meetforever/GRPO_CL huggingface dataset:
user: There are $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find the absolute value of the difference $|P_s - P_d|$. The original answer is in the form \(\frac{m}{n}\), where m and n are integers. Please calculate the value of m + n.
Answer: 2002
user: How many positive integers $n$ satisfy \[ \frac{n+1000}{70} = \lfloor \sqrt{n} \rfloor? \]
(Recall that $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.)
Answer: 6
user: A polynomial $P$ with degree exactly $3$ satisfies $P(0)=1$, $P(1)=3$, and $P(3)=10$. Find a value that cannot be the value of $P(2)$.
Answer: 6
user: In real numbers, find the number of distinct pairs \((x,y)\) that satisfy the system of equations:
\[ x^2 = 4y + 1 \]
and
\[ y^2 = x^3 + 1 \]
Answer: 2
user: The angle bisector of the acute angle formed at the origin by the graphs of the lines \(y = x\) and \(y = 3x\) has equation \(y = kx\). The original answer is in the form \(\frac{k+\sqrt{n}}{m}\). Please find the value of k + m + n.
Answer: 8
user: Let $N = 34 \times 34 \times 63 \times 270$. If the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$ is $1:x$, what is the value of $x$?
Answer: 14
user: At most, how many positive integers less than $51$ are there such that no one is a triple of another one? Provide your answer as an integer.
Answer: 38
user: Two strips of width 1 overlap at an angle of \(\alpha\) as shown. Find the area of the overlap (shown shaded) in terms of \(\alpha\). The answer is in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please provide the value of \(k + m\).
Answer: 2
user: Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^{\circ}$ and $\measuredangle ADC=68^{\circ}$, find $\measuredangle EBC$. If the answer is $x^\circ$, what is the value of $x$?
Answer: 68
user: Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city?
Answer: 23
user: Square \(ABCD\) has an area of \(36\), and \(\overline{AB}\) is parallel to the x-axis. Vertices \(A\), \(B\), and \(C\) are on the graphs of \(y = \log_{a}x\), \(y = 2\log_{a}x\), and \(y = 3\log_{a}x\), respectively. What is the value of \(a\)? The original answer is in the form \(k^{m/n}\), please give the value of k + m + n.
Answer: 10
user: If an integer $n > 8$ is a solution of the equation $x^2 - ax + b = 0$ and the representation of $a$ in the base-$n$ number system is $18$, what is the base-$n$ representation of $b$?
Answer: 80
user: Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, find Ann's age.
Answer: 24
user: Two boys start moving from the same point $A$ on a circular track but in opposite directions. Their speeds are $5$ ft. per second and $9$ ft. per second. If they start at the same time and finish when they first meet at the point $A$ again, find the number of times they meet, excluding the start and finish.
Answer: 13
user: A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline, traveling $85$ meters in $17$ seconds. Ignore air resistance. If the coefficient of kinetic friction between the sled and the slope is $x$, what is the value of $100x$?
Answer: 40
user: Let \(a, b, c, d, e\) be non-negative real numbers such that \(a + b + c + d + e > 0\). Find the least real number \(t\) such that \(a + c = tb\), \(b + d = tc\), \(c + e = td\). The original answer is in the form \(\sqrt{m}\). Please find the value of \(m\).
Answer: 2
user: A three-quarter sector of a circle with a radius of \(4\) inches, together with its interior, can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? Provide your answer in the form of \(a\pi \sqrt{b}\), where \(a\) and \(b\) are integers. Please calculate the value of \(a + b\).
Answer: 10
user: In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?
Answer: 7
user: For each integer $n \geq 4$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_4 a_5 \cdots a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$?
Answer: 962
user: What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$?
Answer: 970
user: Find the value(s) of $y$ for which the following pair of equations \[x^2+y^2-16=0 \text{ and } x^2-3y+12=0\] may have a real common solution. Please provide the sum of all such $y$ values.
Answer: 4
user: The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length of $\overline{AB}$?
Answer: 20
user: If $60^a = 3$ and $60^b = 5$, find the value of $12^{\frac{1-a-b}{2(1-b)}}$.
Answer: 2
user: If we write $|x^2 - 4| < N$ for all $x$ such that $|x - 2| < 0.01$, what is the smallest value we can use for $N$ if the answer is expressed as $N \times 10000$?
Answer: 401
user: The sequence \(\{a_n\}\) is defined as follows: \[ a_n = \sqrt{1 + \left(1 + \frac{1}{n} \right)^2} + \sqrt{1 + \left(1 - \frac{1}{n} \right)^2}. \] Calculate the value of the expression given below: \[ \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}. \] The original answer is in the form \(\sqrt{m} - 1\). Please provide the sum of the integer part of \(\sqrt{m}\) and 1.
Answer: 137
user: In the $xy$-plane, find the number of lines whose $x$-intercept is a positive prime number and whose $y$-intercept is a positive integer that pass through the point $(4,3)$.
Answer: 2
user: Points $A, B, Q, D,$ and $C$ lie on the circle shown. The measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$, respectively. Find the sum of the measures of angles $P$ and $Q$ in degrees.
Answer: 40
user: Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. Find the radius of circle $B$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 17
user: If $(0.2)^x = 2$ and $\log 2 = 0.3010$, then if the value of $x$ to the nearest tenth is $y$, what is the value of $10y$?
Answer: -4
user: Let $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Find the value of $n$.
Answer: 19
user: How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
Answer: 382
user: The floor of a rectangular room is covered with square tiles. The room is 10 tiles long and 5 tiles wide. How many tiles touch the walls of the room?
Answer: 26
user: Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is the value of $\frac{AB}{AD}$?

\[
\begin{asy}
unitsize(5mm);
defaultpen(linewidth(0.8pt)+fontsize(10pt));
pair A=(0,3), B=(8,3), C=(8,2), D=(0,2), Ep=(0,4), F=(4,4), G=(4,0), H=(0,0);
fill(shift(0,2)*xscale(4)*unitsquare,grey);
draw(Ep--F--G--H--cycle);
draw(A--B--C--D);
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",Ep,NW);
label("$F$",F,NE);
label("$G$",G,SE);
label("$H$",H,SW);
\end{asy}
\]
Answer: 10
user: Determine the number of distinct points in the $xy$-plane that are common to the graphs of $(x+y-5)(2x-3y+5)=0$ and $(x-y+1)(3x+2y-12)=0$.
Answer: 1
user: Two standard six-sided dice are tossed. What is the probability that the sum of the numbers is greater than \(7\)? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 17
user: What is the sum of real roots of the equation \(x^4 - 7x^3 + 14x^2 - 14x + 4 = 0\)?
Answer: 5
user: Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. If the measure of $\widehat{ABC}$ is $x^\circ$, what is the value of $x$?
Answer: 105
user: Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. Find the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$. If the ratio is $x:1$, what is the value of $x$?
Answer: 37
user: How many ordered triples of positive integers \((a,b,c)\) are there such that \((2a+b)(2b+a)=2^c\)? Provide your answer as an integer.
Answer: 0
user: When the circumference of a toy balloon is increased from \(20\) inches to \(25\) inches, the radius is increased by how many inches? If the answer is in the form of \(\frac{a}{b}\), what is the value of \(a + b\)?
Answer: 7
user: In the figure, it is given that:
- \(\angle C = 90^\circ\)
- \(\overline{AD} = \overline{DB}\)
- \(DE \perp AB\)
- \(\overline{AB} = 20\)
- \(\overline{AC} = 12\)

Find the area of quadrilateral \(ADEC\). The answer should be in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).
Answer: 119
user: A child has a set of $96$ distinct blocks. Each block is one of $2$ materials (plastic, wood), $3$ sizes (small, medium, large), $4$ colors (blue, green, red, yellow), and $4$ shapes (circle, hexagon, square, triangle). How many blocks in the set are different from the "plastic medium red circle" in exactly two ways? (The "wood medium red square" is such a block.) Find the number of such blocks.
Answer: 29
user: Suppose \(2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.\) The original answer is in \(\frac{k}{m}\) format. What is the value of \(k+m\)?
Answer: 7
user: Lines from the vertices of a unit square are drawn to the midpoints of the sides as shown in the figure below. What is the area of quadrilateral \(ABCD\)? Express your answer in the form \(\frac{k}{m}\) where the fraction is in simplest terms, and give the value of \(k + m\).
Answer: 6
user: Mary chose an even 4-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \ldots, \frac{n}{2}, n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
Answer: 340
user: A circle of radius \(r\) passes through both foci of, and exactly four points on, the ellipse with equation \(x^2 + 16y^2 = 16\). The set of all possible values of \(r\) is an interval \([a, b)\). The original answer is in the form \(\sqrt{a} + b\). Please find the value of \(a + b\).
Answer: 23
user: In a table tennis tournament, every participant played every other participant exactly once. There were twice as many right-handed players as left-handed players. However, the number of games won by left-handed players was 40\% more than the number of games won by right-handed players. (Note: There were no ties and no ambidextrous players.) Find the total number of games played.
Answer: 36
user: For how many ordered triples $(x, y, z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1 + i)^y, z\}$, where $i = \sqrt{-1}$?
Answer: 225
user: For real numbers $x$, let \[P(x)=1+\cos(x)+i \sin(x)-\cos(2x)-i \sin(2x)+\cos(3x)+i \sin(3x)\] where $i=\sqrt{-1}$. Find the number of values of $x$ with $0 \leq x<2 \pi$ such that $P(x)=0$.
Answer: 0
user: Let $ABC$ be a triangle with $m(\widehat{A}) = 70^\circ$ and the incenter $I$. If $|BC| = |AC| + |AI|$, then what is the value of $m(\widehat{B})$ in degrees? If the answer is $x^\circ$, what is the value of $x$?
Answer: 35
user: Let \( A = \frac{2^2 + 3 \cdot 2 + 1}{3! \cdot 4!} + \frac{3^2 + 3 \cdot 3 + 1}{4! \cdot 5!} + \frac{4^2 + 3 \cdot 4 + 1}{5! \cdot 6!} + \dots + \frac{10^2 + 3 \cdot 10 + 1}{11! \cdot 12!} \). Find the remainder when \( 11! \cdot 12! \cdot A \) is divided by 11.
Answer: 10
user: Triangles $ABC$ and $ABD$ are isosceles with $AB = AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then what is the value of $\angle C + \angle D$ if the answer is in degrees?
Answer: 135
user: The base three representation of $x$ is 
\[ 12112211122211112222. \]
What is the first digit (on the left) of the base nine representation of $x$?
Answer: 5
user: The circumference of a circle is \(100\) inches. Find the length of the side of a square inscribed in this circle, expressed in inches. The original answer is in the form \(\frac{k\sqrt{m}}{n\pi}\). Please provide the value of \(k + m + n\).
Answer: 53
user: For all positive numbers \(x, y, z\), evaluate the product:
\[
(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]
\]
What is the simplified expression for this product? The simplified expression is in the form \(x^{-a}y^{-b}z^{-c}\). Please provide the value of \(a+b+c\).
Answer: 6
user: How many prime divisors does the number $1\cdot 2003 + 2\cdot 2002 + 3\cdot 2001 + \cdots + 2001 \cdot 3 + 2002 \cdot 2 + 2003 \cdot 1$ have? Fill in the blank with the number of prime divisors.
Answer: 5
user: A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers. For each ball, the probability that it is tossed into bin \(k\) is \(2^{-k}\) for \(k=1,2,3,\ldots\). What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? Express your answer in the form \(\frac{k}{m}\), where k and m are integers, and find the value of \(k + m\).
Answer: 4
user: What is the remainder when \(2011^{(2011^{(2011^{(2011^{2011})})})}\) is divided by \(19\)?
Answer: 5
user: Given the function $f$ defined as follows: \[ f(0) = 0, \] \[ f(1) = 1, \] and for every integer $n \geq 1$, \[ f(3n-1) = f(n) - 1, \] \[ f(3n) = f(n), \] \[ f(3n+1) = f(n) + 1. \] Find the value of $f(2011)$.
Answer: 3
user: A circle with area \(A_1\) is contained in the interior of a larger circle with area \(A_1 + A_2\). If the radius of the larger circle is \(3\), and if \(A_1, A_2, A_1 + A_2\) is an arithmetic progression, find the radius of the smaller circle. The original answer is in the form \(\sqrt{k}\), please give the value of \(k\).
Answer: 3
user: The weight of \( \frac{1}{3} \) of a large pizza together with \( 3 \frac{1}{2} \) cups of orange slices is the same as the weight of \( \frac{3}{4} \) of a large pizza together with \( \frac{1}{2} \) cup of orange slices. A cup of orange slices weighs \( \frac{1}{4} \) of a pound. What is the weight, in pounds, of a large pizza? Express your answer in the form of \(\frac{k}{m}\) and find the value of \(k + m\).
Answer: 14
user: An integer $x$, with $10 \le x \le 99$, is to be chosen. If all choices are equally likely, find the probability that at least one digit of $x$ is a $7$. The original answer is in the form \frac{k}{m}, where k and m are coprime. Please give the value of k + m.
Answer: 6
user: Find the smallest positive number from the following expressions:

\[ 10 - 3\sqrt{11}, \ 3\sqrt{11} - 10, \ 18 - 5\sqrt{13}, \ 51 - 10\sqrt{26}, \ 10\sqrt{26} - 51 \]

The original answer is in the form of k - m\sqrt{n}, please give the value of k + m.
Answer: 61
user: What is the sum of all $k \leq 25$ such that one can completely cover a $k \times k$ square with $T$ tetrominos without any overlap?
Answer: 84
user: Let $P(x) = x^{2022} + x^{1011} + 1$. Find the polynomial of the form ax^6 + bx^3 + c that divides $P(x)$. Please provide the sum of the coefficients a + b + c.
Answer: 3
user: In the eight-term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. Find the value of $A + H$.
Answer: 25
user: Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\overline{AB}$, and $\overline{CD}$ bisects the right angle. The inscribed circles of $\triangle ADC$ and $\triangle BCD$ have radii $r_a$ and $r_b$, respectively. The value of \frac{r_a}{r_b} is given in the form \frac{k}{m}(n-\sqrt{p}). Please calculate the value of k+m+n+p.
Answer: 43
user: Given a rectangle \(R_1\) with one side measuring \(2\) inches and an area of \(12\) square inches. Rectangle \(R_2\) has a diagonal of \(15\) inches and is similar to \(R_1\). What is the area of \(R_2\), expressed in the form \(\frac{k}{m}\) square inches? Please provide the value of \(k + m\).
Answer: 137
user: In $\triangle ABC$, $\angle ABC = 45^\circ$. Point $D$ is on $\overline{BC}$ such that $2 \cdot BD = CD$ and $\angle DAB = 15^\circ$. Find the value of $\angle ACB$ in degrees. If the answer is $x^\circ$, what is the value of $x$?
Answer: 75
user: If \(i^2 = -1\), then find the sum 
\[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \]
\[ + \cdots + i^{40}\cos{3645^\circ} \]
The original answer is in the form \(\frac{k}{m}(\sqrt{2})(a - bi)\), please calculate the value of a + b + k + m.
Answer: 44
user: An unfair coin lands on heads with a probability of $\frac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the value of $n$?
Answer: 11
user: A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. The rotational inertia of the rod about its center is $md^2$.

Find the ratio $\frac{L}{d}$. The original answer is in the form k\sqrt{m}, please provide the value of k + m.
Answer: 5
user: For any positive integer \(n\), let \[ f(n) = \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\ 0, & \text{otherwise.} \end{cases} \] Find the value of \(\sum_{n = 1}^{1997}{f(n)}\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 58
user: Convex polygons \(P_1\) and \(P_2\) are drawn in the same plane with \(n_1\) and \(n_2\) sides, respectively, where \(n_1 \le n_2\). If \(P_1\) and \(P_2\) do not share any line segment, what is the maximum number of intersections between \(P_1\) and \(P_2\)? The answer is in the form \(kn_1 + m\). Please provide the value of \(k + m\).
Answer: 2
user: Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode?
Answer: 11
user: Points \(A(6,13)\) and \(B(12,11)\) lie on circle \(\omega\) in the plane. Suppose that the tangent lines to \(\omega\) at \(A\) and \(B\) intersect at a point on the \(x\)-axis. Find the value of \(\frac{\text{Area of }\omega}{\pi}\) in the format \(\frac{k}{m}\), and give the sum of k and m.
Answer: 93
user: Find the product of the real roots of the equation $x^4+3x^3+5x^2 + 21x -14=0$. The answer is an integer. Please provide the integer value.
Answer: -2
user: A particle is placed on the parabola $y = x^2 - x - 6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. Find the horizontal distance traveled by the particle, which is the numerical value of the difference in the $x$-coordinates of $P$ and $Q$.
Answer: 3
user: How many ordered pairs of real numbers \((x,y)\) satisfy the following system of equations?
\[
\begin{align*}
x+3y &= 3 \\
\big||x|-|y|\big| &= 1
\end{align*}
\]
Answer: 3
user: A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning 8 times, came home by bus in the afternoon 15 times, and commuted by train (either morning or afternoon) 9 times. Find the total number of working days $x$.
Answer: 16
user: How many positive integers $n < 10^6$ are there such that $n$ is equal to twice the square of an integer and is equal to three times the cube of an integer?
Answer: 3
user: Let $C_{1}$ and $C_{2}$ be two externally tangent circles with diameters $[AB]$ and $[BC]$, and centers $D$ and $E$, respectively. Let $F$ be the intersection point of the tangent line from $A$ to $C_{2}$ and the tangent line from $C$ to $C_{1}$ (both tangent lines are on the same side of $AC$). If $|DB| = |BE| = \sqrt{2}$, find the area of triangle $AFC$. The answer is in the form k\sqrt{m}, where m is the smallest possible integer. Please provide the value of k + m.
Answer: 4
user: Tetrahedron $ABCD$ has the following side lengths: $AB = 5$, $AC = 3$, $BC = 4$, $BD = 4$, $AD = 3$, and $CD = \frac{12}{5}\sqrt{2}$. Find the volume of the tetrahedron. The answer is in the form \(\frac{m}{n}\), please calculate the value of m + n.
Answer: 29
user: In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC = DC = 3$, $OD = 2$, and $AB = 6$, find the radius of the circle. If the radius is \(\sqrt{x}\), what is the value of \(x\)?
Answer: 22
user: Let $ABC$ be a triangle such that angle $C$ is obtuse. Let $D \in [AB]$ and $[DC] \perp [BC]$. If $m(\widehat{ABC}) = \alpha$, $m(\widehat{BCA}) = 3\alpha$, and $|AC| - |AD| = 10$, find the length of $|BD|$.
Answer: 20
user: If \(x \ge 0\), then find the value of \(\sqrt{x \sqrt{x \sqrt{x}}}\). The original answer is in the format \(\sqrt[m]{x^{k}}\). Please give the value of \(k + m\).
Answer: 15
user: A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$, she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$, and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. Find the value of $a+b+c+d$.
Answer: 2024
user: The adjacent sides of the decagon shown meet at right angles. What is the perimeter of the decagon?

\[ \begin{asy} \defaultpen(linewidth(.8pt)); \dotfactor=4; \dot(origin); \dot((12,0)); \dot((12,1)); \dot((9,1)); \dot((9,7)); \dot((7,7)); \dot((7,10)); \dot((3,10)); \dot((3,8)); \dot((0,8)); \draw(origin--(12,0)--(12,1)--(9,1)--(9,7)--(7,7)--(7,10)--(3,10)--(3,8)--(0,8)--cycle); \label("$8$",midpoint(origin--(0,8)),W); \label("$2$",midpoint((3,8)--(3,10)),W); \label("$12$",midpoint(origin--(12,0)),S); \end{asy} \]
Answer: 44
user: For the infinite series \(1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} - \frac{1}{32} + \frac{1}{64} - \frac{1}{128} - \cdots\), let \(S\) be the (limiting) sum. The original answer is in \(\frac{k}{m}\) format, where \(k\) and \(m\) are integers. Please find the value of \(k + m\).
Answer: 9
user: The sequence $a_0, a_1, a_2, \ldots$ is a strictly increasing arithmetic sequence of positive integers such that:
\[
2^{a_7} = 2^{27} \cdot a_7.
\]
What is the minimum possible value of $a_2$?
Answer: 12
user: For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot \left(\frac{2}{5}\right)^n$ an integer? Provide your answer as an integer.
Answer: 9
user: A large rectangle is partitioned into four smaller rectangles by two segments parallel to its sides. The areas of three of these rectangles are given. Determine the area of the fourth rectangle.

```
\begin{asy}
draw((0,0)--(10,0)--(10,7)--(0,7)--cycle);
draw((0,5)--(10,5));
draw((3,0)--(3,7));
label("6", (1.5,6));
label("?", (1.5,2.5));
label("14", (6.5,6));
label("35", (6.5,2.5));
\end{asy}
```

Find the area of the fourth rectangle.
Answer: 15