problem stringlengths 10 5.15k | answer dict |
|---|---|
For positive integers \( n \), let \( g(n) \) return the smallest positive integer \( k \) such that \( \frac{1}{k} \) has exactly \( n \) digits after the decimal point in base 6 notation. Determine the number of positive integer divisors of \( g(2023) \). | {
"answer": "4096576",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ that has center $I$ . If $IA = 5, IB = 7, IC = 4, ID = 9$ , find the value of $\frac{AB}{CD}$ . | {
"answer": "35/36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\bigtriangleup ACE$. Express your answer in simplest radical form. | {
"answer": "\\frac{9\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the xy-plane with a rectangular coordinate system, the terminal sides of angles $\alpha$ and $\beta$ intersect the unit circle at points $A$ and $B$, respectively.
1. If point $A$ is in the first quadrant with a horizontal coordinate of $\frac{3}{5}$ and point $B$ has a vertical coordinate of $\frac{12}{13}$, find the value of $\sin(\alpha + \beta)$.
2. If $| \overrightarrow{AB} | = \frac{3}{2}$ and $\overrightarrow{OC} = a\overrightarrow{OA} + \overrightarrow{OB}$, where $a \in \mathbb{R}$, find the minimum value of $| \overrightarrow{OC} |$. | {
"answer": "\\frac{\\sqrt{63}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle XYZ has legs XY = YZ = 8 cm. In each step of an iterative process, the triangle is divided into four smaller right triangles by joining the midpoints of the sides. However, for this problem, the area of the shaded triangle in each iteration is reduced by a factor of 3 rather than 4. If this process is repeated indefinitely, calculate the total area of the shaded triangles. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the digits 0, 1, 2, 3, 4, 5, if repetition of digits is not allowed, the number of different five-digit numbers that can be formed, which are divisible by 5 and do not have 3 as the hundred's digit, is ______. | {
"answer": "174",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\sin (\omega x+\dfrac{\pi }{3})$ ($\omega > 0$), $f(\dfrac{\pi }{6})=f(\dfrac{\pi }{3})$, and $f(x)$ has a minimum value but no maximum value in the interval $(\dfrac{\pi }{6},\dfrac{\pi }{3})$, find the value of $\omega$. | {
"answer": "\\dfrac{14}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than $1000$ are either a perfect cube or a perfect square? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \ge 4. $$ What is the maximum possible size for the set $S?$ | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M$ denote the number of $8$-digit positive integers where the digits are in non-decreasing order. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed, and the digit zero can now be used.) | {
"answer": "310",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many multiples of 5 are between 100 and 400? | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, let the left and right foci of the ellipse be $F_1$ and $F_2$, respectively. The line passing through $F_1$ and perpendicular to the x-axis intersects the ellipse at points $A$ and $B$. If the line $AF_2$ intersects the ellipse at another point $C$, and the area of triangle $\triangle ABC$ is three times the area of triangle $\triangle BCF_2$, determine the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What number corresponds to the point \( P \) indicated on the given scale? | {
"answer": "12.50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer $n$ such that there are at least $1000$ unordered pairs of diagonals in a regular polygon with $n$ vertices that intersect at a right angle in the interior of the polygon. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of \(x\) if \(x\) is positive and \(x \cdot \lfloor x \rfloor = 90\). Express your answer as a decimal. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least real number $K$ such that for all real numbers $x$ and $y$ , we have $(1 + 20 x^2)(1 + 19 y^2) \ge K xy$ . | {
"answer": "8\\sqrt{95}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius \( 2 \) cm is inscribed in \( \triangle ABC \). Let \( D \) and \( E \) be the points of tangency of the circle with the sides \( AC \) and \( AB \), respectively. If \( \angle BAC = 45^{\circ} \), find the length of the minor arc \( DE \). | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\triangle GHI$ is inscribed inside $\triangle XYZ$ such that $G, H, I$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle GZC, \triangle HYD, \triangle IXF$ have centers $O_1, O_2, O_3$, respectively. Also, $XY = 26, YZ = 28, XZ = 27$, and $\stackrel{\frown}{YI} = \stackrel{\frown}{GZ},\ \stackrel{\frown}{XI} = \stackrel{\frown}{HZ},\ \stackrel{\frown}{XH} = \stackrel{\frown}{GY}$. The length of $GY$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$. | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two distinct positive integers \( x \) and \( y \) are factors of 48. If \( x \cdot y \) is not a factor of 48, what is the smallest possible value of \( x \cdot y \)? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{28}/R_8$ when $R_{28}$ is divided by $R_8$. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express \( 0.3\overline{45} \) as a common fraction. | {
"answer": "\\frac{83}{110}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|\vec{a}| = 2$, $|\vec{b}| = 1$, and $(2\vec{a} - 3\vec{b}) \cdot (2\vec{a} + \vec{b}) = 9$.
(I) Find the angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$;
(II) Find $|\vec{a} + \vec{b}|$ and the projection of vector $\vec{a}$ in the direction of $\vec{a} + \vec{b}$. | {
"answer": "\\frac{5\\sqrt{7}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I randomly choose an integer \( p \) between \( 1 \) and \( 20 \) inclusive. What is the probability that \( p \) is such that there exists an integer \( q \) so that \( p \) and \( q \) satisfy the equation \( pq - 6p - 3q = 3 \)? Express your answer as a common fraction. | {
"answer": "\\frac{3}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the diagram, there is a sequence of curves \(P_{0}, P_{1}, P_{2}, \cdots\). It is given that \(P_{0}\) is an equilateral triangle with an area of 1. Each \(P_{k+1}\) is obtained from \(P_{k}\) by performing the following operations: each side of \(P_{k}\) is divided into three equal parts, an equilateral triangle is constructed outwards on the middle segment of each side, and the middle segments are then removed (\(k=0,1,2, \cdots\)). Let \(S_{n}\) denote the area enclosed by the curve \(P_{n}\).
1. Find a general formula for the sequence \(\{S_{n}\}\).
2. Evaluate \(\lim _{n \rightarrow \infty} S_{n}\). | {
"answer": "\\frac{8}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let point $O$ be the origin of a two-dimensional coordinate system, and let points $A$ and $B$ be located on positive $x$ and $y$ axes, respectively. If $OA = \sqrt[3]{54}$ and $\angle AOB = 45^\circ,$ compute the length of the line segment $AB.$ | {
"answer": "54^{1/3} \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets \( P=\left\{m^{2}-4, m+1,-3\right\} \) and \( Q=\{m-3, 2m-1, 3m+1\} \), if \( P \cap Q = \{-3\} \), find the value of the real number \( m \). | {
"answer": "-\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the non-negative real numbers $x$ and $y$ satisfy $x^{2}+4y^{2}+4xy+4x^{2}y^{2}=32$, find the minimum value of $x+2y$, and the maximum value of $\sqrt{7}(x+2y)+2xy$. | {
"answer": "4\\sqrt{7}+4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\omega$ be a circle with radius $1$ . Equilateral triangle $\vartriangle ABC$ is tangent to $\omega$ at the midpoint of side $BC$ and $\omega$ lies outside $\vartriangle ABC$ . If line $AB$ is tangent to $\omega$ , compute the side length of $\vartriangle ABC$ . | {
"answer": "\\frac{2 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4x$, and the eccentricity of ellipse $E$ is $\frac{\sqrt{2}}{2}$. A line $l$ with a non-zero slope passes through point $M(m, 0)$ ($m > \frac{3}{4}$) and intersects ellipse $E$ at points $A$ and $B$. Point $P(\frac{5}{4}, 0)$, and $\overrightarrow{PA} \cdot \overrightarrow{PB}$ is a constant.
(Ⅰ) Find the equation of ellipse $E$;
(Ⅱ) Find the maximum value of the area of $\triangle OAB$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), it is given that \( \angle C=90^\circ \), \( \angle A=60^\circ \), and \( AC=1 \). Points \( D \) and \( E \) are on sides \( BC \) and \( AB \) respectively such that triangle \( \triangle ADE \) is an isosceles right triangle with \( \angle ADE=90^\circ \). Find the length of \( BE \). | {
"answer": "4-2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Equilateral $\triangle DEF$ has side length $300$. Points $R$ and $S$ lie outside the plane of $\triangle DEF$ and are on opposite sides of the plane. Furthermore, $RA=RB=RC$, and $SA=SB=SC$, and the planes containing $\triangle RDE$ and $\triangle SDE$ form a $150^{\circ}$ dihedral angle. There is a point $M$ whose distance from each of $D$, $E$, $F$, $R$, and $S$ is $k$. Find $k$. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 1 belongs to $\mathcal{T}$ and that 1764 is the largest element of $\mathcal{T},$ what is the greatest number of elements that $\mathcal{T}$ can have? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least $n$ such that any subset of ${1,2,\dots,100}$ with $n$ elements has 2 elements with a difference of 9. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree. | {
"answer": "2500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ be an odd function defined on $(-\infty, +\infty)$, and $f(x+2) = -f(x)$. Given that $f(x) = x$ for $0 \leq x \leq 1$, find $f(3\pi)$. | {
"answer": "10 - 3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall, and there is exactly one player with
every possible height-weight combination.
The USAMTS wants to field a competitive team, so there are some strict requirements.
- If person $P$ is on the team, then anyone who is at least as tall and at most as heavy as $P$ must also be on the team.
- If person $P$ is on the team, then no one whose weight is the same as $P$ ’s height can also be on the team.
Assuming the USAMTS team can have any number of members (including zero), how many different basketball teams can be constructed? | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In circle $O$ with radius 10 units, chords $AC$ and $BD$ intersect at right angles at point $P$. If $BD$ is a diameter of the circle, and the length of $PC$ is 3 units, calculate the product $AP \cdot PB$. | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. Calculate the length of $QR$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
*2021 CCA Math Bonanza Individual Round #8* | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest natural number from which it is impossible to obtain a number divisible by 11 by deleting some of its digits. | {
"answer": "987654321",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than $1000$ are either a perfect cube or a perfect square? | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set of positive integers is said to be pilak if it can be partitioned into 2 disjoint subsets \(F\) and \(T\), each with at least 2 elements, such that the elements of \(F\) are consecutive Fibonacci numbers, and the elements of \(T\) are consecutive triangular numbers. Find all positive integers \(n\) such that the set containing all the positive divisors of \(n\) except \(n\) itself is pilak. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a normal vector of the straight line $l$ is $\overrightarrow{n} = (1, -\sqrt{3})$, find the size of the inclination angle of this straight line. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle $DEF$ where leg $DE = 30$ and leg $EF = 40$, determine the number of line segments with integer length that can be drawn from vertex $E$ to a point on hypotenuse $\overline{DF}$. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select the shape of diagram $b$ from the regular hexagonal grid of diagram $a$. There are $\qquad$ different ways to make the selection (note: diagram $b$ can be rotated). | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of the first is at least half the mass of the second. What is the maximum number of snowmen that the children will be able to make? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A frog starts climbing up a 12-meter deep well at 8 AM. For every 3 meters it climbs up, it slips down 1 meter. The time it takes to slip 1 meter is one-third of the time it takes to climb 3 meters. At 8:17 AM, the frog reaches 3 meters from the top of the well for the second time. How many minutes does it take for the frog to climb from the bottom of the well to the top? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jennifer wants to do origami, and she has a square of side length $ 1$ . However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the octagon Jennifer obtains? | {
"answer": "1 - \\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a committee composed of 10 members who meet around a table: 7 women, 2 men, and 1 child. The women sit in indistinguishable rocking chairs, the men on indistinguishable stools, and the child on a bench, also indistinguishable from any other benches. How many distinct ways are there to arrange the seven rocking chairs, two stools, and one bench for a meeting? | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a table consisting of $n$ rows and $m$ columns, numbers are written such that the sum of the elements in each row is 1248, and the sum of the elements in each column is 2184. Find the numbers $n$ and $m$ for which the expression $2n - 3m$ takes the smallest possible natural value. In the answer, specify the value of $n + m$. | {
"answer": "143",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \ldots + k^2$ is a multiple of $360$. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In any isosceles triangle $PQR$ with $PQ = PR$, the altitude $PS$ bisects the base $QR$ so that $QS = RS$. Given that the triangle sides are $PQ = PR = 15$ and the entire base length $QR = 20$.
[asy]
draw((0,0)--(20,0)--(10,36)--cycle,black+linewidth(1));
draw((10,36)--(10,0),black+linewidth(1)+dashed);
draw((10,0)--(10,1)--(9,1)--(9,0)--cycle,black+linewidth(1));
draw((4.5,-4)--(0,-4),black+linewidth(1));
draw((4.5,-4)--(0,-4),EndArrow);
draw((15.5,-4)--(20,-4),black+linewidth(1));
draw((15.5,-4)--(20,-4),EndArrow);
label("$P$",(10,36),N);
label("$Q$",(0,0),SW);
label("$R$",(20,0),SE);
label("$S$",(10,0),S);
label("15",(0,0)--(10,36),NW);
label("15",(10,36)--(20,0),NE);
label("20",(10,-4));
[/asy] | {
"answer": "50\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of rooks that can be placed on a $300 \times 300$ chessboard such that each rook attacks at most one other rook? (A rook attacks all the squares it can reach according to chess rules without passing through other pieces.) | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is a moving point on the curve $y= \frac {3-e^{x}}{e^{x}+1}$, find the minimum value of the slant angle $\alpha$ of the tangent line at point $P$. | {
"answer": "\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a positive integer, and let $S_n = \{1, 2, \ldots, n\}$ . For a permutation $\sigma$ of $S_n$ and an integer $a \in S_n$ , let $d(a)$ be the least positive integer $d$ for which \[\underbrace{\sigma(\sigma(\ldots \sigma(a) \ldots))}_{d \text{ applications of } \sigma} = a\](or $-1$ if no such integer exists). Compute the value of $n$ for which there exists a permutation $\sigma$ of $S_n$ satisfying the equations \[\begin{aligned} d(1) + d(2) + \ldots + d(n) &= 2017, \frac{1}{d(1)} + \frac{1}{d(2)} + \ldots + \frac{1}{d(n)} &= 2. \end{aligned}\]
*Proposed by Michael Tang* | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(n) \) be a function where, for an integer \( n \), \( f(n) = k \) and \( k \) is the smallest integer such that \( k! \) is divisible by \( n \). If \( n \) is a multiple of 20, determine the smallest \( n \) such that \( f(n) > 20 \). | {
"answer": "420",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill in the 3x3 grid with 9 different natural numbers such that for each row, the sum of the first two numbers equals the third number, and for each column, the sum of the top two numbers equals the bottom number. What is the smallest possible value for the number in the bottom right corner? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F_{1}$ and $F_{2}$ be the two foci of the hyperbola $C$: $\frac{x^{2}}{a^{2}}- \frac{y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), and let $P$ be a point on $C$. If $|PF_{1}|+|PF_{2}|=6a$ and the smallest angle of $\triangle PF_{1}F_{2}$ is $30^{\circ}$, then the eccentricity of $C$ is ______. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$ , with $AB=6$ , $BC=7$ , $CD=8$ . Find $AD$ . | {
"answer": "\\sqrt{51}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference of the circle rotated $75$ degrees clockwise from the first triangle so that it shares a side with the first triangle. We draw a third, fourth, and fifth such triangle each rotated $75$ degrees clockwise from the previous triangle. The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by the intersection? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A necklace is strung with gems in the order of A, B, C, D, E, F, G, H. Now, we want to select 8 gems from it in two rounds, with the requirement that only 4 gems can be taken each time, and at most two adjacent gems can be taken (such as A, B, E, F). How many ways are there to do this (answer with a number)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Empire of Westeros, there were 1000 cities and 2017 roads (each road connects two cities). From each city, it was possible to travel to every other city. One day, an evil wizard cursed $N$ roads, making them impassable. As a result, 7 kingdoms emerged, such that within each kingdom, one can travel between any pair of cities on the roads, but it is impossible to travel from one kingdom to another using the roads. What is the maximum possible value of $N$ for which this is possible? | {
"answer": "2011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We color certain squares of an $8 \times 8$ chessboard red. How many squares can we color at most if we want no red trimino? How many squares can we color at least if we want every trimino to have at least one red square? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a square with side length 8, four congruent equilateral triangles are drawn such that each triangle shares one side with a side of the square and each has a vertex at one of the square's vertices. What is the side length of the largest square that can be inscribed in the space inside the square and outside of the triangles? | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pyramid is constructed on a $5 \times 12$ rectangular base. Each of the four edges joining the apex to the corners of the rectangular base has a length of $15$. Determine the volume of this pyramid. | {
"answer": "270",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex 10-gon \(A_{1} A_{2} \ldots A_{10}\), all sides and all diagonals connecting vertices skipping one (i.e., \(A_{1} A_{3}, A_{2} A_{4},\) etc.) are drawn, except for the side \(A_{1} A_{10}\) and the diagonals \(A_{1} A_{9}\), \(A_{2} A_{10}\).
A path from \(A_{1}\) to \(A_{10}\) is defined as a non-self-intersecting broken line (i.e., a line such that no two nonconsecutive segments share a common point) with endpoints \(A_{1}\) and \(A_{10}\), where each segment coincides with one of the drawn sides or diagonals. Determine the number of such paths. | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a}|^2, \overrightarrow{a}\cdot \overrightarrow{b} = \frac{3}{2}, |\overrightarrow{a}+ \overrightarrow{b}| = 2\sqrt{2}$, find $|\overrightarrow{b}| = \_\_\_\_\_\_\_.$ | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is divided into three congruent rectangles. The middle rectangle is removed and replaced on the side of the original square to form an octagon as shown.
What is the ratio of the length of the perimeter of the square to the length of the perimeter of the octagon?
A $3: 5$
B $2: 3$
C $5: 8$
D $1: 2$
E $1: 1$ | {
"answer": "3:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider $\triangle \natural\flat\sharp$ . Let $\flat\sharp$ , $\sharp\natural$ and $\natural\flat$ be the answers to problems $4$ , $5$ , and $6$ , respectively. If the incircle of $\triangle \natural\flat\sharp$ touches $\natural\flat$ at $\odot$ , find $\flat\odot$ .
*Proposed by Evan Chen* | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, four different numbers are selected, denoted as $a$, $b$, $c$, $d$ respectively. If the parity of $a+b$ is the same as the parity of $c+d$, then the total number of ways to select $a$, $b$, $c$, $d$ is ______ (provide the answer in numerical form). | {
"answer": "912",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of ways to arrange the letters of the word PERSEVERANCE. | {
"answer": "19,958,400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e. a uniformly random bearing $\theta\in [0,2\pi]$ ) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$ . | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the digits 0 to 9, how many three-digit even numbers can be formed without repeating any digits? | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a 50-term sequence $(b_1, b_2, \dots, b_{50})$, the Cesaro sum is 500. What is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \dots, b_{50})$? | {
"answer": "492",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. An altitude $\overline{ER}$ from $E$ to line $DF$ is such that $ER = 15$. Given $DP= 27$ and $EQ = 36$, determine the length of ${DF}$. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a_n\}$ such that the sum of the first $n$ terms is $S_n$, $S_1=6$, $S_2=4$, $S_n>0$ and $S_{2n}$, $S_{2n-1}$, $S_{2n+2}$ form a geometric progression, while $S_{2n-1}$, $S_{2n+2}$, $S_{2n+1}$ form an arithmetic progression. Determine the value of $a_{2016}$.
Options:
A) $-1009$
B) $-1008$
C) $-1007$
D) $-1006$ | {
"answer": "-1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular tetrahedron with edge length $2\sqrt{6}$, the total length of the intersection between the sphere with center $O$ and radius $\sqrt{3}$ and the surface of the tetrahedron is ______. | {
"answer": "8\\sqrt{2}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square and a regular octagon have equal perimeters. If the area of the square is 16, what is the area of the octagon?
A) $8 + 4\sqrt{2}$
B) $4 + 2\sqrt{2}$
C) $16 + 8\sqrt{2}$
D) $4\sqrt{2}$
E) $8\sqrt{2}$ | {
"answer": "8 + 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else inde nitely. If Nathaniel goes fi rst, determine the probability that he ends up winning. | {
"answer": "5/11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hemisphere-shaped bowl with radius 1 foot is filled full with chocolate. All of the chocolate is then evenly distributed between 36 cylindrical molds, each having a height equal to their diameter. What is the diameter of each cylinder? | {
"answer": "\\frac{2^{1/3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right-angled triangle $ABC$, where $AB = AC = 1$, an ellipse is constructed with point $C$ as one of its foci. The other focus of the ellipse lies on side $AB$, and the ellipse passes through points $A$ and $B$. Determine the focal length of the ellipse. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To obtain the graph of the function $y=\sin\left(2x+\frac{\pi}{3}\right)$, find the transformation required to obtain the graph of the function $y=\cos\left(2x-\frac{\pi}{3}\right)$. | {
"answer": "\\left(\\frac{\\pi}{12}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $5\,41G\,507\,2H6$ is divisible by $40.$ Determine the sum of all distinct possible values of the product $GH.$ | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the total number of digits used when the first 2500 positive even integers are written? | {
"answer": "9449",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert $645_{10}$ to base 5. | {
"answer": "10400_5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $(a+i)i=b+ai$, solve for $|a+bi|$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice cycled 240 miles in 4 hours, 30 minutes. Then, she cycled another 300 miles in 5 hours, 15 minutes. What was Alice's average speed in miles per hour for her entire journey? | {
"answer": "55.38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of a geometric sequence is an integer. We know that there is a term in the sequence which is equal to the sum of some other terms of the sequence. What can the ratio of the sequence be? | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular tetrahedron is formed by joining the centers of four non-adjacent faces of a cube. Determine the ratio of the volume of the tetrahedron to the volume of the cube. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {m}=(\sin x，-1)$ and $\overrightarrow {n}=( \sqrt {3}\cos x，- \frac {1}{2})$, and the function $f(x)= \overrightarrow {m}^{2}+ \overrightarrow {m}\cdot \overrightarrow {n}-2$.
(I) Find the maximum value of $f(x)$ and the set of values of $x$ at which the maximum is attained.
(II) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ of triangle $ABC$, respectively, and that they form a geometric sequence. Also, angle $B$ is acute, and $f(B)=1$. Find the value of $\frac{1}{\tan A} + \frac{1}{\tan C}$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student travels from his university to his hometown, a distance of 150 miles, in a sedan that averages 25 miles per gallon. For the return trip, he drives his friend's truck, which averages only 15 miles per gallon. Additionally, they make a detour of 50 miles at an average of 10 miles per gallon in the truck. Calculate the average fuel efficiency for the entire trip. | {
"answer": "16.67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the line $x+ay+6=0$ is parallel to the line $(a-2)x+3y+2a=0$, determine the value of $a$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the six digits $0$, $1$, $2$, $3$, $4$, $5$, form integers without repeating any digit. Determine how many such integers satisfy the following conditions:
$(1)$ How many four-digit even numbers can be formed?
$(2)$ How many five-digit numbers that are multiples of $5$ and have no repeated digits can be formed?
$(3)$ How many four-digit numbers greater than $1325$ and with no repeated digits can be formed? | {
"answer": "270",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\{1,2,3,4,5,6,7,8,9,10,11,12\}$. What is the probability that $xyz-xy-xz-yz$ is even? | {
"answer": "\\frac{10}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(n) = \sum_{k=2}^{\infty} \frac{1}{k^n \cdot k!} \). Calculate \( \sum_{n=2}^{\infty} f(n) \). | {
"answer": "3 - e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $z$ is a complex number such that
\[
z + z^{-1} = 2\sqrt{2},
\]
what is the value of
\[
z^{100} + z^{-100} \, ?
\] | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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