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 |
|---|---|---|---|---|---|---|
Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits. | 972538 | 0 | 8,192 | -1 | 8,192 | |
There is a unique positive integer $n$ such that $\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.$ What is the sum of the digits of $n?$ | 13 | 1. **Rewrite the equation using logarithmic identities:**
Given the equation \(\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}\), we use the identity \(\log_{a^b} c = \frac{1}{b} \log_a c\) to simplify the bases:
\[
\log_2{(\log_{2^4}{n})} = \log_{2^2}{(\log_{2^2}{n})}
\]
which simplifies to:
\[
\log_... | 1 | 2,783.9375 | 2,783.9375 | -1 |
In the prime factorization of $24!$, what is the exponent of $3$? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\cdot 4\cdot3\cdot2\cdot 1= 120$.) | 10 | 0.9375 | 2,084.5625 | 1,677.4 | 8,192 | |
For what value of $x$ will $\frac{3+x}{5+x}$ and $\frac{1+x}{2+x}$ be equal? | 1 | 1 | 2,126 | 2,126 | -1 | |
A number of tourists want to take a cruise, and it is required that the number of people on each cruise ship is the same. If each cruise ship carries 12 people, there will be 1 person left who cannot board. If one cruise ship leaves empty, then all tourists can be evenly distributed among the remaining ships. It is kno... | 169 | 0.6875 | 4,594.875 | 3,536.090909 | 6,924.2 | |
The polynomial $-400x^5+2660x^4-3602x^3+1510x^2+18x-90$ has five rational roots. Suppose you guess a rational number which could possibly be a root (according to the rational root theorem). What is the probability that it actually is a root? | 1/36 | 0.0625 | 6,955.4375 | 3,609 | 7,178.533333 | |
In the production of a certain item, its weight \( X \) is subject to random fluctuations. The standard weight of the item is 30 g, its standard deviation is 0.7, and the random variable \( X \) follows a normal distribution. Find the probability that the weight of a randomly selected item is within the range from 28 t... | 0.9215 | 0.3125 | 6,456.9375 | 5,392.8 | 6,940.636364 | |
Four teams, including Quixajuba, are competing in a volleyball tournament where:
- Each team plays against every other team exactly once;
- Any match ends with one team winning;
- In any match, the teams have an equal probability of winning;
- At the end of the tournament, the teams are ranked by the number of victori... | \frac{1}{8} | 0 | 7,925.625 | -1 | 7,925.625 | |
Estimate the time it takes to send $60$ blocks of data over a communications channel if each block consists of $512$ "chunks" and the channel can transmit $120$ chunks per second. | 240 | 1. **Calculate the total number of chunks**:
Each block consists of $512$ chunks, and there are $60$ blocks. Therefore, the total number of chunks is:
\[
60 \times 512 = 30720
\]
2. **Determine the transmission time**:
The channel can transmit $120$ chunks per second. To find the time it takes to trans... | 0 | 3,735.4375 | -1 | 3,735.4375 |
The function $g(x)$ satisfies
\[g(x + g(x)) = 5g(x)\] for all \(x\), and \(g(1) = 5\). Find \(g(26)\). | 125 | 0 | 6,134.875 | -1 | 6,134.875 | |
Given that $\tan \alpha +\tan \beta -\tan \alpha \tan \beta +1=0$, and $\alpha ,\beta \in \left(\frac{\pi }{2},\pi \right)$, calculate $\alpha +\beta$. | \frac{7\pi}{4} | 0.875 | 5,034.75 | 4,583.714286 | 8,192 | |
In triangle $ABC$, $AC = \sqrt{7}$, $BC = 2$, and $\angle B = 60^\circ$. What is the area of $\triangle ABC$? | \frac{3\sqrt{3}}{2} | 0 | 3,818.3125 | -1 | 3,818.3125 | |
What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $0.1\overline{35}$? | \frac{67}{495} | 0.8125 | 4,551.875 | 3,711.846154 | 8,192 | |
Given a function \\(f(x)\\) defined on \\(\mathbb{R}\\) that satisfies: the graph of \\(y=f(x-1)\\) is symmetric about the point \\((1,0)\\), and when \\(x \geqslant 0\\), it always holds that \\(f(x+2)=f(x)\\). When \\(x \in [0,2)\\), \\(f(x)=e^{x}-1\\), where \\(e\\) is the base of the natural logarithm, evaluate \\(... | 1-e | 0.375 | 6,654.3125 | 5,064.333333 | 7,608.3 | |
Let $A,B,C$ be angles of a triangle, where angle $B$ is obtuse, and \begin{align*}
\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\
\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}.
\end{align*}There are positive integers $p$, $q$, $r$, and $s$ for which \[ \cos^2 C + \cos^2 A + 2... | 222 | 0 | 8,192 | -1 | 8,192 | |
A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\frac{m}{n}$ where $m$ and $n$ are relativel... | 16 | 0.1875 | 7,780.4375 | 5,997 | 8,192 | |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a\sin A + c\sin C - \sqrt{2}a\sin C = b\sin B$.
1. Find $B$.
2. If $\cos A = \frac{1}{3}$, find $\sin C$. | \frac{4 + \sqrt{2}}{6} | 0 | 5,998 | -1 | 5,998 | |
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1... | 961 | Note that every $B_nC_n$ is parallel to each other for any nonnegative $n$. Also, the area we seek is simply the ratio $k=\frac{B_0B_1C_1}{B_0B_1C_1+C_1C_0B_0}$, because it repeats in smaller and smaller units. Note that the area of the triangle, by Heron's formula, is 90.
For ease, all ratios I will use to solve this... | 0 | 8,192 | -1 | 8,192 |
In the diagram, \(PRTY\) and \(WRSU\) are squares. Point \(Q\) is on \(PR\) and point \(X\) is on \(TY\) so that \(PQXY\) is a rectangle. Also, point \(T\) is on \(SU\), point \(W\) is on \(QX\), and point \(V\) is the point of intersection of \(UW\) and \(TY\), as shown. If the area of rectangle \(PQXY\) is 30, the le... | 5.5 | 0 | 8,192 | -1 | 8,192 | |
Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4$($a_{n-1}$ $+$ $a_{n-2}$ $+$ $a_{n-3}$) is divided by $11$. Find $a_{2018} \cdot a_{2020} \cdot a_{2022}$. | 112 | When given a sequence problem, one good thing to do is to check if the sequence repeats itself or if there is a pattern.
After computing more values of the sequence, it can be observed that the sequence repeats itself every 10 terms starting at $a_{0}$.
$a_{0} = 2$, $a_{1} = 5$, $a_{2} = 8$, $a_{3} = 5$, $a_{4} = 6$, ... | 0.375 | 6,239.4375 | 5,973.166667 | 6,399.2 |
The sum of the dimensions of a rectangular prism is the sum of the number of edges, corners, and faces, where the dimensions are 2 units by 3 units by 4 units. Calculate the resulting sum. | 26 | 0.625 | 3,944 | 3,697 | 4,355.666667 | |
Let $x$ and $y$ be two distinct positive real numbers. We define three sequences $(A_n),$ $(G_n),$ and $(H_n)$ as follows. First, $A_1,$ $G_1,$ and $H_1$ are the arithmetic mean, geometric mean, and harmonic mean of $x$ and $y,$ respectively. Then for $n \ge 2,$ $A_n,$ $G_n,$ $H_n$ are the arithmetic mean, geometric... | 273 | 0.0625 | 7,392.4375 | 4,569 | 7,580.666667 | |
Three identical square sheets of paper each with side length \(8\) are stacked on top of each other. The middle sheet is rotated clockwise \(20^\circ\) about its center and the top sheet is rotated clockwise \(50^\circ\) about its center. Determine the area of the resulting polygon.
A) 178
B) 192
C) 204
D) 216 | 192 | 0 | 8,192 | -1 | 8,192 | |
In triangle \(ABC\), a circle \(\omega\) with center \(O\) passes through \(B\) and \(C\) and intersects segments \(\overline{AB}\) and \(\overline{AC}\) again at \(B'\) and \(C'\), respectively. Suppose that the circles with diameters \(BB'\) and \(CC'\) are externally tangent to each other at \(T\). If \(AB = 18\), \... | 65/3 | 0 | 8,192 | -1 | 8,192 | |
Let's define a calendar week as even or odd according to whether the sum of the day numbers within the month in that week is even or odd. Out of the 52 consecutive weeks starting from the first Monday of January, how many can be even? | 30 | 0.0625 | 8,034.375 | 8,192 | 8,023.866667 | |
Given the cubic equation
\[
x^3 + Ax^2 + Bx + C = 0 \quad (A, B, C \in \mathbb{R})
\]
with roots \(\alpha, \beta, \gamma\), find the minimum value of \(\frac{1 + |A| + |B| + |C|}{|\alpha| + |\beta| + |\gamma|}\). | \frac{\sqrt[3]{2}}{2} | 0 | 8,192 | -1 | 8,192 | |
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n\,$ fish for various values of $n\,$.
$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fi... | 943 | Let $f$ be the total number of fish caught by the contestants who didn't catch $0, 1, 2, 3, 13, 14$, or $15$ fish and let $a$ be the number of contestants who didn't catch $0, 1, 2, 3, 13, 14$, or $15$ fish. From $\text{(b)}$, we know that $\frac{69+f+65+28+15}{a+31}=6\implies f=6a+9$. From $\text{(c)}$ we have $\frac{... | 0 | 7,169.3125 | -1 | 7,169.3125 |
Over two days, 100 bankers collected funds to fight a new virus. Each banker contributed a whole number of thousands of rubles, not exceeding 200. Contributions on the first day did not exceed 100 thousand, and contributions on the second day were greater than this amount. Additionally, no pair of all 100 contribution... | 10050 | 0.0625 | 7,661.125 | 5,167 | 7,827.4 | |
The spikiness of a sequence $a_{1}, a_{2}, \ldots, a_{n}$ of at least two real numbers is the sum $\sum_{i=1}^{n-1}\left|a_{i+1}-a_{i}\right|$. Suppose $x_{1}, x_{2}, \ldots, x_{9}$ are chosen uniformly and randomly from the interval $[0,1]$. Let $M$ be the largest possible value of the spikiness of a permutation of $x... | \frac{79}{20} | Our job is to arrange the nine numbers in a way that maximizes the spikiness. Let an element be a peak if it is higher than its neighbor(s) and a valley if it is lower than its neighbor(s). It is not hard to show that an optimal arrangement has every element either a peak or a valley (if you have some number that is ne... | 0 | 7,854.1875 | -1 | 7,854.1875 |
In some cases, it is not necessary to calculate the result to remove the absolute value. For example: $|6+7|=6+7$, $|6-7|=7-6$, $|7-6|=7-6$, $|-6-7|=6+7.\left(1\right)$ According to the above rule, express the following expressions in the form without absolute value symbols: <br/>①$|\frac{7}{17}-\frac{7}{18}|=$______；<... | \frac{505}{1011} | 0.75 | 4,017.625 | 3,801.916667 | 4,664.75 | |
Let $ABCDEFGH$ be a regular octagon, and let $I, J, K$ be the midpoints of sides $AB, DE, GH$ respectively. If the area of $\triangle IJK$ is $144$, what is the area of octagon $ABCDEFGH$? | 1152 | 0 | 8,192 | -1 | 8,192 | |
The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | 125 | To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard, or ${9\choose 2} = 36$. Similarly, there are ${9\choose 2}$ ways to pick the vertical sides, giving us $r = 1296$ rectangles.
For $s$, there are $8^2$ unit squares, $7^2$ of the $2\times2$ squares, an... | 1 | 2,747.25 | 2,747.25 | -1 |
Given that plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero vectors, $|\overrightarrow{a}|=2$, and $\overrightarrow{a} \bot (\overrightarrow{a}+2\overrightarrow{b})$, calculate the projection of vector $\overrightarrow{b}$ in the direction of vector $\overrightarrow{a}$. | -1 | 0.875 | 3,710.4375 | 3,425.928571 | 5,702 | |
From a point \( M \) on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\), two tangent lines are drawn to the circle with the minor axis of the ellipse as its diameter. The points of tangency are \( A \) and \( B \). The line \( AB \) intersects the \(x\)-axis and \(y\)-axis at points \( P \) and \( Q \) respectively. ... | 10/3 | 0.1875 | 7,083.6875 | 6,075 | 7,316.461538 | |
Find the smallest positive integer $n$ such that the divisors of $n$ can be partitioned into three sets with equal sums. | 120 | I claim the answer is 120. First, note that $120=2^{3} \cdot 3 \cdot 5$, so the sum of divisors is $(1+2+4+8)(1+3)(1+5)=15 \cdot 4 \cdot 6=360$. Thus, we need to split the divisors into groups summing to 120 . But then we can just take $\{120\},\{20,40,60\},\{1,2,3,4,5,6,8,10,12,15,24,30\}$. Thus, 120 works. Now we nee... | 0 | 8,157.0625 | -1 | 8,157.0625 |
A store sold an air conditioner for 2000 yuan and a color TV for 2000 yuan. The air conditioner made a 30% profit, while the color TV incurred a 20% loss. Could you help the store owner calculate whether the store made a profit or a loss on this transaction, and by how much? | 38.5 | 0 | 918.1875 | -1 | 918.1875 | |
In triangle $ABC$, $BC = 1$ unit and $\measuredangle BAC = 40^\circ$, $\measuredangle ABC = 90^\circ$, hence $\measuredangle ACB = 50^\circ$. Point $D$ is midway on side $AB$, and point $E$ is the midpoint of side $AC$. If $\measuredangle CDE = 50^\circ$, compute the area of triangle $ABC$ plus twice the area of triang... | \frac{5}{16} | 0 | 8,192 | -1 | 8,192 | |
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin ^{2}\left(\frac{x}{4}\right) \cos ^{6}\left(\frac{x}{4}\right) d x
$$ | \frac{5\pi}{64} | 0.3125 | 7,337.1875 | 6,218.6 | 7,845.636364 | |
The sum of 36 consecutive integers is $6^4$. What is their median? | 36 | 1 | 3,621.625 | 3,621.625 | -1 | |
Find the sum of the digits of the number $\underbrace{44 \ldots 4}_{2012 \text { times}} \cdot \underbrace{99 \ldots 9}_{2012 \text { times}}$. | 18108 | 0.4375 | 6,931.25 | 5,310.285714 | 8,192 | |
Define the sequence $b_{0}, b_{1}, \ldots, b_{59}$ by $$ b_{i}= \begin{cases}1 & \text { if } \mathrm{i} \text { is a multiple of } 3 \\ 0 & \text { otherwise }\end{cases} $$ Let \left\{a_{i}\right\} be a sequence of elements of \{0,1\} such that $$ b_{n} \equiv a_{n-1}+a_{n}+a_{n+1} \quad(\bmod 2) $$ for $0 \leq n \le... | 0, 3, 5, 6 | Try the four possible combinations of values for $a_{0}$ and $a_{1}$. Since we can write $a_{n} \equiv$ $b_{n-1}-a_{n-2}-a_{n-1}$, these two numbers completely determine the solution $\left\{a_{i}\right\}$ beginning with them (if there is one). For $a_{0}=a_{1}=0$, we can check that the sequence beginning $0,0,0,0,1,1$... | 0 | 8,192 | -1 | 8,192 |
Given a parabola $C$ that passes through the point $(4,4)$ and its focus lies on the $x$-axis.
$(1)$ Find the standard equation of parabola $C$.
$(2)$ Let $P$ be any point on parabola $C$. Find the minimum distance between point $P$ and the line $x - y + 4 = 0$. | \frac{3\sqrt{2}}{2} | 0 | 6,361.875 | -1 | 6,361.875 | |
The quadrilateral \(ABCD\) is inscribed in a circle. \(I\) is the incenter of triangle \(ABD\). Find the minimum value of \(BD\) given that \(AI = BC = CD = 2\). | 2\sqrt{3} | 0 | 8,192 | -1 | 8,192 | |
Let $z$ be a nonreal complex number such that $|z| = 1.$ Find the real part of $\frac{1}{1 - z}.$ | \frac{1}{2} | 0.875 | 4,259.875 | 3,698.142857 | 8,192 | |
There are 200 students enrolled at Memorial Middle School. Seventy of the students are in band and 95 are in chorus. If only 150 students are in band and/or chorus, how many students are in both band and chorus? | 15 | 1 | 1,361.75 | 1,361.75 | -1 | |
Given that the function $f(x+2)$ is an odd function and it satisfies $f(6-x)=f(x)$, and $f(3)=2$, determine the value of $f(2008)+f(2009)$. | -2 | 0.75 | 5,394.8125 | 4,462.416667 | 8,192 | |
Find the sum of the first seven prime numbers that have a units digit of 7. | 379 | 0.875 | 3,418.4375 | 3,201.214286 | 4,939 | |
Let $a, b, c, d$ be real numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a, b, c, d)$ such that the minimum value is achieved. | -\frac{1}{8} | The minimum value is $-\frac{1}{8}$. There are eight equality cases in total. The first one is $$\left(\frac{1}{4}+\frac{\sqrt{3}}{4},-\frac{1}{4}-\frac{\sqrt{3}}{4}, \frac{1}{4}-\frac{\sqrt{3}}{4},-\frac{1}{4}+\frac{\sqrt{3}}{4}\right)$$ Cyclic shifting all the entries give three more quadruples. Moreover, flipping th... | 0 | 8,177.75 | -1 | 8,177.75 |
In right triangle $PQR$, we have $\angle Q = \angle R$ and $PR = 6\sqrt{2}$. What is the area of $\triangle PQR$? | 36 | 0.8125 | 2,958.0625 | 2,817.307692 | 3,568 | |
If $f(x) = 3-\!\sqrt{x}$ and $g(x) = 5x +2x^2$, what is $f(g(-5))$? | -2 | 1 | 1,580.3125 | 1,580.3125 | -1 | |
Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \div 18$ and $X$ is an integer, what is the smallest possible value of $X$? | 6172839500 | 0 | 7,893.125 | -1 | 7,893.125 | |
A basketball team has 18 players, including two sets of twins: Ben & Jerry and Tom & Tim. In how many ways can we choose 5 starters if at least one member from each set of twins must be included in the starting lineup? | 1834 | 0.1875 | 7,354.5 | 5,435 | 7,797.461538 | |
A random walk is a process in which something moves from point to point, and where the direction of movement at each step is randomly chosen. Suppose that a person conducts a random walk on a line: he starts at $0$ and each minute randomly moves either $1$ unit in the positive direction or $1$ unit in the negativ... | 15/8 | 0.375 | 7,087.25 | 5,246 | 8,192 | |
A bag contains four balls of the same shape and size, numbered $1$, $2$, $3$, $4$ respectively.
$(1)$ Randomly draw two balls from the bag, and calculate the probability that the sum of the numbers on the drawn balls is even.
$(2)$ First, randomly draw a ball from the bag, and record its number as $m$. Then, return the... | \frac{5}{8} | 1 | 3,308.8125 | 3,308.8125 | -1 | |
Mandy Monkey starts hopping on a number line at 0. She aims to reach 1, but can only hop 1/4 of the distance towards the goal at each jump. Each subsequent hop proceeds as 1/4 of the remaining distance to 1. Calculate the total distance Mandy has traveled after six hops. Express your answer as a common fraction. | \frac{3367}{4096} | 0.5625 | 5,684.3125 | 4,259 | 7,516.857143 | |
How many pairs of positive integers $(x,y)$ satisfy $x^2-y^2=51$? | 2 | 1 | 2,368.125 | 2,368.125 | -1 | |
Given that positive real numbers \( a \) and \( b \) satisfy \( ab(a+b)=4 \), find the minimum value of \( 2a + b \). | 2\sqrt{3} | 0.3125 | 7,497.1875 | 5,968.6 | 8,192 | |
The vertices of the broken line $A B C D E F G$ have coordinates $A(-1, -7), B(2, 5), C(3, -8), D(-3, 4), E(5, -1), F(-4, -2), G(6, 4)$.
Find the sum of the angles with vertices at points $B, E, C, F, D$. | 180 | 0 | 7,688.875 | -1 | 7,688.875 | |
Let the function $f(x)=2x-\cos x$, and $\{a_n\}$ be an arithmetic sequence with a common difference of $\dfrac{\pi}{8}$. If $f(a_1)+f(a_2)+\ldots+f(a_5)=5\pi$, calculate $\left[f(a_3)\right]^2-a_2a_3$. | \dfrac{13}{16}\pi^2 | 0.1875 | 7,761.875 | 6,549 | 8,041.769231 | |
On the board, six numbers are written in a row. It is known that each number, starting from the third, is equal to the product of the two preceding numbers, and the fifth number is equal to 108. Find the product of all six numbers in this row. | 136048896 | 0.25 | 7,903.9375 | 7,214.75 | 8,133.666667 | |
Find the largest real number $c$ such that $$\sum_{i=1}^{101} x_{i}^{2} \geq c M^{2}$$ whenever $x_{1}, \ldots, x_{101}$ are real numbers such that $x_{1}+\cdots+x_{101}=0$ and $M$ is the median of $x_{1}, \ldots, x_{101}$. | \frac{5151}{50} | Suppose without loss of generality that $x_{1} \leq \cdots \leq x_{101}$ and $M=x_{51} \geq 0$. Note that $f(t)=t^{2}$ is a convex function over the reals, so we may "smooth" to the case $x_{1}=\cdots=x_{50}=-51 r$ and $x_{51}=\cdots=x_{101}=50 r$ for some $r \geq 0$, and by homogeneity, $C$ works if and only if $C \le... | 0 | 8,192 | -1 | 8,192 |
The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 481^2 | 0 | 8,192 | -1 | 8,192 | |
The measure of the angles of a pentagon are in the ratio of 3:3:3:4:5. What is the number of degrees in the measure of the largest angle? | 150^\circ | 1 | 1,260.25 | 1,260.25 | -1 | |
Convert the point $( -5, 0, -8 )$ in rectangular coordinates to cylindrical coordinates. Enter your answer in the form $(r,\theta,z),$ where $r > 0$ and $0 \le \theta < 2 \pi.$ | (5,\pi,-8) | 1 | 1,289.3125 | 1,289.3125 | -1 | |
Given the function $f(x)=2\sin x\cos (x+ \frac {π}{3})+ \frac { \sqrt {3}}{2}$.
(I) Find the interval(s) where the function $f(x)$ is monotonically decreasing;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac {π}{2}]$. | -\frac{\sqrt{3}}{2} | 0 | 4,444 | -1 | 4,444 | |
In triangle $XYZ$, $XY = 15$, $XZ = 35$, $YZ = 42$, and $XD$ is an angle bisector of $\angle XYZ$. Find the ratio of the area of triangle $XYD$ to the area of triangle $XZD$, and find the lengths of segments $XD$ and $ZD$. | 29.4 | 0 | 8,192 | -1 | 8,192 | |
Let $\triangle ABC$ be an equilateral triangle with side length $55.$ Points $D,$ $E,$ and $F$ lie on $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively, with $BD = 7,$ $CE=30,$ and $AF=40.$ Point $P$ inside $\triangle ABC$ has the property that \[\angle AEP = \angle BFP = \angle CDP.\] Find $\tan^2(\a... | 075 | We begin by using the fact stated in Solution 3 that, for any point in an equilateral triangle, the lengths of the three perpendicular lines dropped to the sides of the triangle add up to the altitude of that triangle. To make things simple, let's assign $\angle AEP = \angle BFP = \angle CDP = \alpha$. We can label the... | 0 | 7,873.0625 | -1 | 7,873.0625 |
Form a three-digit number without repeating digits using 1, 2, 3, 4, where the number of odd numbers is a certain number. | 12 | 0.625 | 5,559 | 4,612.7 | 7,136.166667 | |
A digital clock shows the time 4:56. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and are in increasing order? | 458 | We would like to find the first time after 4:56 where the digits are consecutive digits in increasing order. It would make sense to try 5:67, but this is not a valid time. Similarly, the time cannot start with 6, 7, 8, or 9. No time starting with 10 or 11 starts with consecutive increasing digits. Starting with 12, we ... | 0.0625 | 8,125.9375 | 7,135 | 8,192 |
Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then pours half the coffee from the first cup to the second and, after stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first cup ... | \frac{2}{5} | We will analyze the problem step by step to determine the fraction of cream in the first cup after the described operations.
#### Initial Setup:
- **Cup 1:** 4 ounces of coffee.
- **Cup 2:** 4 ounces of cream.
#### Step 1: Pour half the coffee from Cup 1 to Cup 2
- Amount of coffee transferred from Cup 1 to Cup 2 = $... | 0.75 | 4,902.6875 | 4,072.333333 | 7,393.75 |
Given:
$$
\begin{array}{l}
A \cup B \cup C=\{a, b, c, d, e, f\}, \\
A \cap B=\{a, b, c, d\}, \\
c \in A \cap B \cap C .
\end{array}
$$
How many sets $\{A, B, C\}$ satisfy the given conditions? | 200 | 0.0625 | 7,722.375 | 8,192 | 7,691.066667 | |
The solution of the equation $\dfrac{1+2^{x}}{1+x^{-x}}= \dfrac{1}{4}$ is $x=$ . | -2 | 0.6875 | 6,313.1875 | 5,459.181818 | 8,192 | |
Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$. | \frac{5}{3} | 1. **Identify Key Points and Relationships**:
- Points $A, B, C, D, E,$ and $F$ are collinear on line $\overline{AF}$, and each segment between consecutive points is of length 1.
- Point $G$ is not on line $AF$, and points $H$ and $J$ lie on lines $\overline{GD}$ and $\overline{GF}$ respectively.
- Lines $\ov... | 0.75 | 5,115.0625 | 4,089.416667 | 8,192 |
Given the parabola $y^{2}=2px\left(p \gt 0\right)$ with focus $F$ and the intersection point $E$ of its directrix with the $x$-axis, let line $l$ pass through $E$.<br/>$(1)$ If line $l$ is tangent to the parabola at point $M$, then $\angle EMF=$____.<br/>$(2)$ Let $p=6$. If line $l$ intersects the parabola at points $A... | 12 | 0.0625 | 8,139.3125 | 7,349 | 8,192 | |
Let $\alpha$ be a root of $x^6-x-1$ , and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$ . It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$ . Find the largest integer $n<728$ for which ... | 727 | 0 | 8,192 | -1 | 8,192 | |
Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation
\[x^2(x^2+2z) - y^2(y^2+2z)=a.\]
*Ray Li.*
<details><summary>Clarifications</summary>[list=1][*] $x,y,z$ are not necessarily distinct.[/list]</details> | 1257 | 0 | 8,117.6875 | -1 | 8,117.6875 | |
Given that the angles of a triangle at points \( A, B \), and \( C \) are such that \( \angle ABC = 50^\circ \) and \( \angle ACB = 30^\circ \), calculate the value of \( x \). | 80 | 0.0625 | 4,634.3125 | 5,453 | 4,579.733333 | |
An isosceles triangle has its vertex at $(0,5)$ and a base between points $(3,5)$ and $(13,5)$. The two equal sides are each 10 units long. If the third vertex (top vertex) is in the first quadrant, what is the y-coordinate? | 5 + 5\sqrt{3} | 0.1875 | 7,531.1875 | 4,691.333333 | 8,186.538462 | |
Given the curve E with the polar coordinate equation 4(ρ^2^-4)sin^2^θ=(16-ρ^2)cos^2^θ, establish a rectangular coordinate system with the non-negative semi-axis of the polar axis as the x-axis and the pole O as the coordinate origin.
(1) Write the rectangular coordinate equation of the curve E;
(2) If point P is a movi... | \sqrt{10} | 0.25 | 6,943.0625 | 5,758.5 | 7,337.916667 | |
If the solution set of the system of linear inequalities in one variable $x$, $\left\{{\begin{array}{l}{x-1≥2x+1}\\{2x-1＜a}\end{array}}\right.$, is $x\leqslant -2$, and the solution of the fractional equation in variable $y$, $\frac{{y-1}}{{y+1}}=\frac{a}{{y+1}}-2$, is negative, then the sum of all integers $a$ that sa... | -8 | 0.375 | 4,382.3125 | 5,474.5 | 3,727 | |
It is known that the equation $ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$ . | 1001000 | 0.875 | 3,362.375 | 2,901.285714 | 6,590 | |
\frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{17}{16} + \frac{33}{32} + \frac{65}{64} - 7 = | -\frac{1}{64} | 1. **Convert each fraction to have a common denominator of 64:**
- $\frac{3}{2} = \frac{3 \times 32}{2 \times 32} = \frac{96}{64}$
- $\frac{5}{4} = \frac{5 \times 16}{4 \times 16} = \frac{80}{64}$
- $\frac{9}{8} = \frac{9 \times 8}{8 \times 8} = \frac{72}{64}$
- $\frac{17}{16} = \frac{17 \times 4}{16 \times... | 1 | 990.6875 | 990.6875 | -1 |
The base of pyramid \( T ABCD \) is an isosceles trapezoid \( ABCD \) with the length of the shorter base \( BC \) equal to \( \sqrt{3} \). The ratio of the areas of the parts of the trapezoid \( ABCD \), divided by the median line, is \( 5:7 \). All the lateral faces of the pyramid \( T ABCD \) are inclined at an ang... | 0.875 | 0 | 8,192 | -1 | 8,192 | |
Find the sum of all real values of $x$ that satisfy
\[x = 1 - x + x^2 - x^3 + x^4 - x^5 + \dotsb.\] | \frac{-1 + \sqrt{5}}{2} | 0 | 4,551.5625 | -1 | 4,551.5625 | |
A volleyball net is in the shape of a rectangle with dimensions of $50 \times 600$ cells.
What is the maximum number of strings that can be cut so that the net does not fall apart into pieces? | 30000 | 0.0625 | 7,201.375 | 4,299 | 7,394.866667 | |
On the lateral edges \(AA_1\), \(BB_1\), and \(CC_1\) of a triangular prism \(ABC A_1 B_1 C_1\), points \(M\), \(N\), and \(P\) are located respectively such that \(AM: AA_1 = B_1N: BB_1 = C_1P: CC_1 = 3:4\). On the segments \(CM\) and \(A_1N\), points \(E\) and \(F\) are located respectively such that \(EF \parallel ... | 1/3 | 0.0625 | 7,636.6875 | 5,653 | 7,768.933333 | |
A certain brand specialty store is preparing to hold a promotional event during New Year's Day. Based on market research, the store decides to select 4 different models of products from 2 different models of washing machines, 2 different models of televisions, and 3 different models of air conditioners (different model... | 100 | 0.125 | 7,266.5625 | 4,658.5 | 7,639.142857 | |
The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and $x$ are all equal. What is the value of $x$? | 11 | To solve this problem, we need to find the value of $x$ such that the mean, median, and mode of the set $\{3, 4, 5, 6, 6, 7, x\}$ are all equal.
1. **Mode**: The mode is the number that appears most frequently in the set. Since $6$ appears twice and no other number appears more than once (unless $x$ is one of these nu... | 0.875 | 4,268.375 | 3,933.071429 | 6,615.5 |
Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:
[list]
[*]If at some stage the cards are in $m$ piles, she chooses $1\leq k<m$ and arranges the car... | 2^{n-2} |
Fiona starts with \( n \) cards labeled from 1 to \( n \) lined up in increasing order. Her task is to perform a series of pile rearrangements until she consolidates them into a single pile. Given the specific operation she can perform, we explore how many distinct final single pile sequences she can achieve.
**Under... | 0 | 8,192 | -1 | 8,192 |
Let \( S_{n} \) be the sum of the first \( n \) terms of an arithmetic sequence \( \{a_{n}\} \). Given \( S_{6}=36 \) and \( S_{n}=324 \). If \( S_{n-6} = 144 \) for \( n > 6 \), then \( n \) equals \(\qquad\). | 18 | 0.875 | 5,668.5625 | 5,308.071429 | 8,192 | |
How many positive odd integers greater than 1 and less than $200$ are square-free? | 80 | 0.1875 | 7,911.0625 | 7,503.333333 | 8,005.153846 | |
On the Cartesian grid, Johnny wants to travel from $(0,0)$ to $(5,1)$, and he wants to pass through all twelve points in the set $S=\{(i, j) \mid 0 \leq i \leq 1,0 \leq j \leq 5, i, j \in \mathbb{Z}\}$. Each step, Johnny may go from one point in $S$ to another point in $S$ by a line segment connecting the two points. H... | 252 | Observe that Johnny needs to pass through the points $(0,0),(1,0),(2,0), \ldots,(5,0)$ in that order, and he needs to pass through $(0,1),(1,1),(2,1), \ldots,(5,1)$ in that order, or else he will intersect his own path. Then, the problem is equivalent to interlacing those two sequence together, so that the first term i... | 0 | 8,140.75 | -1 | 8,140.75 |
Select three different numbers from the set {-8, -6, -4, 0, 3, 5, 7} to form a product and determine the smallest possible value. | -280 | 0.4375 | 1,891.5625 | 2,503.285714 | 1,415.777778 | |
Line segment $\overline{AB}$ is a diameter of a circle with $AB = 36$. Point $C$, not equal to $A$ or $B$, lies on the circle in such a manner that $\overline{AC}$ subtends a central angle less than $180^\circ$. As point $C$ moves within these restrictions, what is the area of the region traced by the centroid (center ... | 18\pi | 0.8125 | 4,968.4375 | 4,456.923077 | 7,185 | |
The fraction \(\frac{p}{q}\) is in its simplest form. If \(\frac{7}{10} < \frac{p}{q} < \frac{11}{15}\), where \(q\) is the smallest possible positive integer and \(c = pq\), find the value of \(c\). | 35 | 0.6875 | 5,594.75 | 4,414.181818 | 8,192 | |
Rectangle $PQRS$ lies in a plane with $PQ=RS=2$ and $QR=SP=6$. The rectangle is rotated $90^\circ$ clockwise about $R$, then rotated $90^\circ$ clockwise about the point $S$ moved to after the first rotation. What is the length of the path traveled by point $P$? | $(3+\sqrt{10})\pi$ | 1. **Identify the dimensions of the rectangle and the rotations:**
- Rectangle $PQRS$ has dimensions $PQ = RS = 2$ and $QR = SP = 6$.
- The rectangle undergoes two $90^\circ$ clockwise rotations: first about point $R$, then about the new position of point $S$.
2. **Calculate the radius of the circle traced by po... | 0 | 5,772.3125 | -1 | 5,772.3125 |
On the sides $AB$ and $AC$ of triangle $ABC$ lie points $K$ and $L$, respectively, such that $AK: KB = 4:7$ and $AL: LC = 3:2$. The line $KL$ intersects the extension of side $BC$ at point $M$. Find the ratio $CM: BC$. | 8 : 13 | 0 | 7,264.5 | -1 | 7,264.5 | |
Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\frac {m\sqrt {n}}{p},$ where $m,$ $n,$ and $p$ are positive integers... | 578 | We use the Pythagorean Theorem on $ABC$ to determine that $AB=25.$
Let $N$ be the orthogonal projection from $C$ to $AB.$ Thus, $[CDM]=\frac{(DM)(MN)} {2}$, $MN=AM-AN$, and $[ABC]=\frac{24 \cdot 7} {2} =\frac{25 \cdot (CN)} {2}.$
From the third equation, we get $CN=\frac{168} {25}.$
By the Pythagorean Theorem in $\D... | 0.3125 | 7,960.875 | 7,452.4 | 8,192 |
**polyhedral**
we call a $12$ -gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points $(0,1)$ , $(1,0)$ , $(-1,0)$ and $(0,-1)$ .
find the faces of the <u>massivest</u> polyh... | 36 | 0 | 8,150.8125 | -1 | 8,150.8125 | |
Given the function $f(x)$ defined on $\mathbb{R}$ and satisfying the condition $f(x+2) = 3f(x)$, when $x \in [0, 2]$, $f(x) = x^2 - 2x$. Find the minimum value of $f(x)$ when $x \in [-4, -2]$. | -\frac{1}{9} | 0.4375 | 5,800.1875 | 4,313.571429 | 6,956.444444 |
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