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 |
|---|---|---|---|---|---|---|
A balloon that inflates into the shape of a perfect cube is being blown up at a rate such that at time \( t \) (in fortnights), it has a surface area of \( 6t \) square furlongs. At what rate, in cubic furlongs per fortnight, is the air being pumped in when the surface area is 144 square furlongs? | 3\sqrt{6} | 1 | 3,014.9375 | 3,014.9375 | -1 | |
Five years from now, Billy's age will be twice Joe's current age. Currently, the sum of their ages is 60. How old is Billy right now? | 38\frac{1}{3} | 0.1875 | 4,960.375 | 3,637 | 5,265.769231 | |
The rules for a race require that all runners start at $A$, touch any part of the 1200-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. [asy]
import olympiad; import geometry; size(250);
defaultpen(linewidth(0.8));
draw((... | 1442 | 0.625 | 6,986.4375 | 6,263.1 | 8,192 | |
A fair coin is flipped $7$ times. What is the probability that at least $5$ consecutive flips come up heads? | \frac{1}{16} | 0 | 8,192 | -1 | 8,192 | |
Compute
\[\sum_{n = 1}^\infty \frac{1}{n(n + 2)}.\] | \frac{3}{4} | 1 | 4,517.6875 | 4,517.6875 | -1 | |
The magnitude of the vector $\overset{→}{a} +2 \overset{→}{b}$, where $\overset{→}{a} =(2,0)$, $\overset{→}{b}$ is a unit vector with a magnitude of 1 and the angle between the two vectors is $60^{\circ}$. | 2\sqrt{3} | 0.9375 | 3,734.625 | 3,437.466667 | 8,192 | |
Given the variance of a sample is $$s^{2}= \frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \_\_\_\_\_\_. | 60 | 0.5 | 3,258.6875 | 3,141.75 | 3,375.625 | |
On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to | 2.1 | 1. **Identify the vertices and their relationships**: In a $4 \times 4 \times 3$ rectangular parallelepiped, let's assume $D$ is at the origin $(0,0,0)$, and $A$, $B$, $C$ are at $(4,0,0)$, $(0,4,0)$, and $(0,0,3)$ respectively. These vertices are adjacent to $D$.
2. **Volume of pyramid $ABCD$**: The volume $V$ of a p... | 0 | 8,192 | -1 | 8,192 |
Given that the radius of circle \( \odot O \) is 1, the quadrilateral \( ABCD \) is an inscribed square, \( EF \) is a diameter of \( \odot O \), and \( M \) is a point moving along the boundary of the square \( ABCD \). Find the minimum value of \(\overrightarrow{ME} \cdot \overrightarrow{MF} \). | -1/2 | 0 | 7,992.375 | -1 | 7,992.375 | |
Given $f(\alpha)=\frac{2\sin(2\pi-\alpha)\cos(2\pi+\alpha)-\cos(-\alpha)}{1+\sin^{2}\alpha+\sin(2\pi+\alpha)-\cos^{2}(4\pi-\alpha)}$, find the value of $f\left(-\frac{23}{6}\pi \right)$. | -\sqrt{3} | 0.5625 | 6,836.5625 | 6,161.777778 | 7,704.142857 | |
In a geometric sequence of real numbers, the sum of the first $2$ terms is $7$, and the sum of the first $6$ terms is $91$. The sum of the first $4$ terms is | 32 | Let the first term of the geometric sequence be $a$ and the common ratio be $r$. Then the terms of the sequence are $a, ar, ar^2, ar^3, \ldots$.
1. **Sum of the first 2 terms:**
\[
a + ar = 7
\]
2. **Sum of the first 6 terms:**
\[
a + ar + ar^2 + ar^3 + ar^4 + ar^5 = 91
\]
We can express the sum... | 0 | 5,578.875 | -1 | 5,578.875 |
In the Cartesian coordinate system, given points A(1, -3), B(4, -1), P(a, 0), and N(a+1, 0), if the perimeter of the quadrilateral PABN is minimal, then find the value of a. | a = \frac{5}{2} | 0.5625 | 6,701.75 | 5,739.666667 | 7,938.714286 | |
Xiaoqiang conducts an experiment while drinking a beverage. He inserts a chopstick vertically into the bottom of the cup and measures the wetted part, which is exactly 8 centimeters. He then turns the chopstick around and inserts the other end straight into the bottom of the cup. He finds that the dry part of the chops... | 24 | 0.25 | 4,357.5 | 3,647.5 | 4,594.166667 | |
Let \( n \) be a two-digit number such that the square of the sum of the digits of \( n \) is equal to the sum of the digits of \( n^2 \). Find the sum of all possible values of \( n \). | 139 | 0 | 6,325.375 | -1 | 6,325.375 | |
Joe's quiz scores were 88, 92, 95, 81, and 90, and then he took one more quiz and scored 87. What was his mean score after all six quizzes? | 88.83 | 0.4375 | 4,108.875 | 2,709.142857 | 5,197.555556 | |
How many numbers are in the first $20$ rows of Pascal's Triangle (from the $0$th row to the $19$th row)? | 210 | 1 | 1,789.8125 | 1,789.8125 | -1 | |
Analogical reasoning is an important method of reasoning. Based on the similarity of two things in some characteristics, conclusions can be drawn that they may be similar in other characteristics. Reading perception: In addition and subtraction of fractions with different denominators, it is often necessary to first co... | -\frac{1011}{2023} | 0.4375 | 6,394.3125 | 5,379.142857 | 7,183.888889 | |
Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. What is the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$? | 8 \sqrt{2} | 0 | 8,192 | -1 | 8,192 | |
Convert the point $\left( 5, \frac{3 \pi}{2} \right)$ in polar coordinates to rectangular coordinates. | (0,-5) | 1 | 1,427.875 | 1,427.875 | -1 | |
Integer \( n \) such that the polynomial \( f(x) = 3x^3 - nx - n - 2 \) can be factored into a product of two non-constant polynomials with integer coefficients. Find the sum of all possible values of \( n \). | 192 | 0.1875 | 7,545.875 | 4,746 | 8,192 | |
Five positive integers from $1$ to $15$ are chosen without replacement. What is the probability that their sum is divisible by $3$ ? | 1/3 | 0.1875 | 7,951.6875 | 6,910.333333 | 8,192 | |
Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$.
| 60 | 0.0625 | 7,836.6875 | 6,923 | 7,897.6 | |
Given \( DE = 9 \), \( EB = 6 \), \( AB = \frac{1}{3}AD \), and \( FD = \frac{3}{4}AD \), find \( FC \). Assume triangle \( ADE \) is similar to triangle \( AFB \) and triangle \( AFC \). | 16.875 | 0 | 7,191.625 | -1 | 7,191.625 | |
The line passing through the points (3, 9) and (-1, 1) intersects the x-axis at a point whose x-coordinate is $\frac{9-1}{3-(-1)}$ | - \frac{3}{2} | 0.375 | 7,499.875 | 6,346.333333 | 8,192 | |
If \(a\), \(b\), and \(c\) are distinct positive integers such that \(abc = 16\), then the largest possible value of \(a^b - b^c + c^a\) is: | 263 | 1 | 5,599.75 | 5,599.75 | -1 | |
What is the sum of all positive integer solutions less than or equal to $20$ to the congruence $13(3x-2)\equiv 26\pmod 8$? | 36 | 1 | 2,276.8125 | 2,276.8125 | -1 | |
Solve the following quadratic equation: $x^2 + 5x - 4 = 0.$ | \frac{-5 - \sqrt{41}}{2} | 0 | 2,517.125 | -1 | 2,517.125 | |
The dimensions of a triangle are tripled to form a new triangle. If the area of the new triangle is 54 square feet, how many square feet were in the area of the original triangle? | 6 | 1 | 1,517.6875 | 1,517.6875 | -1 | |
What is the sum of the positive factors of 48? | 124 | 1 | 2,008.4375 | 2,008.4375 | -1 | |
Let $S = 1 - 2 + 3 - 4 + \cdots + 2009 - 2010$. What is the residue of $S$, modulo 2010? | 1005 | 1 | 3,295.125 | 3,295.125 | -1 | |
Given $tanA=\frac{2}{3}$, find the value of $\cos A$. | \frac{3\sqrt{13}}{13} | 0 | 1,145.375 | -1 | 1,145.375 | |
Evaluate $y(y-3x)$ for $x=3$ and $y=0$. | 0 | 1 | 1,326.125 | 1,326.125 | -1 | |
How many ways are there to divide a group of 6 friends among the basketball team, the soccer team, and the track team? (Each team could have anywhere from 0 to 6 of the friends on it. Assume the friends are distinguishable.) | 729 | 0.9375 | 1,991.3125 | 1,577.933333 | 8,192 | |
To test the quality of a certain product, it was decided to use the random number table method to draw 5 samples from 300 products for inspection. The products are numbered from 000, 001, 002, ..., to 299. The following are the 7th and 8th rows of the random number table. If we start from the 5th number in the 7th row ... | 057 | 0 | 6,212.375 | -1 | 6,212.375 | |
Let \( x, y, z \) be nonnegative real numbers. Define:
\[
A = \sqrt{x + 3} + \sqrt{y + 6} + \sqrt{z + 12},
\]
\[
B = \sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}.
\]
Find the minimum value of \( A^2 - B^2 \). | 36 | 0 | 8,192 | -1 | 8,192 | |
If several students participate in three competitions where the champion earns 5 points, the runner-up earns 3 points, and the third-place finisher earns 1 point, and there are no ties, what is the minimum score a student must achieve to definitely have a higher score than any other student?
(The 7th American Junior ... | 13 | 0 | 7,809.9375 | -1 | 7,809.9375 | |
Find the $y$-intercepts of the following system of equations:
1. $2x - 3y = 6$
2. $x + 4y = -8$ | -2 | 0.5 | 1,800.5625 | 1,257.75 | 2,343.375 | |
If $ rac{x-y}{z-y}=-10$, what is the value of $ rac{x-z}{y-z}$? | 11 | Since the problem asks us to find the value of $ rac{x-z}{y-z}$, then this value must be the same no matter what $x, y$ and $z$ we choose that satisfy $ rac{x-y}{z-y}=-10$. Thus, if we can find numbers $x, y$ and $z$ that give $ rac{x-y}{z-y}=-10$, then these numbers must give the desired value for $ rac{x-z}{y-z}$. If... | 1 | 1,562.375 | 1,562.375 | -1 |
When three positive integers are added in pairs, the resulting sums are 998, 1050, and 1234. What is the difference between the largest and smallest of the three original positive integers? | 236 | Suppose that the three integers are $x, y$ and $z$ where $x+y=998$, $x+z=1050$, and $y+z=1234$. From the first two equations, $(x+z)-(x+y)=1050-998$ or $z-y=52$. Since $z+y=1234$ and $z-y=52$, then $(z+y)+(z-y)=1234+52$ or $2z=1286$ and so $z=643$. Since $z=643$ and $z-y=52$, then $y=z-52=643-52=591$. Since $x+y=998$ a... | 1 | 2,407.9375 | 2,407.9375 | -1 |
Simplify $\cos 36^\circ - \cos 72^\circ.$ | \frac{1}{2} | 0.75 | 6,926.8125 | 6,505.083333 | 8,192 | |
On each side of an equilateral triangle, a point is taken. The sides of the triangle with vertices at these points are perpendicular to the sides of the original triangle. In what ratio does each of these points divide the side of the original triangle? | 1:2 | 0 | 6,578.375 | -1 | 6,578.375 | |
Rationalize the denominator of \(\frac{5}{4\sqrt{7} - 3\sqrt{2}}\) and write your answer in the form \(\displaystyle \frac{A\sqrt{B} + C\sqrt{D}}{E}\), where \(B < D\), the fraction is in lowest terms, and all radicals are in simplest radical form. What is \(A+B+C+D+E\)? | 138 | 0.9375 | 4,331.5625 | 4,074.2 | 8,192 | |
Find the area of the circle inscribed in a right triangle if the projections of the legs onto the hypotenuse are 9 meters and 16 meters, respectively. | 25 \pi | 0.75 | 4,199 | 3,982.333333 | 4,849 | |
There are 100 black balls and 100 white balls in a box. What is the minimum number of balls that need to be drawn, without looking, to ensure that there are at least 2 balls of the same color? To ensure that there are at least 2 white balls? | 102 | 0.4375 | 4,223.4375 | 3,476.714286 | 4,804.222222 | |
If $\log_3 (x+5)^2 + \log_{1/3} (x - 1) = 4,$ compute $x.$ | \frac{71 + \sqrt{4617}}{2} | 0 | 7,402.5 | -1 | 7,402.5 | |
A set \( \mathcal{T} \) of distinct positive integers has the following property: for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 1024 is the largest element of ... | 15 | 0 | 8,192 | -1 | 8,192 | |
Given that point \( P \) lies on the hyperbola \( \Gamma: \frac{x^{2}}{463^{2}} - \frac{y^{2}}{389^{2}} = 1 \). A line \( l \) passes through point \( P \) and intersects the asymptotes of hyperbola \( \Gamma \) at points \( A \) and \( B \), respectively. If \( P \) is the midpoint of segment \( A B \) and \( O \) is ... | 180107 | 0.125 | 8,192 | 8,192 | 8,192 | |
How many triangles with positive area can be formed where each vertex is at point $(i,j)$ in the coordinate grid, with integers $i$ and $j$ ranging from $1$ to $4$ inclusive? | 516 | 0.0625 | 8,163.4375 | 7,735 | 8,192 | |
Let $A, E, H, L, T$, and $V$ be chosen independently and at random from the set $\left\{0, \frac{1}{2}, 1\right\}$. Compute the probability that $\lfloor T \cdot H \cdot E\rfloor=L \cdot A \cdot V \cdot A$. | \frac{55}{81} | There are $3^{3}-2^{3}=19$ ways to choose $L, A$, and $V$ such that $L \cdot A \cdot V \cdot A=0$, since at least one of $\{L, A, V\}$ must be 0 , and $3^{3}-1=26$ ways to choose $T, H$, and $E$ such that $\lfloor T \cdot H \cdot E\rfloor=0$, since at least one of $\{T, H, E\}$ must not be 1 , for a total of $19 \cdot ... | 0 | 8,032.625 | -1 | 8,032.625 |
A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \leq 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$? | \frac{50}{99} | 1. **Understanding the Problem**: We need to find the maximum value of $a$ such that the line $y = mx + 2$ does not pass through any lattice points for $0 < x \leq 100$ and $\frac{1}{2} < m < a$.
2. **Impact of the Constant Term**: The constant term "+2" shifts the line vertically but does not affect the slope. Theref... | 0.1875 | 8,087.75 | 7,663.666667 | 8,185.615385 |
Determine the product of the real parts of the solutions to the equation \(2x^2 + 4x = 1 + i\). | \frac{1 - 3}{4} | 0 | 7,978.75 | -1 | 7,978.75 | |
The letter T is formed by placing two $3\:\text{inch}\!\times\!5\:\text{inch}$ rectangles to form a T shape. The vertical rectangle is placed in the middle of the horizontal one, overlapping it by $1.5$ inches on both sides. What is the perimeter of the new T, in inches? | 20 | 0.125 | 7,936.375 | 7,229 | 8,037.428571 | |
Compute
\[\sin^2 6^\circ + \sin^2 12^\circ + \sin^2 18^\circ + \dots + \sin^2 174^\circ.\] | \frac{31}{2} | 0 | 6,946.5625 | -1 | 6,946.5625 | |
Given the function $f(x) = x^3 + ax^2 + bx + a^2$ has an extremum at $x = 1$ with the value of 10, find the values of $a$ and $b$. | -11 | 0.75 | 4,227.1875 | 3,896.666667 | 5,218.75 | |
The lock opens only if a specific three-digit number is entered. An attempt consists of randomly selecting three digits from a given set of five. The code was guessed correctly only on the last of all attempts. How many attempts preceded the successful one? | 124 | 0.75 | 3,333.8125 | 2,426.333333 | 6,056.25 | |
Seven standard dice are glued together to make a solid. The pairs of faces of the dice that are glued together have the same number of dots on them. How many dots are on the surface of the solid? | 105 | 0.0625 | 7,682.8125 | 7,854 | 7,671.4 | |
Crestwood Elementary School has an active four-square league, consisting of ten players, including Justin and Tim. Each day at recess, the ten players split into two four-square games, each with five players in no relevant order. Over the course of a semester, each possible match-up of five players occurs once. How man... | 56 | 0.6875 | 4,789.4375 | 3,764.090909 | 7,045.2 | |
It takes Pearl 7 days to dig 4 holes. It takes Miguel 3 days to dig 2 holes. If they work together and each continues digging at these same rates, how many holes in total will they dig in 21 days? | 26 | Since Pearl digs 4 holes in 7 days and $\frac{21}{7}=3$, then in 21 days, Pearl digs $3 \cdot 4=12$ holes. Since Miguel digs 2 holes in 3 days and $\frac{21}{3}=7$, then in 21 days, Miguel digs $7 \cdot 2=14$ holes. In total, they dig $12+14=26$ holes in 21 days. | 1 | 562.625 | 562.625 | -1 |
Observe the sequence: (1), (4, 7), (10, 13, 16), (19, 22, 25, 28), ..., then 2008 is in the $\boxed{\text{th}}$ group. | 37 | 0.375 | 7,367.25 | 6,659.5 | 7,791.9 | |
Given a parabola $y^2=2px$ ($p>0$) with focus $F$, a circle is drawn with $F$ as the center and $p$ as the radius, intersecting the $y$-axis at points $A$ and $B$. Connect $F$ to $A$, intersecting the parabola at point $D$ (which lies on segment $FA$). Extend $FA$ to intersect the parabola's directrix at point $C$. If ... | 32 | 0.875 | 5,591.875 | 5,220.428571 | 8,192 | |
Let $S$ be the set of all points in the plane whose coordinates are positive integers less than or equal to 100 (so $S$ has $100^{2}$ elements), and let $\mathcal{L}$ be the set of all lines $\ell$ such that $\ell$ passes through at least two points in $S$. Find, with proof, the largest integer $N \geq 2$ for which it ... | 4950 | Let the lines all have slope $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Without loss of generality, let this slope be positive. Consider the set of points that consists of the point of $S$ with the smallest coordinates on each individual line in the set $L$. Consider a point $(x, y)$ in this, because there i... | 0 | 7,853.875 | -1 | 7,853.875 |
There are exactly 120 ways to color five cells in a \( 5 \times 5 \) grid such that each row and each column contains exactly one colored cell.
There are exactly 96 ways to color five cells in a \( 5 \times 5 \) grid without the corner cell such that each row and each column contains exactly one colored cell.
How man... | 78 | 0.25 | 7,483 | 5,356 | 8,192 | |
Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated. | 300 | 0.25 | 5,951.5 | 5,284 | 6,174 | |
The radius of a sphere is $p$ units and the radius of a hemisphere is $2p$ units. What is the ratio of the volume of the sphere to the volume of the hemisphere? | \frac{1}{4} | 1 | 2,555.875 | 2,555.875 | -1 | |
What is the largest four-digit number whose digits add up to 16? | 9700 | 0.5 | 7,321.3125 | 6,450.625 | 8,192 | |
A spinner is divided into six congruent sectors, numbered from 1 to 6. Jane and her brother each spin the spinner once. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? | \frac{5}{6} | 0.5625 | 7,006.875 | 6,396.777778 | 7,791.285714 | |
There is a positive integer $n$ such that $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$? | 10 |
We start with the given equation:
\[
(n+1)! + (n+2)! = n! \cdot 440
\]
First, we express the factorials in terms of \(n!\):
\[
(n+1)! = (n+1)n!
\]
\[
(n+2)! = (n+2)(n+1)n!
\]
Substituting these into the equation, we get:
\[
(n+1)n! + (n+2)(n+1)n! = 440n!
\]
Factoring out \((n+1)n!\) from the left side:
\[
(n+1)n!(1... | 0.75 | 3,720.4375 | 2,229.916667 | 8,192 |
In the diagram, \(\triangle ABC\) is right-angled at \(C\). Point \(D\) is on \(AC\) so that \(\angle ABC = 2 \angle DBC\). If \(DC = 1\) and \(BD = 3\), determine the length of \(AD\). | \frac{9}{7} | 0.875 | 5,435.875 | 5,042.142857 | 8,192 | |
In the triangular prism $P-ABC$, the three edges $PA$, $PB$, and $PC$ are mutually perpendicular, with $PA=1$, $PB=2$, and $PC=2$. If $Q$ is any point on the circumsphere of the triangular prism $P-ABC$, what is the maximum distance from $Q$ to the plane $ABC$? | \frac{3}{2} + \frac{\sqrt{6}}{6} | 0 | 8,192 | -1 | 8,192 | |
Among all the simple fractions with a numerator and denominator that are two-digit numbers, find the smallest fraction greater than $\frac{3}{4}$. Provide its numerator in the answer. | 73 | 0 | 8,192 | -1 | 8,192 | |
The MathMatters competition consists of 10 players $P_1$ , $P_2$ , $\dots$ , $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$ : the loser is ranked 10th, while the winner plays $P_8$ . The loser of that game is ranked 9th, while the winner plays $P_7$ . They keep rep... | 512 | 0.4375 | 7,225.0625 | 6,002.428571 | 8,176 | |
If the domain of functions $f(x)$ and $g(x)$ is $R$, and $\frac{f(x)}{g(x)}=\frac{g(x+2)}{f(x-2)}$, and $\frac{f(2022)}{g(2024)}=2$, then $\sum_{k=0}^{23}\frac{f(2k)}{g(2k+2)}=\_\_\_\_\_\_$. | 30 | 0.1875 | 7,403.4375 | 5,485 | 7,846.153846 | |
What is the smallest base-10 integer that can be represented as $XX_6$ and $YY_8$, where $X$ and $Y$ are valid digits in their respective bases? | 63 | 0 | 7,597.9375 | -1 | 7,597.9375 | |
What is the largest four-digit negative integer congruent to $1 \pmod{17}?$ | -1002 | 0.25 | 6,728.8125 | 5,106.5 | 7,269.583333 | |
For how many pairs of consecutive integers in $\{3000,3001,3002,\ldots,4000\}$ is no borrowing required when the first integer is subtracted from the second? | 1000 | 0.0625 | 8,077.75 | 8,192 | 8,070.133333 | |
How many integers fall between $\sqrt5$ and $\sqrt{50}$ on a number line? | 5 | 0.9375 | 3,095.375 | 3,227.066667 | 1,120 | |
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square? | \frac{\pi}{8} | 1 | 2,020.1875 | 2,020.1875 | -1 | |
Trevor and Edward play a game in which they take turns adding or removing beans from a pile. On each turn, a player must either add or remove the largest perfect square number of beans that is in the heap. The player who empties the pile wins. For example, if Trevor goes first with a pile of 5 beans, he can either add ... | 0, 5, 20, 29, 45, 80, 101, 116, 135, 145, 165, 173, 236, 257, 397, 404, 445, 477, 540, 565, 580, 629, 666, 836, 845, 885, 909, 944, 949, 954, 975 | The correct answers are 0 (worth imaginary points), 5 (worth 0 points), 20 (4 points), 29, 45 (5 points), 80 (6 points), 101, 116, 135, 145, 165, 173 (7 points), 236, 257 (8 points), 397, 404, 445, 477, 540, 565, 580, 629, 666 (9 points), 836, 845, 885, 909, 944, 949, 954, 975 (10 points). This game is called Epstein's... | 0 | 8,192 | -1 | 8,192 |
Given the function $f(x)$ that satisfies $f\left(\frac{\pi}{2} - x\right) + f(x) = 0$ and $f(\pi + x) = f(-x)$, calculate the value of $f\left(\frac{79\pi}{24}\right)$. | \frac{\sqrt{2} - \sqrt{6}}{4} | 0 | 7,877.9375 | -1 | 7,877.9375 | |
In a square of side length 4 , a point on the interior of the square is randomly chosen and a circle of radius 1 is drawn centered at the point. What is the probability that the circle intersects the square exactly twice? | \frac{\pi+8}{16} | Consider the two intersection points of the circle and the square, which are either on the same side of the square or adjacent sides of the square. In order for the circle to intersect a side of the square twice, it must be at distance at most 1 from that side and at least 1 from all other sides. The region of points w... | 0 | 8,164.4375 | -1 | 8,164.4375 |
Points $A$, $B$, $Q$, $D$, and $C$ lie on the circle shown and the measures of arcs $BQ$ and $QD$ are $42^\circ$ and $38^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$, in degrees.
[asy]
import graph;
unitsize(2 cm);
pair A, B, C, D, P, Q;
A = dir(160);
B = dir(45);
C = dir(190);
D = dir(... | 40^\circ | 0 | 7,807.1875 | -1 | 7,807.1875 | |
In triangle \(XYZ,\) \(XY = 5,\) \(XZ = 7,\) \(YZ = 9,\) and \(W\) lies on \(\overline{YZ}\) such that \(\overline{XW}\) bisects \(\angle YXZ.\) Find \(\cos \angle YXW.\) | \frac{3\sqrt{5}}{10} | 0 | 6,356.0625 | -1 | 6,356.0625 | |
Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \times 3$ grid, and then removing the center copy. Let $a_{n}$ be the expected value of $\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|$, whe... | 1217 | By symmetry, we only need to consider the $x$-distance, then we can multiply our answer by 2. Let this quantity be $g(n)=a_{n} / 2$. Divide the $n$th iteration fractal into three meta-columns of equal width. Then the probability that a random point is in the first, second, and third meta-columns is $\frac{3}{8}, \frac{... | 0 | 8,192 | -1 | 8,192 |
Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to det... | 2020 | 0 | 7,942.625 | -1 | 7,942.625 | |
Three cards are dealt successively without replacement from a standard deck of 52 cards. What is the probability that the first card is a $\heartsuit$, the second card is a King, and the third card is a $\spadesuit$? | \frac{13}{2550} | 0.125 | 7,798.3125 | 5,502.5 | 8,126.285714 | |
Given points E and D are on sides AB and BC of triangle ABC, where AE:EB=1:3 and CD:DB=1:2, find the value of EF/FC + AF/FD. | \frac{3}{2} | 0.0625 | 7,672.6875 | 7,552 | 7,680.733333 | |
Sides $AB$, $BC$, and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4$, $5$, and $20$, respectively.
If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B = \frac{3}{5}$, then side $AD$ has length
A polygon is called “simple” if it is not self intersecting. | 25 | 1. **Given Information and Angle Relationships**:
- Sides $AB = 4$, $BC = 5$, and $CD = 20$.
- Angles $B$ and $C$ are obtuse.
- $\sin C = -\cos B = \frac{3}{5}$.
Since $B$ and $C$ are obtuse, we have:
\[
\sin(180^\circ - C) = \sin C = \frac{3}{5} \quad \text{and} \quad \cos(180^\circ - B) = -\cos B =... | 0.375 | 7,547.8125 | 6,474.166667 | 8,192 |
Robert likes chocolate milk, so he decides to visit the milk bottling plant every day for a week to get the free samples. Unfortunately for him, the bottling plant sometimes bottles regular milk instead of chocolate milk, so each day the plant has a 2/3 chance of bottling chocolate milk. What is the probability that th... | \frac{80}{243} | 1 | 2,901.4375 | 2,901.4375 | -1 | |
What is the greatest integer $x$ for which $\frac79 > \frac{x}{13}$? | 10 | 1 | 1,863.3125 | 1,863.3125 | -1 | |
A card is chosen at random from a standard deck of 52 cards, and then it is replaced and another card is chosen. What is the probability that at least one of the cards is a diamond or an ace? | \frac{88}{169} | 0.8125 | 3,972.5 | 2,998.769231 | 8,192 | |
Given the function $f(x) = e^{x} \cos x - x$.
(I) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$. | -\frac{\pi}{2} | 0.8125 | 5,454.75 | 4,823.076923 | 8,192 | |
Hooligan Vasya loves running on the metro escalator, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving down, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down on an e... | 324 | 0.1875 | 7,445.8125 | 4,843 | 8,046.461538 | |
A merchant buys goods at $25\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\%$ on the marked price and still clear a profit of $25\%$ on the selling price. What percent of the list price must he mark the goods? | 125\% | 1. **Setting the List Price**: Assume the list price of the goods is $L = 100$ units. This simplification does not affect the generality of the problem since we are asked for a percentage.
2. **Calculating the Purchase Price**: The merchant buys the goods at a $25\%$ discount. Therefore, the purchase price is:
\[
... | 1 | 3,261 | 3,261 | -1 |
Shown below are rows 1, 2, and 3 of Pascal's triangle.
\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]Let $(a_i),$ $(b_i),$ $(c_i)$ be the sequence, from left to right, of elements in the 2005th, 2006th, and 2007th rows, respectively, with the leftmost element occur... | \frac{1}{2} | 0.625 | 6,338.875 | 5,247 | 8,158.666667 | |
For how many real numbers $a^{}_{}$ does the quadratic equation $x^2 + ax^{}_{} + 6a=0$ have only integer roots for $x^{}_{}$? | 10 | 0.5625 | 6,642.375 | 5,437.111111 | 8,192 | |
Consider a new infinite geometric series: $$\frac{7}{4} + \frac{28}{9} + \frac{112}{27} + \dots$$
Determine the common ratio of this series. | \frac{16}{9} | 0 | 8,192 | -1 | 8,192 | |
Suppose I have 6 shirts, 4 ties, and 3 pairs of pants. If an outfit requires a shirt and pants, and can either have a tie or not have a tie, how many outfits can I make? | 90 | 0.75 | 3,745.3125 | 2,298.166667 | 8,086.75 | |
In Rivertown, car plates each contain three symbols: two letters followed by a digit. The first letter is chosen from the set ${A, B, G, H, T}$, the second letter from ${E, I, O, U}$, and the digit from $0$ to $9$.
To accommodate an increase in the number of cars, Rivertown decides to expand each set by adding new sy... | 130 | 0.0625 | 7,967.1875 | 6,741 | 8,048.933333 | |
If the function $f(x) = \tan(2x - \frac{\pi}{6})$, then the smallest positive period of $f(x)$ is \_\_\_\_\_\_; $f\left(\frac{\pi}{8}\right)=$ \_\_\_\_\_\_. | 2 - \sqrt{3} | 1 | 2,297.75 | 2,297.75 | -1 | |
How many positive 3-digit numbers are divisible by 11? | 81 | 0.9375 | 2,586.3125 | 2,212.6 | 8,192 |
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