled \( B \), and are only allowed to take steps along the edges down to the left, down to the right, or horizo... | 22 | 0 | 8,192 | -1 | 8,192 | |
A factory produced an original calculator that performs two operations:
(a) the usual addition, denoted by \( + \)
(b) an operation denoted by \( \circledast \).
We know that, for any natural number \( a \), the following hold:
\[
(i) \quad a \circledast a = a \quad \text{ and } \quad (ii) \quad a \circledast 0 =... | 2000 | 0.9375 | 2,960.5625 | 2,771.8 | 5,792 | |
If \[f(x) =
\begin{cases}
x^2-4 &\quad \text{if } x \ge -4, \\
x + 3 &\quad \text{otherwise},
\end{cases}
\]then for how many values of $x$ is $f(f(x)) = 5$? | 5 | 0.8125 | 4,322.0625 | 3,695.692308 | 7,036.333333 | |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\] | \frac{131}{11} | 0 | 8,192 | -1 | 8,192 | |
Given $\cos \left( \frac {π}{4}-α \right) = \frac {3}{5}$, and $\sin \left( \frac {5π}{4}+β \right) = - \frac {12}{13}$, with $α \in \left( \frac {π}{4}, \frac {3π}{4} \right)$ and $β \in (0, \frac {π}{4})$, find the value of $\sin (α+β)$. | \frac {56}{65} | 0.5625 | 6,667.1875 | 6,247.666667 | 7,206.571429 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively, and $C= \frac{3}{4}\pi$, $\sin A= \frac{\sqrt{5}}{5}$.
(I) Find the value of $\sin B$;
(II) If $c-a=5-\sqrt{10}$, find the area of $\triangle ABC$. | \frac{5}{2} | 0.875 | 4,762.625 | 4,409.5 | 7,234.5 | |
What is the side length of the larger square if a small square is drawn inside a larger square, and the area of the shaded region and the area of the unshaded region are each $18 \mathrm{~cm}^{2}$? | 6 \mathrm{~cm} | Since the area of the larger square equals the sum of the areas of the shaded and unshaded regions inside, then the area of the larger square equals $2 \times 18 \mathrm{~cm}^{2}=36 \mathrm{~cm}^{2}$.
Since the larger square has an area of $36 \mathrm{~cm}^{2}$, then its side length is $\sqrt{36 \mathrm{~cm}^{2}}=6 \... | 0 | 3,118 | -1 | 3,118 |
650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti? | \frac{5}{2} | 1. **Identify the number of students who preferred each pasta type:** According to the problem, the number of students who preferred spaghetti is 250 and the number of students who preferred manicotti is 100.
2. **Set up the ratio:** The ratio of the number of students who preferred spaghetti to the number of students... | 0 | 6,597.0625 | -1 | 6,597.0625 |
Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression
\[ \frac{a^m+3^m}{a^2-3a+1} \]
does not take any integer values for $(a,m) \in A \times \mathbb{Z^+}.$ | 11 | 0 | 8,161.6875 | -1 | 8,161.6875 | |
The volume of a regular octagonal prism is $8 \, \mathrm{m}^{3}$, and its height is $2.2 \, \mathrm{m}$. Find the lateral surface area of the prism. | 16 \sqrt{2.2 \cdot (\sqrt{2}-1)} | 0 | 7,767.8125 | -1 | 7,767.8125 | |
For how many positive integers \( n \) is the sum
\[
(-n)^{3} + (-n+1)^{3} + \cdots + (n-2)^{3} + (n-1)^{3} + n^{3} + (n+1)^{3}
\]
less than \( 3129 \)? | 13 | 0.75 | 6,441.1875 | 5,857.583333 | 8,192 | |
Suppose \(a\), \(b\), and \(c\) are three positive numbers that satisfy \(abc = 1\), \(a + \frac{1}{c} = 7\), and \(b + \frac{1}{a} = 34\). Find \(c + \frac{1}{b}\). | \frac{43}{237} | 0.8125 | 6,392.125 | 5,976.769231 | 8,192 | |
Let $\alpha$ be an acute angle. If $\sin\left(\alpha - \frac{\pi}{4}\right) = \frac{1}{3}$, then $\cos2\alpha = \_\_\_\_\_\_$. | -\frac{4\sqrt{2}}{9} | 0 | 4,745.5625 | -1 | 4,745.5625 | |
Let \(ABCDEF\) be a regular hexagon and let point \(O\) be the center of the hexagon. How many ways can you color these seven points either red or blue such that there doesn't exist any equilateral triangle with vertices of all the same color? | 6 | Without loss of generality, let \(O\) be blue. Then we can't have any two adjacent blues on the perimeter of \(ABCDEF\). However, because of the two larger equilateral triangles \(ACE\) and \(BDF\), we need at least two blues to keep us from having an all red equilateral triangle. We can't have three blues on the perim... | 0 | 7,755.5 | -1 | 7,755.5 |
Given a sequence of positive terms $\{a\_n\}$, where $a\_2=6$, and $\frac{1}{a\_1+1}$, $\frac{1}{a\_2+2}$, $\frac{1}{a\_3+3}$ form an arithmetic sequence, find the minimum value of $a\_1a\_3$. | 19+8\sqrt{3} | 0.625 | 7,317.25 | 6,792.4 | 8,192 | |
Find the maximum possible number of diagonals of equal length in a convex hexagon. | 7 | First, we will prove that 7 is possible. Consider the following hexagon \(A B C D E F\) whose vertices are located at \(A(0,0), B\left(\frac{1}{2}, 1-\frac{\sqrt{3}}{2}\right), C\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right), D(0,1), E\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right), F\left(-\frac{1}{2}, 1-\frac{\sqrt{3}}{2... | 0 | 7,960.875 | -1 | 7,960.875 |
A convex quadrilateral is drawn in the coordinate plane such that each of its vertices $(x, y)$ satisfies the equations $x^{2}+y^{2}=73$ and $x y=24$. What is the area of this quadrilateral? | 110 | The vertices all satisfy $(x+y)^{2}=x^{2}+y^{2}+2 x y=73+2 \cdot 24=121$, so $x+y= \pm 11$. Similarly, $(x-y)^{2}=x^{2}+y^{2}-2 x y=73-2 \cdot 24=25$, so $x-y= \pm 5$. Thus, there are four solutions: $(x, y)=(8,3),(3,8),(-3,-8),(-8,-3)$. All four of these solutions satisfy the original equations. The quadrilateral is t... | 0.0625 | 7,936.3125 | 8,192 | 7,919.266667 |
Solve for $x$: $x+2x = 400-(3x+4x)$. | 40 | 1 | 1,552.1875 | 1,552.1875 | -1 | |
Given the function $f(x)=\cos ωx \cdot \sin (ωx- \frac {π}{3})+ \sqrt {3}\cos ^{2}ωx- \frac{ \sqrt {3}}{4}(ω > 0,x∈R)$, and the distance from one symmetry center of the graph of the function $y=f(x)$ to the nearest symmetry axis is $\frac {π}{4}$.
(I) Find the value of $ω$ and the equation of the symmetry axis of $f(x)... | \frac {3+2 \sqrt {6}}{3} | 0 | 8,087.1875 | -1 | 8,087.1875 | |
Compute $\tan 3825^\circ$. | 1 | 0.9375 | 3,210 | 2,877.866667 | 8,192 | |
Given that the volume of a regular triangular pyramid $P-ABC$ is $\frac{1}{12}$, the center of its circumscribed sphere is $O$, and it satisfies $\vec{OA} + \vec{OB} + \vec{OC} = \vec{0}$, find the radius of the circumscribed sphere of the regular triangular pyramid $P-ABC$. | \frac{\sqrt{3}}{3} | 0 | 7,258.8125 | -1 | 7,258.8125 | |
Boris was given a Connect Four game set for his birthday, but his color-blindness makes it hard to play the game. Still, he enjoys the shapes he can make by dropping checkers into the set. If the number of shapes possible modulo (horizontal) flips about the vertical axis of symmetry is expressed as $9(1+2+\cdots+n)$, f... | 729 | There are $9^{7}$ total shapes possible, since each of the 7 columns can contain anywhere from 0 to 8 checkers. The number of shapes symmetric with respect to a horizontal flip is the number of shapes of the leftmost four columns, since the configuration of these four columns uniquely determines the configuration of th... | 0 | 7,398.8125 | -1 | 7,398.8125 |
Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x \), \( y \), and \( z \) such that \( x \mid y^{3} \) and \( y \mid z^{3} \) and \( z \mid x^{3} \), it always holds that \( x y z \mid (x+y+z)^{n} \). | 13 | 0.0625 | 8,139.375 | 7,853 | 8,158.466667 | |
Hooligan Vasya likes to run on the escalator in the subway. He runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving downward, it takes him 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down th... | 324 | 0.125 | 7,869.125 | 7,812.5 | 7,877.214286 | |
Let $A B C$ be a triangle with $A B=5, B C=8$, and $C A=7$. Let $\Gamma$ be a circle internally tangent to the circumcircle of $A B C$ at $A$ which is also tangent to segment $B C. \Gamma$ intersects $A B$ and $A C$ at points $D$ and $E$, respectively. Determine the length of segment $D E$. | $\frac{40}{9}$ | First, note that a homothety $h$ centered at $A$ takes $\Gamma$ to the circumcircle of $A B C, D$ to $B$ and $E$ to $C$, since the two circles are tangent. As a result, we have $D E \| B C$. Now, let $P$ be the center of $\Gamma$ and $O$ be the circumcenter of $A B C$: by the homothety $h$, we have $D E / B C=A P / A O... | 0 | 8,066 | -1 | 8,066 |
Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
What is the value of \( f(90) \)? | 999 | 0.125 | 8,192 | 8,192 | 8,192 | |
Given that $a$, $b$, $c \in \mathbb{R}$, $a + b + c = 3$, $a \geqslant b \geqslant c$, the equation $ax^{2}+bx+c=0$ has real roots. Find the minimum value of $a$. | \frac{4}{3} | 0.5 | 7,070.1875 | 6,145.875 | 7,994.5 | |
For every integer $n\ge2$, let $\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides
$\prod_{n=2}^{5300}\text{pow}(n)$? | 77 | To solve this problem, we need to determine how many times the prime factor $67$ appears in the product $\prod_{n=2}^{5300}\text{pow}(n)$, where $\text{pow}(n)$ is defined as the largest power of the largest prime that divides $n$.
1. **Identify the largest prime factor of 2010**: The prime factorization of $2010$ is ... | 0 | 8,027.5625 | -1 | 8,027.5625 |
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