prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes $1$ minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to make a detour, walking $3$ blocks of equal length instead of $1$ block to reach the next corner. From the tim... | 42 |
Define an $upno$ to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. Similarly, define a $downno$ to be a positive integer of $2$ or more digits where the digits are strictly decreasing moving left to right. For instance, the number $258$ is an $upno$ and $8620$... | 511 |
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$. | 21 |
In the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of $x?$ | 11 |
The letters $\text{P}, \text{Q},$ and $\text{R}$ are entered into a $20\times20$ table according to the pattern shown below. Determine how many times each of the letters $\text{P}$, $\text{Q}$, and $\text{R}$ will appear in the completed table. Provide your answer in the format: number of Ps, number of Qs, number of Rs... | 400 |
Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. Find the perimeter of $\triangle{APQ}$. | 84 |
Flora the frog starts at \(0\) on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance \(m\) with probability \(\frac{1}{2^m}\). Find the probability that Flora will eventually land at \(10\). Express your answer as a fractio... | 3 |
Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$? Provide your answer in the form of $a\sqrt{b}$, where $a$ and $b$ are integers. Please find the value of a + b. | 18 |
Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between their centers is \(\frac{1}{2}\). Circle \(C_3\) is the largest circle internally tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). The radius of \(C_4\) is ... | 31 |
Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting position? | 411 |
The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. Find the value of $|r-s|$. | 1 |
Sonya the frog chooses a point uniformly at random lying within the square \([0, 6] \times [0, 6]\) in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from \([0, 1]\) and a direction uniformly at random from \{north, south, east, west\}. All of her choices are indep... | 13 |
What is the product of all solutions to the equation $$\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023?$$ | 1 |
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is \(\frac{2}{3}\) the side length of the larger triangle. What is the ratio of the area of one trapezoid to the ar... | 17 |
Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is the value of $n$? | 32 |
Rows $1, 2, 3, 4,$ and $5$ of a triangular array of integers are shown below. $$\begin{array}{cccccccc}&&&&1&&&&\\&&&1&&1&&&\\&&1&&3&&1&&\\&1&&5&&5&&1&\\1&&7&&11&&7&&1\end{array}$$ Each row after the first row is formed by placing a $1$ at each end of the row, and each interior entry is $1$ greater than the sum of the ... | 5 |
Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers? | 60 |
Let $P(x)$ be the unique polynomial of minimal degree with the following properties: $P(x)$ has a leading coefficient $1$, $1$ is a root of $P(x)-1$, $2$ is a root of $P(x-2)$, $3$ is a root of $P(3x)$, and $4$ is a root of $4P(x)$. The roots of $P(x)$ are integers, with one exception. The root that is not an integer c... | 47 |
If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A,B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\ov... | 29 |
A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is \(3\sqrt{3}\) inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at ... | 10 |
Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done? | 2 |
Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between their centers is \(\frac{1}{2}\). Circle \(C_3\) is the largest circle internally tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). The radius of \(C_4\) is ... | 31 |
How many distinct values of $x$ satisfy $\lfloor{x}\rfloor^2-3x+2=0$, where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to $x$? | 4 |
Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$. What is the value of $f(2023)$? | 96 |
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is $4000$. What is the first term? | 5 |
If the positive integer $n$ has positive integer divisors $a$ and $b$ with $n = ab$, then $a$ and $b$ are said to be complementary divisors of $n$. Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $20$ and another pair of complementary divisors that differ by $23$. Find ... | 15 |
An arithmetic sequence of positive integers has $n \ge 3$ terms, initial term $a$, and common difference $d > 1$. Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$. The sum of the terms he wrote was $222$. What is the value of $a + d + n$? | 20 |
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation \[(1+2a)(2+2b)(2a+b) = 32ab?\] | 1 |
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}... | 5 |
Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations. $abcd=2^6\cdot 3^9\cdot 5^7$ ${lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$ ${lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$ ${lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$ ${lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$ ${lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$ ${lcm}(c,d)=2^2\... | 3 |
A regular pentagon with area \(\sqrt{5}+1\) is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? The original answer is in the form of \(n - \sqrt{m}\). Please find the value of m + n. | 6 |
A regular pentagon with area \(\sqrt{5}+1\) is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? The original answer is in the form \(\sqrt{k} - m\). Please give the value of k + m. | 6 |
Fifteen integers $a_1, a_2, a_3, \dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that \[1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace \text{ and } \thickspace 241 \le a_{15}\le 250.\] Find the sum of the digits of $a_{14}$. | 8 |
Aaliyah rolls two standard $6$-sided dice. She notices that the product of the two numbers rolled is a multiple of $6$. Find the integer that cannot be the sum of the two numbers. | 6 |
On Monday, Taye has $2. Every day, he either gains $3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later? | 6 |
Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.) | 5 |
A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a $3 \times 3$ grid attacks all $8$ other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3 \times 3$ grid so t... | 32 |
Jordan owns $15$ pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. What is the least possible value of the fraction of the collection that consists of red high-top sneakers? The original answer is in \(\frac{k}{m}\) format, please... | 19 |
A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3 : 1$. Then $3$ green frogs moved to the sunny side and $5$ yellow frogs moved to the shady side. Now the ratio is $4 : 1$. What is the differ... | 24 |
Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point $(0,4)$ to point $(2,0)$ and colors the $4$ cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point $(2000,3000)$ to point $(5000,8000)$. How many cells will he color thi... | 7000 |
Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high while the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides form a $45^\circ$ angle with the ground. The artwork has an area of $183$ square f... | 5 |
Find the value of $\left(1+\cos \frac{\pi}{7}\right)\left(1+\cos \frac{3 \pi}{7}\right)\left(1+\cos \frac{5 \pi}{7}\right)$. The original answer is in the form \frac{k}{m}, where k and m are integers. Please provide the value of k + m. | 15 |
Evaluate \(\lim \limits_{n \rightarrow \infty} \sum\limits_{k=1}^{n} \arctan \frac{2}{k^{2}}\). The answer is in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\). | 7 |
A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. Find the smallest value that the range (largest observation minus smallest) can assume for such a sample. | 5 |
If $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, then find the value of \[\displaystyle\sum_{N=1}^{1024} \lfloor \log_{2}N\rfloor.\] | 8204 |
Square \(ABCD\) has side length \(2\). A semicircle with diameter \(\overline{AB}\) is constructed inside the square, and the tangent to the semicircle from \(C\) intersects side \(\overline{AD}\) at \(E\). The length of \(\overline{CE}\) can be expressed in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplifi... | 7 |
Let \(a\) and \(b\) be real numbers such that \(\sin a + \sin b = \frac{\sqrt{2}}{2}\) and \(\cos a + \cos b = \frac{\sqrt{6}}{2}\). Find the value of \(\sin(a+b)\). The original answer is in the form \(\frac{k\sqrt{n}}{m}\), please give the value of \(k + m + n\). | 6 |
In a softball league, after each team has played every other team 4 times, the total accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team received 3 points for a win, 1 point for a tie, and no points for a loss, find the number of games that ended in a tie. | 5 |
The base of a triangle is $80$, and one side of the base angle is $60^\circ$. The sum of the lengths of the other two sides is $90$. Find the length of the shortest side of the triangle. | 17 |
There exist positive integers $A$, $B$, and $C$, with no common factor greater than 1, such that: \[A \log_{200} 5 + B \log_{200} 2 = C. \] Find the value of $A + B + C$. | 6 |
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie, Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved t... | 2 |
In this figure, \( \angle RFS = \angle FDR \), \( FD = 4 \) inches, \( DR = 6 \) inches, \( FR = 5 \) inches, \( FS = 7\frac{1}{2} \) inches. Find the length of \( RS \), in inches, as a decimal. Multiply your final answer by 100 and provide the result. | 625 |
Find the number of real solutions to the equation \( \frac{x}{100} = \sin x \). | 63 |
The six-digit number $20210A$ is prime for only one digit $A$. What is the value of $A$? | 9 |
How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 15\}$ have the following two properties? 1. No two consecutive integers belong to $S$. 2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. Find the number of such subsets. | 405 |
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. Find the probability that no two adjacent people will stand. The original answer is in \frac{k}{m} format, please give the value of k + m... | 303 |
Six distinct integers are picked at random from \(\{1,2,3,\ldots,10\}\). What is the probability that, among those selected, the second smallest is \(3\)? Express your answer in the form of \(\frac{k}{m}\), where the fraction is in simplest form, and find the value of \(k + m\). | 4 |
Triangle \( ABC \) has side lengths in arithmetic progression, and the smallest side has length \( 6 \). If the triangle has an angle of \( 120^\circ \), what is the area of \( ABC \)? The original answer is in the form of \( k\sqrt{m} \). Please provide the value of \( k + m \). | 18 |
Let the sequence \((a_n)\) be defined as \(a_1=\frac{1}{3}\) and \(a_{n+1}=\frac{a_n}{\sqrt{1+13a_n^2}}\) for every \(n\geq 1\). If \(a_k\) is the largest term of the sequence satisfying \(a_k < \frac{1}{50}\), find the value of \(k\). | 193 |
What is the number of terms with rational coefficients among the $1001$ terms of the expression $(x \sqrt[3]{2} + y \sqrt{3})^{1000}$? | 167 |
$ABCD$ is a square with a side of unit length. Points $E$ and $F$ are taken respectively on sides $AB$ and $AD$ such that $AE = AF$, and the quadrilateral $CDFE$ has maximum area. The original answer is in \frac{k}{m} format, please give the value of k + m. | 13 |
In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle, and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees? If the answer is $x^\circ$, what is the value of $x$? | 15 |
For any complex number \(w = a + bi\), \(|w|\) is defined to be the real number \(\sqrt{a^2 + b^2}\). If \(w = \cos{40^\circ} + i\sin{40^\circ}\), find the value of \[ x = |w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1} \] in the form \(\frac{k}{l}\sin{\phi^\circ}\), where \(k\), \(l\) are integers, and \(\phi\) is an angle in ... | 31 |
In the figure, $ABCD$ is a $2 \times 2$ square, $E$ is the midpoint of $\overline{AD}$, and $F$ is on $\overline{BE}$. If $\overline{CF}$ is perpendicular to $\overline{BE}$, find the area of quadrilateral $CDEF$. The original answer is in \frac{k}{m} format, where k and m are coprime integers. Please give the value of... | 16 |
The cost $C$ of sending a parcel post package weighing $P$ pounds, where $P$ is an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. What is the formula for the cost $C$ in terms of $P$? The original answer is in the form $C = a + b(P - 1)$, please give the value of $a + b$. | 13 |
Find the number of distinct solutions of the equation $\big|x-|2x+1|\big| = 3$. | 2 |
Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, find the value of $n$. | 9 |
A circular piece of metal of maximum size is cut out of a square piece, and then a square piece of maximum size is cut out of the circular piece. Find the total amount of metal wasted as a fraction of the area of the original square. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m. | 3 |
Let \((1+x+x^2)^n = a_0 + a_1x + a_2x^2 + \ldots + a_{2n}x^{2n}\) be an identity in \(x\). If we let \(s = a_0 + a_2 + a_4 + \ldots + a_{2n}\), the original answer is in the form \(\frac{k \cdot 3^n + m}{n}\), where n cannot be divisible by 2. Please find the value of \(k + m + n\). | 4 |
Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$? | 15 |
Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$? If the answer is \(\sqrt{x}\), what is the value of \(x\)? | 95 |
Let $P(x) = 1 - x + x^2 - x^3 + \dots + x^{18} - x^{19}$ and $Q(x) = P(x - 1)$. Find the coefficient of $x^2$ in polynomial $Q$. | 1140 |
How many base 10 four-digit numbers, $N = \underline{a} \underline{b} \underline{c} \underline{d}$, satisfy all three of the following conditions? (i) $4000 \leq N < 6000;$ (ii) $N$ is a multiple of 5; (iii) $3 \leq b < c \leq 6$. Find the number of such numbers. | 24 |
Let $ABCD$ be a quadrilateral with sides $|\overline{AB}|=2$, $|\overline{BC}|=|\overline{CD}|=4$, and $|\overline{DA}|=5$. The opposite angles, $\angle A$ and $\angle C$, are equal. What is the length of diagonal $BD$? The original answer is in $k\sqrt{m}$ format, please give the value of k + m. | 8 |
How many ordered pairs of integers $(x, y)$ satisfy the equation $$x^{2020} + y^2 = 2y?$$ | 4 |
Square $BDEC$ with center $F$ is constructed on the outside of triangle $ABC$ such that $\angle A = 90^\circ$, $|AB| = \sqrt{12}$, $|AC| = 2$. If $[AF] \cap [BC] = \{G\}$, find the length of $|BG|$. The original answer is in the form $m - n\sqrt{p}$, please give the value of $m + n + p$. | 11 |
How many ways are there to make change for $55$ cents using any number of pennies, nickels, dimes, and quarters? | 60 |
What is the largest possible area of an isosceles trapezoid in which the largest side is $13$ and the perimeter is $28$? Find the largest possible area. | 27 |
Ali and Veli go hunting. The probability that each will successfully hit a duck is \(\frac{1}{2}\) on any given shot. During the hunt, Ali shoots 12 times, and Veli shoots 13 times. What is the probability that Veli hits more ducks than Ali? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in s... | 3 |
We call a number $10^3 < n < 10^6$ a balanced number if the sum of its last three digits is equal to the sum of its other digits. Find the sum of all balanced numbers in $\bmod{13}$. | 0 |
How many ordered pairs $(s, d)$ of positive integers with $4 \leq s \leq d \leq 2019$ are there such that when $s$ silver balls and $d$ diamond balls are randomly arranged in a row, the probability that the balls on each end have the same color is $\frac{1}{2}$? Provide your answer as an integer. | 60 |
A six-digit number (base 10) is considered "squarish" if it satisfies the following conditions: 1. None of its digits are zero. 2. It is a perfect square. 3. The first two digits, the middle two digits, and the last two digits of the number are all perfect squares when considered as two-digit numbers. How many squarish... | 2 |
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. Find the area of the region in the plane covered by these squares. | 175 |
A pyramid of non-negative integers is constructed as follows: 1. The first row consists of only $0$. 2. The second row consists of $1$ and $1$. 3. The $n^{th}$ row (for $n > 2$) is an array of $n$ integers, where the leftmost and rightmost elements are equal to $n - 1$, and the interior numbers are equal to the sum of ... | 2 |
Find the number of 3-digit numbers not containing the digit 0 and such that one of the digits is the sum of the two others. | 108 |
How many integer pairs $(m,n)$ are there satisfying the equation $4mn(m+n-1)=(m^2+1)(n^2+1)$? Provide your answer as an integer. | 5 |
A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two-hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchant... | 35 |
Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each. Along this line, the midpoint of the base of one triangle is a vertex of the next. What is the area of the region of the plane that is covered by the union of the five triangular ... | 15 |
Each of 100 students sends messages to 50 different students. Find the least number of pairs of students who send messages to each other. | 50 |
Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. The original answer is in \(... | 7 |
The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex? | 12 |
An observer stands on the side of the front of a stationary train. When the train starts moving with constant acceleration, it takes $5$ seconds for the first car to pass the observer. If the time it takes for the $10^{\text{th}}$ car to pass is $x$ seconds, find the value of $100x$. | 81 |
When simplified, \( (x^{-1} + y^{-1})^{-1} \) is equal to what expression in the form \(\frac{kx}{my}\)? Please find the value of k + m. | 2 |
Square $PQRS$ lies in the first quadrant. Points $(3,0)$, $(5,0)$, $(7,0)$, and $(13,0)$ lie on lines $SP$, $RQ$, $PQ$, and $SR$, respectively. If the sum of the coordinates of the center of the square $PQRS$ is $x$, what is the value of $10x$? | 64 |
What is the $33$-rd number after the decimal point of $(\sqrt{10} + 3)^{2001}$? | 0 |
A box contains 11 balls, numbered 1, 2, 3, ..., 11. If 6 balls are drawn simultaneously at random, find the probability that the sum of the numbers on the balls drawn is odd. Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and provide the value of k + m. | 349 |
If two poles $20''$ and $80''$ high are $100''$ apart, find the height (in inches) of the intersection of the lines joining the top of each pole to the foot of the opposite pole. | 16 |
What is the largest volume of a sphere which touches a unit sphere internally and touches externally a regular tetrahedron whose corners are over the unit sphere? Express your answer as a fraction in the form \(\frac{k}{m}\), and give the value of \(k + m\). | 4 |
Corners are sliced off a unit cube so that the six faces each become regular octagons. The original answer for the total volume of the removed tetrahedra is in the form \(\frac{k - m\sqrt{2}}{n}\). Please find the value of k + m + n. | 20 |
What is the radius of a circle inscribed in a rhombus with diagonals of length \(10\) and \(24\)? Express your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please find the value of \(k + m\). | 73 |
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