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 |
|---|---|---|---|---|---|---|
Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? | \frac{11}{60} | To solve this problem, we need to determine the total number of outcomes and the number of favorable outcomes where the product of the numbers on the tile and the die is a perfect square.
1. **Total Outcomes**:
- There are 10 tiles, each with a number from 1 to 10.
- A standard die has 6 faces, numbered from 1 ... | 0.25 | 6,695.625 | 5,045 | 7,245.833333 |
In the diagram, each of the three identical circles touch the other two. The circumference of each circle is 36. What is the perimeter of the shaded region? [asy]
defaultpen(1);
path p = (1, 0){down}..{-dir(30)}dir(-60){dir(30)}..{dir(-30)}((2, 0) + dir(-120)){-dir(-30)}..{up}(1, 0)--cycle;
fill(p, gray(0.75));
dr... | 18 | 0.9375 | 3,605.125 | 3,422.933333 | 6,338 | |
Jo adds up all the positive integers from 1 to 100. Kate does a similar thing with the first 100 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 100 values. What is the positive difference between Jo's sum and Kate's sum? | 50 | 0.3125 | 7,530.4375 | 6,075 | 8,192 | |
Given an arithmetic sequence $\left\{a_{n}\right\}$ with the first term $a_{1}>0$, and the following conditions:
$$
a_{2013} + a_{2014} > 0, \quad a_{2013} a_{2014} < 0,
$$
find the largest natural number $n$ for which the sum of the first $n$ terms, $S_{n}>0$, holds true. | 4026 | 0.125 | 7,922.875 | 6,510.5 | 8,124.642857 | |
How many positive even multiples of $3$ less than $2020$ are perfect squares? | 7 | To find the number of positive even multiples of $3$ less than $2020$ that are perfect squares, we start by considering the form of such numbers.
1. **Form of the number**: A number that is both an even multiple of $3$ and a perfect square can be written as $36k^2$, where $k$ is an integer. This is because the number... | 0.8125 | 4,762.25 | 3,970.769231 | 8,192 |
Seven distinct integers are picked at random from the set {1,2,3,...,12}. What is the probability that among those selected, the second smallest number is 4? | \frac{7}{33} | 0.375 | 5,674.625 | 4,417.166667 | 6,429.1 | |
A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be... | 81 | Let $k$ be the number of women selected. Then, the number of men not selected is $11-(k-1)=12-k$. Note that the sum of the number of women selected and the number of men not selected is constant at $12$. Each combination of women selected and men not selected corresponds to a committee selection. Since choosing 12 indi... | 0.3125 | 7,543 | 6,115.2 | 8,192 |
Let \( X = \{1, 2, \ldots, 2001\} \). Find the smallest positive integer \( m \) such that in any \( m \)-element subset \( W \) of \( X \), there exist \( u, v \in W \) (where \( u \) and \( v \) are allowed to be the same) such that \( u + v \) is a power of 2. | 1000 | 0 | 8,192 | -1 | 8,192 | |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$, respectively, and it is given that $a < b < c$ and $$\frac{a}{\sin A} = \frac{2b}{\sqrt{3}}$$.
(1) Find the size of angle $B$;
(2) If $a=2$ and $c=3$, find the length of side $b$ and the area of $\triangle ABC$. | \frac{3\sqrt{3}}{2} | 0 | 5,856 | -1 | 5,856 | |
Given the planar vectors $\overrightarrow{a}, \overrightarrow{b}$, with $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $\overrightarrow{a} \cdot \overrightarrow{b}=1$, let $\overrightarrow{e}$ be a unit vector in the plane. Find the maximum value of $y=\overrightarrow{a} \cdot \overrightarrow{e} + \overrightar... | \sqrt{7} | 0.875 | 3,520.125 | 2,852.714286 | 8,192 | |
Points \( M \) and \( N \) are the midpoints of the sides \( AC \) and \( CB \) of the isosceles triangle \( ACB \). Point \( L \) lies on the median \( BM \) such that \( BL : BM = 4 : 9 \). A circle with center at point \( L \) is tangent to the line \( MN \) and intersects the line \( AB \) at points \( Q \) and \( ... | 2(2 + \sqrt{13}) | 0 | 5,799.375 | -1 | 5,799.375 | |
Kevin has an elm tree in his yard that is $11\frac{2}{3}$ feet tall and an oak tree that is $17\frac{5}{6}$ feet tall. How much taller is the oak tree than the elm tree? Express your answer as a simplified mixed number. | 6\frac{1}{6}\text{ feet} | 1 | 1,505.8125 | 1,505.8125 | -1 | |
Call a set of positive integers good if there is a partition of it into two sets $S$ and $T$, such that there do not exist three elements $a, b, c \in S$ such that $a^{b}=c$ and such that there do not exist three elements $a, b, c \in T$ such that $a^{b}=c$ ( $a$ and $b$ need not be distinct). Find the smallest positiv... | 65536 | First, we claim that the set $\{2,4,8,256,65536\}$ is not good. Assume the contrary and say $2 \in S$. Then since $2^{2}=4$, we have $4 \in T$. And since $4^{4}=256$, we have $256 \in S$. Then since $256^{2}=65536$, we have $65536 \in T$. Now, note that we cannot place 8 in either $S$ or $T$, contradiction. Hence $n \l... | 0 | 8,192 | -1 | 8,192 |
In Papa Carlo's room, there is a clock on each wall, and they all show incorrect times: the first clock is off by 2 minutes, the second by 3 minutes, the third by 4 minutes, and the fourth by 5 minutes. One day, Papa Carlo decided to find out the exact time before leaving the house, and he saw the following times on th... | 14:58 | 0 | 8,119.625 | -1 | 8,119.625 | |
Altitudes $\overline{AX}$ and $\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\angle BAC = 61^\circ$ and $\angle ABC = 73^\circ$, then what is $\angle CHX$? | 73^\circ | 0.375 | 7,913.625 | 7,601.833333 | 8,100.7 | |
The quadratic $x^2 + 5x + c$ has roots in the form of $x = \frac{-5 \pm \sqrt{c}}{2}$. What is the value of $c$? | 5 | 1 | 1,559.0625 | 1,559.0625 | -1 | |
The largest three-digit number divided by an integer, with the quotient rounded to one decimal place being 2.5, will have the smallest divisor as: | 392 | 0.3125 | 2,566.9375 | 1,800.2 | 2,915.454545 | |
What is the least positive integer greater than 1 that leaves a remainder of 1 when divided by each of 2, 3, 4, 5, 6, 7, 8 and 9? | 2521 | 0.9375 | 3,403.8125 | 3,084.6 | 8,192 | |
Let $A B C$ be a triangle with incircle tangent to the perpendicular bisector of $B C$. If $B C=A E=$ 20, where $E$ is the point where the $A$-excircle touches $B C$, then compute the area of $\triangle A B C$. | 100 \sqrt{2} | Let the incircle and $B C$ touch at $D$, the incircle and perpendicular bisector touch at $X, Y$ be the point opposite $D$ on the incircle, and $M$ be the midpoint of $B C$. Recall that $A, Y$, and $E$ are collinear by homothety at $A$. Additionally, we have $M D=M X=M E$ so $\angle D X Y=\angle D X E=90^{\circ}$. Ther... | 0 | 7,768.4375 | -1 | 7,768.4375 |
What is the distance between the two (non-intersecting) face diagonals on adjacent faces of a unit cube? | \frac{\sqrt{3}}{3} | 0 | 6,055.625 | -1 | 6,055.625 | |
We have an $n$-gon, and each of its vertices is labeled with a number from the set $\{1, \ldots, 10\}$. We know that for any pair of distinct numbers from this set there is at least one side of the polygon whose endpoints have these two numbers. Find the smallest possible value of $n$. | 50 | Each number be paired with each of the 9 other numbers, but each vertex can be used in at most 2 different pairs, so each number must occur on at least $\lceil 9 / 2\rceil=5$ different vertices. Thus, we need at least $10 \cdot 5=50$ vertices, so $n \geq 50$. To see that $n=50$ is feasible, let the numbers $1, \ldots, ... | 0 | 8,006.625 | -1 | 8,006.625 |
Given the function \(f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}\), where \(k\) is a positive integer, if for any real number \(a\), it holds that \(\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbb{R}\}\), find the minimum value of \(k\). | 16 | 0.5 | 7,291.25 | 6,390.5 | 8,192 | |
Given an ellipse $C$: $\frac{{x}^{2}}{3}+{y}^{2}=1$ with left focus and right focus as $F_{1}$ and $F_{2}$ respectively. The line $y=x+m$ intersects $C$ at points $A$ and $B$. If the area of $\triangle F_{1}AB$ is twice the area of $\triangle F_{2}AB$, find the value of $m$. | -\frac{\sqrt{2}}{3} | 0 | 6,666.3125 | -1 | 6,666.3125 | |
Let \( f(n) \) be the integer closest to \( \sqrt[4]{n} \). Then, \( \sum_{k=1}^{2018} \frac{1}{f(k)} = \) ______. | \frac{2823}{7} | 0.25 | 8,133.4375 | 8,122.75 | 8,137 | |
The diagonal lengths of a rhombus are 18 units and 26 units. Calculate both the area and the perimeter of the rhombus. | 20\sqrt{10} | 0.8125 | 1,494.8125 | 1,664.538462 | 759.333333 | |
A bowl contains 10 jellybeans (four red, one blue and five white). If you pick three jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red? Express your answer as a common fraction. | \frac{3}{10} | 1 | 2,380 | 2,380 | -1 | |
Lisa, a child with strange requirements for her projects, is making a rectangular cardboard box with square bases. She wants the height of the box to be 3 units greater than the side of the square bases. What should the height be if she wants the surface area of the box to be at least 90 square units while using the le... | 6 | 1 | 2,094.375 | 2,094.375 | -1 | |
If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $43^\circ,$ what is the angle between the vectors $-\mathbf{a}$ and $\mathbf{b}$? | 137^\circ | 0.9375 | 2,789.125 | 2,428.933333 | 8,192 | |
The greatest integer function, $\lfloor x\rfloor$, denotes the largest integer less than or equal to $x$. For example, $\lfloor3.5\rfloor=3$, $\lfloor\pi\rfloor=3$ and $\lfloor -\pi\rfloor=-4$. Find the sum of the three smallest positive solutions to $x-\lfloor x\rfloor=\frac1{\lfloor x\rfloor}.$ Express your answer as... | 10\frac{1}{12} | 1 | 3,146.875 | 3,146.875 | -1 | |
The product of two consecutive even negative integers is 2496. What is the sum of these two integers? | -102 | 0 | 8,144.25 | -1 | 8,144.25 | |
Given the function \( y = \frac{1}{2}\left(x^{2}-100x+196+\left|x^{2}-100x+196\right|\right) \), what is the sum of the function values when the variable \( x \) takes on the 100 natural numbers \( 1, 2, 3, \ldots, 100 \)? | 390 | 0.6875 | 5,739.5625 | 4,918.272727 | 7,546.4 | |
Let \( ABCD \) be a square with side length \( 5 \), and \( E \) be a point on \( BC \) such that \( BE = 3 \) and \( EC = 2 \). Let \( P \) be a variable point on the diagonal \( BD \). Determine the length of \( PB \) if \( PE + PC \) is minimized. | \frac{15 \sqrt{2}}{8} | 0 | 6,606.6875 | -1 | 6,606.6875 | |
Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $I_{A}, I_{B}, I_{C}$ be the $A, B, C$ excenters of this triangle, and let $O$ be the circumcenter of the triangle. Let $\gamma_{A}, \gamma_{B}, \gamma_{C}$ be the corresponding excircles and $\omega$ be the circumcircle. $X$ is one of the intersections betwe... | -\frac{49}{65} | Let $r_{A}, r_{B}, r_{C}$ be the exradii. Using $O X=R, X I_{A}=r_{A}, O I_{A}=\sqrt{R\left(R+2 r_{A}\right)}$ (Euler's theorem for excircles), and the Law of Cosines, we obtain $$\cos \angle O X I_{A}=\frac{R^{2}+r_{A}^{2}-R\left(R+2 r_{A}\right)}{2 R r_{A}}=\frac{r_{A}}{2 R}-1$$ Therefore it suffices to compute $\fra... | 0 | 8,192 | -1 | 8,192 |
Let $a,$ $b,$ $c$ be a three-term arithmetic series where all the terms are positive, such that $abc = 64.$ Find the smallest possible value of $b.$ | 4 | 1 | 4,469.375 | 4,469.375 | -1 | |
In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board? | 480 | 0 | 8,012.375 | -1 | 8,012.375 | |
Given the function $f(x) = \cos(\omega x - \frac{\pi}{3}) - \cos(\omega x)$ $(x \in \mathbb{R}, \omega$ is a constant, and $1 < \omega < 2)$, the graph of function $f(x)$ is symmetric about the line $x = \pi$.
(Ⅰ) Find the smallest positive period of the function $f(x)$.
(Ⅱ) In $\triangle ABC$, the sides opposite a... | \frac{\sqrt{3}}{4} | 0 | 7,693.5625 | -1 | 7,693.5625 | |
Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.) | 59 |
Given an \(8 \times 8\) chessboard where each square initially contains a rook, we need to determine the maximal number of rooks that can be removed such that each removed rook initially attacked an odd number of other rooks. A rook attacks another rook if they are positioned in the same row or column and there are no... | 0 | 8,192 | -1 | 8,192 |
Given that a triangular corner with side lengths DB=EB=1.5 is cut from an equilateral triangle ABC of side length 4.5, determine the perimeter of the remaining quadrilateral. | 12 | 0.5625 | 5,388.375 | 4,791 | 6,156.428571 | |
Find the area of the region bounded by a function $y=-x^4+16x^3-78x^2+50x-2$ and the tangent line which is tangent to the curve at exactly two distinct points.
Proposed by Kunihiko Chikaya | 1296/5 | 0.125 | 8,124.6875 | 7,653.5 | 8,192 | |
Find the sum of the solutions to
\[\frac{1}{\sin x} + \frac{1}{\cos x} = 2 \sqrt{2}\]in the interval $0 \le x \le 2 \pi.$ | \frac{11 \pi}{4} | 0.125 | 7,878.6875 | 6,006 | 8,146.214286 | |
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^{2} - a^{2} = ac \), then the radian measure of angle \( B \) is equal to ________. | \frac{2\pi}{7} | 0.5 | 7,326.5 | 6,461 | 8,192 | |
Vasya wrote a note on a piece of paper, folded it in quarters, and wrote "MAME" on top. He then unfolded the note, added something more, folded it again randomly along the crease lines (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" re... | 1/8 | 0.0625 | 7,630.1875 | 7,673 | 7,627.333333 | |
Given the set S={1, 2, 3, ..., 40}, and a subset A⊆S containing three elements, find the number of such sets A that can form an arithmetic progression. | 380 | 0.625 | 6,255.3125 | 5,093.3 | 8,192 | |
20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.
Two neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ competes with each of the 20 fighters from vill... | 290 | 0 | 7,866.9375 | -1 | 7,866.9375 | |
Find the positive integer $n$ such that
$$\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.$$ | 47 | 0.9375 | 3,459.4375 | 3,143.933333 | 8,192 | |
Seven members of the family are each to pass through one of seven doors to complete a challenge. The first person can choose any door to activate. After completing the challenge, the adjacent left and right doors will be activated. The next person can choose any unchallenged door among the activated ones to complete th... | 64 | 0.1875 | 7,849 | 7,755.666667 | 7,870.538462 | |
Given $a$, $b$, $c \in \mathbb{R}$, and $2a+2b+c=8$, find the minimum value of $(a-1)^2+(b+2)^2+(c-3)^2$. | \frac{49}{9} | 0.9375 | 5,327.75 | 5,136.8 | 8,192 | |
In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{{\begin{array}{l}{x=2\cos\alpha,}\\{y=\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar ... | \frac{8\sqrt{5}}{15} | 0 | 7,033.125 | -1 | 7,033.125 | |
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? | 43 | 0.875 | 3,854.6875 | 3,235.071429 | 8,192 | |
Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x.$ | 8 | 0 | 8,192 | -1 | 8,192 | |
The game of rock-scissors is played just like rock-paper-scissors, except that neither player is allowed to play paper. You play against a poorly-designed computer program that plays rock with $50 \%$ probability and scissors with $50 \%$ probability. If you play optimally against the computer, find the probability tha... | \frac{163}{256} | Since rock will always win against scissors, the optimum strategy is for you to always play rock; then, you win a game if and only if the computer plays scissors. Let $p_{n}$ be the probability that the computer plays scissors $n$ times; we want $p_{0}+p_{1}+p_{2}+p_{3}+p_{4}$. Note that by symmetry, $p_{n}=p_{8-n}$ fo... | 0.875 | 5,107.1875 | 4,666.5 | 8,192 |
A point is randomly thrown on the segment [12, 17] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+k-90\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | 2/3 | 0 | 8,141.6875 | -1 | 8,141.6875 | |
How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0? | 90 | We can think about it as shading a $4 \times 4$ array so that there are exactly two shaded unit squares in each row and each column. Then the answer is $\boxed{90}$. | 0.125 | 8,042.25 | 6,994 | 8,192 |
Define the function \(f(n)\) on the positive integers such that \(f(f(n)) = 3n\) and \(f(3n + 1) = 3n + 2\) for all positive integers \(n\). Find \(f(729)\). | 729 | 0 | 8,192 | -1 | 8,192 | |
The maximum value of the real number $k$ for which the inequality $\sqrt{x-3}+\sqrt{6-x} \geqslant k$ has a solution with respect to $x$ is: | $\sqrt{6}$ | 0 | 3,180.75 | -1 | 3,180.75 | |
Triangle $ABC$ with right angle at $C$, $\angle BAC < 45^\circ$ and $AB = 4$. Point $P$ on $\overline{AB}$ is chosen such that $\angle APC = 2\angle ACP$ and $CP = 1$. The ratio $\frac{AP}{BP}$ can be represented in the form $p + q\sqrt{r}$, where $p$, $q$, $r$ are positive integers and $r$ is not divisible by the squa... | 7 | 0.0625 | 8,192 | 8,192 | 8,192 | |
The function $y= |x-1|+|2x-1|+|3x-1|+ |4x-1|+|5x-1|$ achieves its minimum value when the variable $x$ equals what value? | \frac{1}{3} | 0.125 | 7,151.6875 | 4,732.5 | 7,497.285714 | |
Angie's class has 2 girls for every 3 boys. If there are 20 students in the class, how many girls are in Angie's class? | 8 | 1 | 696.0625 | 696.0625 | -1 | |
Find the $3 \times 3$ matrix $\mathbf{M}$ such that for a $3 \times 3$ matrix $\mathbf{N},$ $\mathbf{M} \mathbf{N}$ is the result of swapping the first row and second row of $\mathbf{N},$ and doubling the third row of $\mathbf{N}.$ In other words,
\[\mathbf{M} \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{p... | \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{pmatrix} | 0.25 | 7,058.6875 | 6,850.5 | 7,128.083333 | |
Square $XYZW$ has area $144$. Point $P$ lies on side $\overline{XW}$, such that $XP = 2WP$. Points $Q$ and $R$ are the midpoints of $\overline{ZP}$ and $\overline{YP}$, respectively. Quadrilateral $XQRW$ has an area of $20$. Calculate the area of triangle $RWP$. | 12 | 0 | 8,192 | -1 | 8,192 | |
We start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\frac{1}{3}$. What is the probability that we are able to ... | 17/27 | 0 | 8,124.5 | -1 | 8,124.5 | |
In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing? | 32 | To solve this problem, we need to count the number of permutations of the sequence $1, 2, 3, 4, 5$ such that no three consecutive terms are either strictly increasing or strictly decreasing. We analyze the problem by considering the possible patterns of increases and decreases between consecutive terms.
#### Case Anal... | 0 | 8,192 | -1 | 8,192 |
Find the quotient of the division $(3z^4-4z^3+5z^2-11z+2)/(2+3z)$. | z^3 -2z^2+3z-\frac{17}{3} | 0.75 | 5,380.125 | 5,110.916667 | 6,187.75 | |
The sum
$$
\frac{1}{1 \times 2 \times 3}+\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\cdots+\frac{1}{100 \times 101 \times 102}
$$
can be expressed as $\frac{a}{b}$, a fraction in its simplest form. Find $a+b$. | 12877 | 0.3125 | 7,705.875 | 6,636.4 | 8,192 | |
Given points $P(\sqrt{3}, 1)$, $Q(\cos x, \sin x)$, and $O$ as the origin of coordinates, the function $f(x) = \overrightarrow{OP} \cdot \overrightarrow{QP}$
(Ⅰ) Find the smallest positive period of the function $f(x)$;
(Ⅱ) If $A$ is an internal angle of $\triangle ABC$, $f(A) = 4$, $BC = 3$, and the area of $\triang... | 3 + 2\sqrt{3} | 0.875 | 4,426.75 | 4,297.357143 | 5,332.5 | |
In triangle $MPQ$, a line parallel to side $MQ$ intersects side $MP$, the median $MM_1$, and side $PQ$ at points $D$, $E$, and $F$ respectively. It is known that $DE = 5$ and $EF = 7$. What is the length of $MQ$? | 17 | 0.375 | 6,869.375 | 4,665 | 8,192 | |
China's space station has entered the formal construction phase. The Tianhe core module, Wentian experimental module, and Mengtian experimental module will all dock in 2022, forming a "T" shaped structure. During the construction phase of the Chinese space station, there are 6 astronauts staying in the space station. I... | 450 | 0.25 | 6,329.4375 | 4,207.75 | 7,036.666667 | |
Given a matrix $\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$ satisfies: $a_{11}$, $a_{12}$, $a_{21}$, $a_{22} \in \{0,1\}$, and $\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{vmatrix} =0$, determine the total number of distinct matrices. | 10 | 0.125 | 8,006.0625 | 6,704.5 | 8,192 | |
The area of the triangle formed by the tangent line at point $(1,1)$ on the curve $y=x^3$, the x-axis, and the line $x=2$ is $\frac{4}{3}$. | \frac{8}{3} | 0 | 8,192 | -1 | 8,192 | |
Find the number of 7 -tuples $\left(n_{1}, \ldots, n_{7}\right)$ of integers such that $$\sum_{i=1}^{7} n_{i}^{6}=96957$$ | 2688 | Consider the equation in modulo 9. All perfect 6 th powers are either 0 or 1. Since 9 divides 96957, it must be that each $n_{i}$ is a multiple of 3. Writing $3 a_{i}=n_{i}$ and dividing both sides by $3^{6}$, we have $a_{1}^{6}+\cdots+a_{7}^{6}=133$. Since sixth powers are nonnegative, $\left|a_{i}\right| \leq 2$. Aga... | 0 | 8,176.4375 | -1 | 8,176.4375 |
In triangle $PQR$, angle $R$ is a right angle and the altitude from $R$ meets $\overline{PQ}$ at $S$. The lengths of the sides of $\triangle PQR$ are integers, $PS=17^3$, and $\cos Q = a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | 18 | 0 | 7,542.6875 | -1 | 7,542.6875 | |
Equilateral $\triangle A B C$ has side length 6. Let $\omega$ be the circle through $A$ and $B$ such that $C A$ and $C B$ are both tangent to $\omega$. A point $D$ on $\omega$ satisfies $C D=4$. Let $E$ be the intersection of line $C D$ with segment $A B$. What is the length of segment $D E$? | \frac{20}{13} | Let $F$ be the second intersection of line $C D$ with $\omega$. By power of a point, we have $C F=9$, so $D F=5$. This means that $\frac{[A D B]}{[A F B]}=\frac{D E}{E F}=\frac{D E}{5-D E}$. Now, note that triangle $C A D$ is similar to triangle $C F A$, so $\frac{F A}{A D}=\frac{C A}{C D}=\frac{3}{2}$. Likewise, $\fra... | 0.5 | 6,939.4375 | 5,686.875 | 8,192 |
A certain point has rectangular coordinates $(10,3)$ and polar coordinates $(r, \theta).$ What are the rectangular coordinates of the point with polar coordinates $(r^2, 2 \theta)$? | (91,60) | 1 | 3,108.75 | 3,108.75 | -1 | |
Given the ellipse C: $mx^2+3my^2=1$ ($m>0$) with a major axis length of $2\sqrt{6}$, and O as the origin.
(1) Find the equation of ellipse C and its eccentricity.
(2) Let point A be (3,0), point B be on the y-axis, and point P be on ellipse C, with point P on the right side of the y-axis. If $BA=BP$, find the minim... | 3\sqrt{3} | 0.25 | 8,063 | 7,676 | 8,192 | |
Given the function $f(x) = (\sin x + \cos x)^2 + \cos 2x - 1$.
(1) Find the smallest positive period of the function $f(x)$;
(2) Find the maximum and minimum values of $f(x)$ in the interval $\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$. | -\sqrt{2} | 0 | 4,430.5625 | -1 | 4,430.5625 | |
The product of all real roots of the equation $x^{\log_{10}{x}}=10$ is | 1 | 1. **Rewrite the given equation**: We start with the equation \(x^{\log_{10}x} = 10\).
2. **Take the logarithm of both sides**: Applying the logarithm base 10 to both sides, we get:
\[
\log_{10}(x^{\log_{10}x}) = \log_{10}10
\]
Using the power rule of logarithms, \(\log_b(a^c) = c \log_b a\), this simplifi... | 1 | 2,671.5625 | 2,671.5625 | -1 |
The measure of angle $ACB$ is 45 degrees. If ray $CA$ is rotated 510 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? | 75 | 0.25 | 5,694.6875 | 4,497.25 | 6,093.833333 | |
Given that the area of $\triangle ABC$ is $S$, and $\overrightarrow{BA} \cdot \overrightarrow{CA} = S$.
(1) Find the value of $\tan A$;
(2) If $B = \frac{\pi}{4}, c = 6$, find the area of $\triangle ABC$, $S$. | 12 | 0.8125 | 5,577.5 | 5,202.076923 | 7,204.333333 | |
A bug starts at a vertex of a square. On each move, it randomly selects one of the three vertices where it is not currently located and crawls along a side of the square to that vertex. Given that the probability that the bug moves to its starting vertex on its eighth move is \( \frac{p}{q} \), where \( p \) and \( q \... | 2734 | 0.125 | 7,707.9375 | 5,738.5 | 7,989.285714 | |
Given that the sequence \( a_1, a_2, \cdots, a_n, \cdots \) satisfies \( a_1 = a_2 = 1 \) and \( a_3 = 2 \), and for any \( n \in \mathbf{N}^{*} \), it holds that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( \sum_{i=1}^{2023} a_i \). | 4044 | 0.9375 | 4,713.3125 | 4,684.866667 | 5,140 | |
Assume that $f(a+b) = f(a) + f(b) + ab$ , and that $f(75) - f(51) = 1230$ . Find $f(100)$ . | 3825 | 0.9375 | 4,469.125 | 4,587.733333 | 2,690 | |
Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\angle{BAC}$? | 36 | 1. **Identify the Geometry and Key Properties**:
Let the center of the circle be $O$. The lines $AB$ and $AC$ are tangents to the circle at points $B$ and $C$, respectively. By the property of tangents, the radii $OB$ and $OC$ are perpendicular to $AB$ and $AC$, respectively. Therefore, $\angle ABO = \angle ACO = ... | 0.9375 | 3,765.4375 | 3,470.333333 | 8,192 |
Timur and Alexander are counting the trees growing around the house. They move in the same direction but start counting from different trees. How many trees are growing around the house if the tree that Timur counted as the 12th, Alexander counted as the 33rd, and the tree that Timur counted as the 105th, Alexander cou... | 118 | 0.4375 | 5,526.25 | 4,239.142857 | 6,527.333333 | |
Let $x$ be a real number between $0$ and $\tfrac{\pi}2$ such that \[\dfrac{\sin^4(x)}{42}+\dfrac{\cos^4(x)}{75} = \dfrac{1}{117}.\] Find $\tan(x)$ . | \frac{\sqrt{14}}{5} | 0 | 6,486.25 | -1 | 6,486.25 | |
ABCD is a square with side of unit length. Points E and F are taken respectively on sides AB and AD so that AE = AF and the quadrilateral CDFE has maximum area. In square units this maximum area is: | \frac{5}{8} | 1. **Define the problem setup**: Let $ABCD$ be a square with side length 1. Points $E$ and $F$ are on sides $AB$ and $AD$ respectively such that $AE = AF = x$. We need to maximize the area of quadrilateral $CDFE$.
2. **Express the area of $CDFE$**: Consider dropping a perpendicular from $E$ to line $DC$, and let $G$ b... | 0.875 | 5,558 | 5,293.071429 | 7,412.5 |
Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a>0$ and $b>0$. If the point $F_2$ is symmetric with respect to the asymptote line and lies on the hyperbola, calculate the eccentricity of the hyperbola. | \sqrt{5} | 0.4375 | 6,659.5625 | 5,768.142857 | 7,352.888889 | |
Mahmoud tosses three coins. What is the probability that he will get at least one head? | \frac{7}{8} | 1 | 1,701.5625 | 1,701.5625 | -1 | |
Let $ABCD$ be an isosceles trapezoid with $AB$ and $CD$ as parallel bases, and $AB > CD$. A point $P$ inside the trapezoid connects to the vertices $A$, $B$, $C$, $D$, creating four triangles. The areas of these triangles, starting from the triangle with base $\overline{CD}$ moving clockwise, are $3$, $4$, $6$, and $7$... | \frac{7}{3} | 0 | 7,858.8125 | -1 | 7,858.8125 | |
David, Delong, and Justin each showed up to a problem writing session at a random time during the session. If David arrived before Delong, what is the probability that he also arrived before Justin? | \frac{2}{3} | 0.1875 | 6,673.75 | 6,228 | 6,776.615385 | |
Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes, they are still equidistant from $O$. What... | 2: 3 | 0 | 7,262.9375 | -1 | 7,262.9375 | |
Through the vertices \( A \) and \( C \) of triangle \( ABC \), lines are drawn perpendicular to the bisector of angle \( ABC \), intersecting lines \( CB \) and \( BA \) at points \( K \) and \( M \) respectively. Find \( AB \) if \( BM = 10 \) and \( KC = 2 \). | 12 | 0.0625 | 7,512.8125 | 6,583 | 7,574.8 | |
Find the largest negative root of the equation
$$
4 \sin (3 x) + 13 \cos (3 x) = 8 \sin (x) + 11 \cos (x)
$$ | -0.1651 | 0 | 8,192 | -1 | 8,192 | |
Find the number of integers $n$ that satisfy:
\[50 < n^2 < 200.\] | 14 | 1 | 3,485.9375 | 3,485.9375 | -1 | |
Solve the equations:
① $3(x-1)^3 = 24$;
② $(x-3)^2 = 64$. | -5 | 0.9375 | 1,561.375 | 1,445.4 | 3,301 | |
What is the smallest number divisible by integers 1 through 9? | 2520 | 1 | 2,747.9375 | 2,747.9375 | -1 | |
Máté is always in a hurry. He observed that it takes 1.5 minutes to get to the subway when he stands on the moving escalator, while it takes 1 minute to run down the stationary stairs. How long does it take Máté to get down if he can run down the moving escalator? | 36 | 0.125 | 5,589.5 | 2,575.5 | 6,020.071429 | |
Fill in the blanks with appropriate numbers to make the equation true: $x^2+5x+\_\_=(x+\_\_)^2.$ | \frac{5}{2} | 0.25 | 598.4375 | 574.25 | 606.5 | |
Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | 198 | 0 | 8,192 | -1 | 8,192 | |
In a certain number quiz, the test score of a student with seat number $n$ ($n=1,2,3,4$) is denoted as $f(n)$. If $f(n) \in \{70,85,88,90,98,100\}$ and it satisfies $f(1)<f(2) \leq f(3)<f(4)$, then the total number of possible combinations of test scores for these 4 students is \_\_\_\_\_\_\_\_. | 35 | 0.125 | 8,020.25 | 8,183.5 | 7,996.928571 | |
Using the numbers from 1 to 22 exactly once each, Antoine writes 11 fractions. For example, he could write the fractions \(\frac{10}{2}, \frac{4}{3}, \frac{15}{5}, \frac{7}{6}, \frac{8}{9}, \frac{11}{19}, \frac{12}{14}, \frac{13}{17}, \frac{22}{21}, \frac{18}{16}, \frac{20}{1}\).
Antoine wants to have as many fraction... | 10 | 0 | 8,192 | -1 | 8,192 |
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