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 |
|---|---|---|---|---|---|---|
Circle $C$ with radius 2 has diameter $\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$, and can be written in the form $... | 254 | We use the notation of Solution 1 for triangle $\triangle DEC$ \[\sin \angle EDC = \frac {EF}{DE} = \frac {1}{4} \implies \cos \angle EDC = \frac {\sqrt{15}}{4}.\] We use Cosine Law for $\triangle DEC$ and get: \[(4r)^2 +(2 – 3r)^2 – 2 \cdot 4r \cdot (2 – 3r) \cdot \frac {\sqrt{15}}{4} = (2 – r)^2\]. \[(24 + 6 \sqrt{15... | 0.25 | 7,938.3125 | 7,177.25 | 8,192 |
Problem
Solve in integers the equation \[x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.\]
Solution
We first notice that both sides must be integers, so $\frac{x+y}{3}$ must be an integer.
We can therefore perform the substitution $x+y = 3t$ where $t$ is an integer.
Then:
$(3t)^2 - xy = (t+1)^3$
$9t^2 + x (x - 3t) = t^3... | \[
\left( \frac{1}{2} \left(3p(p-1) \pm \sqrt{4(p(p-1)+1)^3 - 27(p(p-1))^2} \right), 3p(p-1) - \frac{1}{2} \left(3p(p-1) \pm \sqrt{4(p(p-1)+1)^3 - 27(p(p-1))^2} \right) \right)
\] | Let $n = \frac{x+y}{3}$ .
Thus, $x+y = 3n$ .
We have \[x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3 \implies (x+y)^2 - xy = \left(\frac{x+y}{3}+1\right)^3\] Substituting $n$ for $\frac{x+y}{3}$ , we have \[9n^2 - x(3n-x) = (n+1)^3\] Treating $x$ as a variable and $n$ as a constant, we have \[9n^2 - 3nx + x^2 = (n+1)^3,\... | 0 | 7,082.1875 | -1 | 7,082.1875 |
The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find, with proof, the maximum number of tiles that... | 12 |
To solve this problem, we need to maximize the number of \(2 \times 2\) tiles that can be placed on a \(8 \times 8\) chessboard, such that the sum of the numbers in each tile is less than 100. The numbers \(1, 2, \ldots, 64\) are written on the chessboard, with each square containing a unique number.
### Step 1: Unde... | 0 | 8,094.25 | -1 | 8,094.25 |
Find the sum of $555_6$, $55_6$ and $5_6$ in base $6$. | 1103_6 | 0.5 | 6,195.5 | 4,556.75 | 7,834.25 | |
How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 0.5 | 1,631.9375 | 2,083.375 | 1,180.5 | |
Cara is sitting at a circular table with her seven friends. Two of her friends, Alice and Bob, insist on sitting together but not next to Cara. How many different possible pairs of people could Cara be sitting between? | 10 | 0 | 6,554.75 | -1 | 6,554.75 | |
A stock investment went up $25\%$ in 2006. Starting at this increased value, what percent would it have to go down in 2007 to be back to its original price at the beginning of 2006? | 20 | 1 | 2,178.9375 | 2,178.9375 | -1 | |
In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5? | 401802 | 0.75 | 6,196.125 | 5,530.833333 | 8,192 | |
Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it to Alice, he keeps it. Alice starts with th... | \frac{9}{20} | 0.8125 | 4,553.625 | 3,714 | 8,192 | |
Suppose that $(a_1, b_1), (a_2, b_2), \ldots , (a_{100}, b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i, j)$ satisfying $1 \le i < j \le 100$ and $|a_ib_j - a_j b_i|=1$ . Determine the largest possible value of $N$ over all possible choices of the $100$ ... | \[\boxed{N=197}\] | Let's start off with just $(a_1, b_1), (a_2, b_2)$ and suppose that it satisfies the given condition. We could use $(1, 1), (1, 2)$ for example. We should maximize the number of conditions that the third pair satisfies. We find out that the third pair should equal $(a_1+a_2, b_1+b_2)$ :
We know this must be true: \[|a_... | 0 | 8,147.5625 | -1 | 8,147.5625 |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $c\cos B + b\cos C = 2a\cos A$, $M$ is the midpoint of $BC$, and $AM=1$, find the maximum value of $b+c$. | \frac{4\sqrt{3}}{3} | 0 | 5,516.1875 | -1 | 5,516.1875 | |
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 3 adjacent place numbers sum to at least 1 and no four consecutive place numbers are all zeroes. Find the number of elements in $\mathcal{T}_{12}$. | 1705 | 0.0625 | 8,093.125 | 6,610 | 8,192 | |
The sequence $\left\{a_{n}\right\}$ is defined such that $a_{n}$ is the last digit of the sum $1 + 2 + \cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Calculate $S_{2016}$. | 7066 | 0.5625 | 6,536.625 | 5,620.555556 | 7,714.428571 | |
There are two docks, $A$ and $B$, on a river, where dock $A$ is upstream and dock $B$ is downstream. There are two boats, Boat 1 and Boat 2. The speed of Boat 1 in still water is twice the speed of Boat 2. Both boats start simultaneously from docks $A$ and $B$, respectively, and move towards each other. When Boat 1 de... | 40 | 0.0625 | 8,073.75 | 6,580 | 8,173.333333 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $c=2$, $2\sin A= \sqrt {3}a\cos C$.
(1) Find the measure of angle $C$;
(2) If $2\sin 2A+ \sin (2B+C)= \sin C$, find the area of $\triangle ABC$. | \dfrac {2 \sqrt {3}}{3} | 0 | 6,359.125 | -1 | 6,359.125 | |
A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time). Furthermore, after exactly 60 seconds, ... | 61 | 0.5 | 7,575.875 | 7,008.625 | 8,143.125 | |
Given that $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are unit vectors with an angle of $\frac {2π}{3}$ between them, and $\overrightarrow {a}$ = 3 $\overrightarrow {e_{1}}$ + 2 $\overrightarrow {e_{2}}$, $\overrightarrow {b}$ = 3 $\overrightarrow {e_{2}}$, find the projection of $\overrightarrow {a}$ onto... | \frac {1}{2} | 0.25 | 4,768.9375 | 3,178 | 5,299.25 | |
Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board? | 13 | 0.875 | 5,677.5 | 5,318.285714 | 8,192 | |
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by \[ q(x) = \sum_{k=1}^{p-1} a_k x^k, \] where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative... | \frac{p-1}{2} | The answer is $\frac{p-1}{2}$. Define the operator $D = x \frac{d}{dx}$, where $\frac{d}{dx}$ indicates formal differentiation of polynomials. For $n$ as in the problem statement, we have $q(x) = (x-1)^n r(x)$ for some polynomial $r(x)$ in $\mathbb{F}_p$ not divisible by $x-1$. For $m=0,\dots,n$, by the product rule we... | 0 | 8,192 | -1 | 8,192 |
Five dice with faces numbered 1 through 6 are stacked in a similar manner to the original problem. Ten of the thirty faces are visible, leaving twenty faces hidden. The visible numbers are 1, 2, 2, 3, 3, 3, 4, 4, 5, and 6. What is the total number of dots NOT visible in this view? | 72 | 0.9375 | 2,560.6875 | 2,606.266667 | 1,877 | |
If only one quarter of the earth's surface is not covered by bodies of water, and only one half of that exposed land area is inhabitable for humans (because of poles, deserts, etc.), what fraction of the earth's surface can humans live on? | \frac{1}{8} | 0.625 | 1,083.0625 | 1,081.2 | 1,086.166667 | |
A certain college student had the night of February 23 to work on a chemistry problem set and a math problem set (both due on February 24, 2006). If the student worked on his problem sets in the math library, the probability of him finishing his math problem set that night is 95% and the probability of him finishing hi... | 95/159 | 0.5 | 6,001.75 | 5,825.25 | 6,178.25 | |
There are 18 identical cars in a train. In some cars, exactly half of the seats are free, in others, exactly one-third of the seats are free, and in the remaining cars, all seats are occupied. At the same time, exactly one-ninth of all seats in the whole train are free. How many cars have all seats occupied? | 13 | 0.125 | 7,725.6875 | 4,461.5 | 8,192 | |
On December 8, 2022, the Joint Prevention and Control Mechanism of the State Council held a press conference to introduce further optimization of the implementation of epidemic prevention and control measures. It was emphasized that individuals are responsible for their own health. Xiao Hua prepared some medicines. The... | \frac{5}{6} | 0.875 | 3,032.125 | 2,866.714286 | 4,190 | |
What is the greatest integer less than 100 for which the greatest common divisor of that integer and 12 is 4? | 92 | 1 | 3,918.625 | 3,918.625 | -1 | |
A "Kiwi" business owner has two types of "Kiwi" for promotion, type \\(A\\) and type \\(B\\). If you buy 2 pieces of type \\(A\\) "Kiwi" and 1 piece of type \\(B\\) "Kiwi", the total cost is 120 yuan; if you buy 3 pieces of type \\(A\\) "Kiwi" and 2 pieces of type \\(B\\) "Kiwi", the total cost is 205 yuan.
\\((1)\\) ... | 1125 | 0.8125 | 3,695.0625 | 3,480 | 4,627 | |
A town's reservoir has a usable water volume of 120 million cubic meters. Assuming the annual precipitation remains unchanged, it can sustain a water supply for 160,000 people for 20 years. After urbanization and the migration of 40,000 new residents, the reservoir will only be able to maintain the water supply for the... | 50 | 0.0625 | 7,329.3125 | 6,002 | 7,417.8 | |
Circles $P$, $Q$, and $R$ are externally tangent to each other and internally tangent to circle $S$. Circles $Q$ and $R$ are congruent. Circle $P$ has radius 2 and passes through the center of $S$. What is the radius of circle $Q$? | \frac{16}{9} | 0.8125 | 5,213.6875 | 4,526.384615 | 8,192 | |
Given that $α \in (0, \frac{π}{2})$, and $\sin (\frac{π}{6} - α) = -\frac{1}{3}$, find the value of $\cos α$. | \frac{2\sqrt{6} - 1}{6} | 0 | 4,678.4375 | -1 | 4,678.4375 | |
In $\triangle ABC$, if $a= \sqrt {5}$, $b= \sqrt {15}$, $A=30^{\circ}$, then $c=$ \_\_\_\_\_\_. | 2 \sqrt {5} | 0 | 7,249.4375 | -1 | 7,249.4375 | |
For each positive integer $n$, an associated non-negative integer $f(n)$ is defined to satisfy the following three rules:
i) $f(a b)=f(a)+f(b)$.
ii) $f(n)=0$ if $n$ is a prime greater than 10.
iii) $f(1)<f(243)<f(2)<11$.
Given that $f(2106)<11$, determine the value of $f(96)$. | 31 | 0.25 | 7,383.375 | 4,957.5 | 8,192 | |
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? | 8 | 1. **Understanding the Problem**: Mary divides a circle into 12 sectors with central angles that are integers and form an arithmetic sequence. The sum of these angles must be $360^\circ$ since they complete a circle.
2. **Setting Up the Equations**: Let $a_1$ be the first term and $d$ be the common difference of the a... | 0.8125 | 5,775.25 | 5,217.538462 | 8,192 |
Find two lines of symmetry of the graph of the function $y=x+\frac{1}{x}$. Express your answer as two equations of the form $y=a x+b$. | $y=(1+\sqrt{2}) x$ and $y=(1-\sqrt{2}) x$ | The graph of the function $y=x+\frac{1}{x}$ is a hyperbola. We can see this more clearly by writing it out in the standard form $x^{2}-x y+1=0$ or $\left(\frac{y}{2}\right)^{2}-\left(x-\frac{1}{2} y\right)^{2}=1$. The hyperbola has asymptotes given by $x=0$ and $y=x$, so the lines of symmetry will be the (interior and ... | 0 | 8,192 | -1 | 8,192 |
The function $f$ is not defined for $x = 0,$ but for all non-zero real numbers $x,$
\[f(x) + 2f \left( \frac{1}{x} \right) = 3x.\]Find the real solutions to $f(x) = f(-x).$ Enter the real solutions, separated by commas. | \sqrt{2},-\sqrt{2} | 0.4375 | 3,021.125 | 3,223.571429 | 2,863.666667 | |
Find the distance between the foci of the hyperbola
\[\frac{y^2}{18} - \frac{x^2}{2} = 1.\] | 4 \sqrt{5} | 1 | 1,796.5 | 1,796.5 | -1 | |
Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$. Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The area of $\triangle ABC$ is $3$. What is the slope of $l_3$? | \tfrac34 | 1 | 3,736.0625 | 3,736.0625 | -1 | |
Given points $A(-2,0)$ and $P(1, \frac{3}{2})$ on the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$, and two lines with slopes $k$ and $-k (k>0)$ passing through point $P$ intersect ellipse $M$ at points $B$ and $C$.
(I) Find the equation of ellipse $M$ and its eccentricity.
(II) If quadrilateral $PAB... | \frac{3}{2} | 0.1875 | 8,118.8125 | 7,801.666667 | 8,192 | |
A cube with side length 2 has vertices $Q_1, Q_2, Q_3, Q_4, Q_1', Q_2', Q_3',$ and $Q_4'$. Vertices $Q_2$, $Q_3$, and $Q_4$ are adjacent to $Q_1$, and for $1\le i\le 4,$ vertices $Q_i$ and $Q_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $\overline{Q_1Q_2}$, $\overline{Q_1Q... | \frac{4\sqrt{2}}{3} | 0 | 8,157.5625 | -1 | 8,157.5625 | |
A floor decoration is a circle with eight rays pointing from the center. The rays form eight congruent central angles. One of the rays points due north. What is the measure in degrees of the smaller angle formed between the ray pointing East and the ray pointing Southwest?
[asy]
size(3cm,3cm);
draw(unitcircle);
... | 135 | 0.9375 | 4,396.125 | 4,143.066667 | 8,192 | |
We need to arrange the performance order for 4 singing programs and 2 skit programs. The requirement is that there must be exactly 3 singing programs between the 2 skit programs. The number of possible arrangements is \_\_\_\_\_\_ . (Answer with a number) | 96 | 0.75 | 5,766.8125 | 5,506.5 | 6,547.75 | |
Given that the side lengths of triangle \( \triangle ABC \) are 6, \( x \), and \( 2x \), find the maximum value of its area \( S \). | 12 | 0.8125 | 6,306.125 | 5,870.923077 | 8,192 | |
Given that point P is on the curve \( y = -x^{2} - 1 \) and point Q is on the curve \( x = 1 + y^{2} \), find the minimum value of \( |PQ| \). | \frac{3\sqrt{2}}{4} | 0 | 7,632.3125 | -1 | 7,632.3125 | |
Find the maximum value of \( x + y \), given that \( x^2 + y^2 - 3y - 1 = 0 \). | \frac{\sqrt{26}+3}{2} | 0 | 4,460.0625 | -1 | 4,460.0625 | |
A natural number \( N \) greater than 20 is a palindrome in both base 14 and base 20 (a palindrome is a number that reads the same forward and backward, such as \( 12321 \) and \( 3443 \), but \( 12331 \) is not a palindrome). What is the smallest value of \( N \) (expressed in base 10)? | 105 | 0 | 8,141.0625 | -1 | 8,141.0625 | |
Consider a string of $n$ $8$'s, $8888\cdots88$, into which $+$ signs are inserted to produce an arithmetic expression. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $8000$? | 1000 | 0 | 8,192 | -1 | 8,192 | |
The absolute value of -1.2 is ____, and its reciprocal is ____. | -\frac{5}{6} | 0.25 | 362.5 | 366.25 | 361.25 | |
Given triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $9\sin ^{2}B=4\sin ^{2}A$ and $\cos C=\frac{1}{4}$, calculate $\frac{c}{a}$. | \frac{\sqrt{10}}{3} | 0 | 4,622.125 | -1 | 4,622.125 | |
Given that $\sin 2α - 2 = 2 \cos 2α$, find the value of ${\sin}^{2}α + \sin 2α$. | \frac{8}{5} | 0.5 | 7,215.625 | 6,239.25 | 8,192 | |
In right triangle $JKL$, angle $J$ measures 60 degrees and angle $K$ measures 30 degrees. When drawn, the angle bisectors of angles $J$ and $K$ intersect at a point $M$. What is the measure of obtuse angle $JMK$?
[asy]
import geometry;
import olympiad;
unitsize(0.8inch);
dotfactor = 3;
defaultpen(linewidth(1pt)+fontsi... | 135 | 0.3125 | 7,607.0625 | 6,320.2 | 8,192 | |
Kelvin the frog currently sits at $(0,0)$ in the coordinate plane. If Kelvin is at $(x, y)$, either he can walk to any of $(x, y+1),(x+1, y)$, or $(x+1, y+1)$, or he can jump to any of $(x, y+2),(x+2, y)$ or $(x+1, y+1)$. Walking and jumping from $(x, y)$ to $(x+1, y+1)$ are considered distinct actions. Compute the num... | 1831830 | Observe there are $\binom{14}{6}=3003$ up-right paths from $(0,0)$ to $(6,8)$, each of which are 14 steps long. Any two of these steps can be combined into one: $UU, RR$, and $RU$ as jumps, and $UR$ as walking from $(x, y)$ to $(x+1, y+1)$. The number of ways to combine steps is the number of ways to group 14 actions i... | 0 | 8,192 | -1 | 8,192 |
The altitudes of an acute-angled triangle \( ABC \) drawn from vertices \( B \) and \( C \) are 7 and 9, respectively, and the median \( AM \) is 8. Points \( P \) and \( Q \) are symmetric to point \( M \) with respect to sides \( AC \) and \( AB \), respectively. Find the perimeter of the quadrilateral \( APMQ \). | 32 | 0.0625 | 8,192 | 8,192 | 8,192 | |
What is the value of $\dfrac{\sqrt[5]{11}}{\sqrt[7]{11}}$ expressed as 11 raised to what power? | \frac{2}{35} | 0.8125 | 2,275.875 | 1,923 | 3,805 | |
Given 6 persons, with the restriction that person A and person B cannot visit Paris, calculate the total number of distinct selection plans for selecting 4 persons to visit Paris, London, Sydney, and Moscow, where each person visits only one city. | 240 | 0.125 | 7,413 | 4,513 | 7,827.285714 | |
Dima took the fractional-linear function \(\frac{a x + 2b}{c x + 2d}\), where \(a, b, c, d\) are positive numbers, and summed it with the remaining 23 functions obtained from it by permuting the numbers \(a, b, c, d\). Find the root of the sum of all these functions, independent of the numbers \(a, b, c, d\). | -1 | 0.0625 | 7,865 | 8,192 | 7,843.2 | |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0), (2010,0), (2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? | \frac{335}{2011} | 0.6875 | 6,248.75 | 5,810 | 7,214 | |
If $a>0$ and $b>0,$ a new operation $\nabla$ is defined as follows: $$a \nabla b = \dfrac{a + b}{1 + ab}.$$For example, $$3 \nabla 6 = \frac{3 + 6}{1 + 3 \times 6} = \frac{9}{19}.$$Calculate $2 \nabla 5.$ | \frac{7}{11} | 0.9375 | 1,829.0625 | 1,404.866667 | 8,192 | |
If $f(n)$ denotes the number of divisors of $2024^{2024}$ that are either less than $n$ or share at least one prime factor with $n$ , find the remainder when $$ \sum^{2024^{2024}}_{n=1} f(n) $$ is divided by $1000$ . | 224 | 0 | 8,192 | -1 | 8,192 | |
Bill buys a stock that decreases by $20\%$ on the first day, and then on the second day the stock increases by $30\%$ of its value at the end of the first day. What was the overall percent increase in Bill's stock over the two days? | 4 | 1 | 1,649.3125 | 1,649.3125 | -1 | |
In a relay race from Moscow to Petushki, two teams of 20 people each participated. Each team divided the distance into 20 segments (not necessarily equal) and assigned them among the participants so that each person ran exactly one segment (each participant's speed is constant, but the speeds of different participants ... | 38 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Determine the minimum possible value of the sum
\[\frac{a}{3b} + \frac{b}{5c} + \frac{c}{7a},\]
where $a,$ $b,$ and $c$ are positive real numbers. | \frac{3}{\sqrt[3]{105}} | 0 | 7,568.375 | -1 | 7,568.375 | |
What is the largest quotient that can be formed using two numbers chosen from the set $\{ -24, -3, -2, 1, 2, 8 \}$? | 12 | To find the largest quotient formed using two numbers from the set $\{-24, -3, -2, 1, 2, 8\}$, we need to consider the absolute values of the numbers and the signs to maximize the quotient $\frac{a}{b}$.
1. **Maximizing the Quotient**:
- The quotient $\frac{a}{b}$ is maximized when $a$ is maximized and $b$ is minim... | 0.6875 | 5,682.375 | 4,808.818182 | 7,604.2 |
Two trains, each composed of 15 identical cars, were moving towards each other at constant speeds. Exactly 28 seconds after their first cars met, a passenger named Sasha, sitting in the compartment of the third car, passed by a passenger named Valera in the opposite train. Moreover, 32 seconds later, the last cars of t... | 12 | 0 | 8,120 | -1 | 8,120 | |
In triangle $XYZ$, the sides are in the ratio $3:4:5$. If segment $XM$ bisects the largest angle at $X$ and divides side $YZ$ into two segments, find the length of the shorter segment given that the length of side $YZ$ is $12$ inches. | \frac{9}{2} | 0 | 5,657.625 | -1 | 5,657.625 | |
Two circles of radius \( r \) are externally tangent to each other and internally tangent to the ellipse \( x^2 + 4y^2 = 8 \). Find \( r \). | \frac{\sqrt{6}}{2} | 0 | 6,421.5 | -1 | 6,421.5 | |
In rectangle $PQRS$, $PQ = 150$. Let $T$ be the midpoint of $\overline{PS}$. Given that line $PT$ and line $QT$ are perpendicular, find the greatest integer less than $PS$. | 212 | 0 | 8,192 | -1 | 8,192 | |
The numbers from 1 to 200, inclusive, are placed in a bag. A number is randomly selected from the bag. What is the probability that it is neither a perfect square, a perfect cube, nor a multiple of 7? Express your answer as a common fraction. | \frac{39}{50} | 0.0625 | 6,005.625 | 8,192 | 5,859.866667 | |
48 blacksmiths need to shoe 60 horses. Each blacksmith takes 5 minutes to make one horseshoe. What is the minimum time they should spend on the job? (Note: A horse cannot stand on two legs.) | 25 | 0 | 7,434.0625 | -1 | 7,434.0625 | |
The expression $\frac{x^2-3x+2}{x^2-5x+6} \div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is: | 1 | 1. **Factorize each quadratic expression** in the given complex fraction:
- Numerator of the first fraction: $x^2 - 3x + 2 = (x-2)(x-1)$.
- Denominator of the first fraction: $x^2 - 5x + 6 = (x-3)(x-2)$.
- Numerator of the second fraction: $x^2 - 5x + 4 = (x-4)(x-1)$.
- Denominator of the second fraction: $... | 0.875 | 3,340.3125 | 2,647.214286 | 8,192 |
Find $(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x^{8}+1\right) \cdots$, where $|x|<1$. | \frac{1}{1-x} | Let $S=(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x^{8}+1\right) \cdots=1+x+x^{2}+x^{3}+\cdots$. Since $x S=x+x^{2}+x^{3}+x^{4}+\cdots$, we have $(1-x) S=1$, so $S=\frac{1}{1-x}$. | 1 | 4,301.375 | 4,301.375 | -1 |
From the numbers 1, 2, 3, 5, 7, 8, two numbers are randomly selected and added together. Among the different sums that can be obtained, let the number of sums that are multiples of 2 be $a$, and the number of sums that are multiples of 3 be $b$. Then, the median of the sample 6, $a$, $b$, 9 is ____. | 5.5 | 0.3125 | 6,179.625 | 6,436.6 | 6,062.818182 | |
On a computer keyboard, the key for the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will actually print.
Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers satisfy this condition? | 28 | 0.25 | 6,799.3125 | 4,984.75 | 7,404.166667 | |
An equilateral triangle $PQR$ is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ so that $Q$ is at $(0,b),$ and $\overline{PR}$ is parallel to the $x$-axis, as shown below. Also, foci $F_1$ and $F_2$ lie on sides $\overline{QR}$ and $\overline{PQ},$ respectively. Find $\frac{PQ}{F_1 F_2}.$
[asy]
un... | \frac{8}{5} | 0.375 | 7,253.625 | 5,689.666667 | 8,192 | |
Given a quadratic function $y=ax^{2}-4ax+3+b\left(a\neq 0\right)$.
$(1)$ Find the axis of symmetry of the graph of the quadratic function;
$(2)$ If the graph of the quadratic function passes through the point $\left(1,3\right)$, and the integers $a$ and $b$ satisfy $4 \lt a+|b| \lt 9$, find the expression of the qu... | t = \frac{5}{2} | 0.1875 | 8,134.6875 | 7,939 | 8,179.846154 | |
At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent were children although there were 12 adult... | 520 | 0.75 | 3,862 | 2,787.166667 | 7,086.5 | |
The absolute value of a number \( x \) is equal to the distance from 0 to \( x \) along a number line and is written as \( |x| \). For example, \( |8|=8, |-3|=3 \), and \( |0|=0 \). For how many pairs \( (a, b) \) of integers is \( |a|+|b| \leq 10 \)? | 221 | 0.3125 | 7,633.25 | 6,404 | 8,192 | |
A sequence of positive integers $a_{1}, a_{2}, \ldots, a_{2017}$ has the property that for all integers $m$ where $1 \leq m \leq 2017,3\left(\sum_{i=1}^{m} a_{i}\right)^{2}=\sum_{i=1}^{m} a_{i}^{3}$. Compute $a_{1337}$. | \[ a_{1337} = 4011 \] | I claim that $a_{i}=3 i$ for all $i$. We can conjecture that the sequence should just be the positive multiples of three because the natural numbers satisfy the property that the square of their sum is the sum of their cubes, and prove this by induction. At $i=1$, we have that $3 a_{i}^{2}=a_{i}^{3}$, so $a_{i}=3$. Now... | 0 | 5,925.875 | -1 | 5,925.875 |
Given that the vector $\overrightarrow {a} = (3\cos\alpha, 2)$ is parallel to the vector $\overrightarrow {b} = (3, 4\sin\alpha)$, find the value of the acute angle $\alpha$. | \frac{\pi}{4} | 0.9375 | 1,906.8125 | 1,945.066667 | 1,333 | |
If $|x| + x + y = 10$ and $x + |y| - y = 12,$ find $x + y.$ | \frac{18}{5} | 1 | 3,498.5625 | 3,498.5625 | -1 | |
2019 students are voting on the distribution of \(N\) items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possible nu... | 1009 | To get an item, a student must receive at least 2 votes on that item. Since each student receives at most 2019 votes, the number of items one student can receive does not exceed \(\frac{2019}{2}=1009.5\). So, the answer is at most 1009. This occurs when \(N=2018\) and item \(i\) was voted to student \(1,1,2,3, \ldots, ... | 0 | 8,192 | -1 | 8,192 |
Let
\[g(x) = \left\{
\begin{array}{cl}
x + 5 & \text{if $x < 15$}, \\
3x - 6 & \text{if $x \ge 15$}.
\end{array}
\right.\]
Find $g^{-1}(10) + g^{-1}(57).$ | 26 | 1 | 2,058.125 | 2,058.125 | -1 | |
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$. | 48 |
We are tasked with finding the smallest positive number \(\lambda\) such that for any 12 points on the plane \(P_1, P_2, \ldots, P_{12}\) (which can overlap), if the distance between any two of them does not exceed 1, then \(\sum_{1 \le i < j \le 12} |P_iP_j|^2 \le \lambda\).
Let \(O\) be an arbitrary point, and let ... | 0 | 8,192 | -1 | 8,192 |
The villages Arkadino, Borisovo, and Vadimovo are connected by straight roads in pairs. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which completely coincides with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangular field, one side of which completely coi... | 135 | 0 | 8,192 | -1 | 8,192 | |
My co-worker Erich is very odd. He only likes numbers that are divisible by 5. How many different last digits are possible in numbers that Erich likes? | 2 | 1 | 1,522 | 1,522 | -1 | |
Compute \[ \left\lfloor \dfrac {1007^3}{1005 \cdot 1006} - \dfrac {1005^3}{1006 \cdot 1007} + 5 \right\rfloor,\] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x.$ | 12 | 0 | 7,764 | -1 | 7,764 | |
Given \(1990 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}}\), where \(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}\) are distinct non-negative integers. Find \(\alpha_{1} + \alpha_{2} + \cdots + \alpha_{n}\). | 43 | 0.8125 | 4,432.625 | 3,565.076923 | 8,192 | |
Count the total number of possible scenarios in a table tennis match between two players, where the winner is the first one to win three games and they play until a winner is determined. | 20 | 0.8125 | 5,016 | 4,283.076923 | 8,192 | |
If the graph of the power function $y=f(x)$ passes through the point $(9, \frac{1}{3})$, find the value of $f(25)$. | \frac{1}{5} | 0.5 | 6,079.0625 | 4,046.125 | 8,112 | |
The ellipse $5x^2 - ky^2 = 5$ has one of its foci at $(0, 2)$. Find the value of $k$. | -1 | 0.6875 | 3,887 | 3,460.818182 | 4,824.6 | |
Twenty gremlins and fifteen imps are at the Annual Mischief Convention. The imps have had a lot of in-fighting lately and refuse to shake hands with each other, but they readily shake hands with all of the gremlins. Meanwhile, all the gremlins are quite friendly and shake hands with all of the other gremlins as well as... | 490 | 0.9375 | 4,557.3125 | 4,315 | 8,192 | |
Masha looked at the drawing and said: "There are seven rectangles here: one big one and six small ones." "There are also various middle-sized rectangles here," said her mother. How many rectangles are there in total in this drawing? Explain your answer. | 18 | 0.125 | 1,857.6875 | 2,323.5 | 1,791.142857 | |
Choose two different numbers from the set of numbers {1, 2, ..., 8, 9}, and find the probability that their product is an odd number. (Express the result as a numerical value). | \frac{5}{18} | 1 | 1,313.1875 | 1,313.1875 | -1 | |
Samantha has 10 different colored marbles in her bag. In how many ways can she choose five different marbles such that at least one of them is red? | 126 | 1 | 2,933.8125 | 2,933.8125 | -1 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a > b$, $a=5$, $c=6$, and $\sin B= \frac{3}{5}$.
(Ⅰ) Find the values of $b$ and $\sin A$;
(Ⅱ) Find the value of $\sin \left(2A+ \frac{\pi}{4}\right)$. | \frac{7\sqrt{2}}{26} | 0 | 4,356.8125 | -1 | 4,356.8125 | |
A right circular cylinder is inscribed in a right circular cone. The cone has a diameter of 14 and an altitude of 20, and the axes of the cylinder and cone coincide. The height of the cylinder is three times its radius. Find the radius of the cylinder. | \frac{140}{41} | 0.8125 | 4,470.8125 | 3,612.076923 | 8,192 | |
If $x+\frac{1}{x}=6$, then what is the value of $x^{2}+\frac{1}{x^{2}}$? | 34 | 1 | 1,748.125 | 1,748.125 | -1 | |
A metallic weight has a mass of 25 kg and is an alloy of four metals. The first metal in this alloy is one and a half times more than the second; the mass of the second metal is related to the mass of the third as \(3: 4\), and the mass of the third metal to the mass of the fourth as \(5: 6\). Determine the mass of the... | 7.36 | 0.25 | 6,651.0625 | 4,303.5 | 7,433.583333 | |
Let \( S = \{1, 2, \cdots, 10\} \). If a subset \( T \) of \( S \) has at least 2 elements and the absolute difference between any two elements in \( T \) is greater than 1, then \( T \) is said to have property \( P \). Find the number of different subsets of \( S \) that have property \( P \). | 133 | 0.8125 | 5,432.3125 | 4,795.461538 | 8,192 | |
If $\left(2x-a\right)^{7}=a_{0}+a_{1}(x+1)+a_{2}(x+1)^{2}+a_{3}(x+1)^{3}+\ldots +a_{7}(x+1)^{7}$, and $a_{4}=-560$.<br/>$(1)$ Find the value of the real number $a$;<br/>$(2)$ Find the value of $|a_{1}|+|a_{2}|+|a_{3}|+\ldots +|a_{6}|+|a_{7}|$. | 2186 | 0.3125 | 6,834.6875 | 5,046.4 | 7,647.545455 | |
To set up for a Fourth of July party, David is making a string of red, white, and blue balloons. He places them according to the following rules: - No red balloon is adjacent to another red balloon. - White balloons appear in groups of exactly two, and groups of white balloons are separated by at least two non-white ba... | 99 | It is possible to achieve 99 red balloons with the arrangement $$\text { WWBBBWW } \underbrace{\text { RBBBWWRBBBWW ...RBBBWW, }}_{99 \text { RBBBWW's }}$$ which contains $99 \cdot 6+7=601$ balloons. Now assume that one can construct a chain with 98 or fewer red balloons. Then there can be 99 blocks of non-red balloons... | 0 | 8,192 | -1 | 8,192 |
A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 10 | 0.8125 | 3,357.25 | 3,558.076923 | 2,487 |
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