Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Calculate the difference between the number of hours in $10$ years and the number of seconds in $1$ day.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which
\[ \log_{10} (x + y) = z \text{ and } \log_{10} (x^2 + y^2) = z + 1. \]
There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have
\[ x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}. \]
The original answer is in \(\frac{m}{n}\) format. Please find the value of \(m + n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A square and a circle have equal perimeters. Find the ratio of the area of the circle to the area of the square. The original answer is in \(\frac{k}{m}\) format, where k and m are integers, and the fraction is in its simplest form. Please provide the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A small block with an initial speed $v_0$ smoothly moves onto a sloped big block of mass $M$. After the small block reaches a height $h$ on the slope, it slides back down. Find the height $h$ in terms of $v_0$, $g$, $m$, and $M$. The original answer is in the form \frac{k}{m} \cdot \frac{n v_0^2}{p + M}. Please give the value of k + m + n + p.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Rectangle \(ABCD\) has \(AB=4\) and \(BC=3\). Segment \(EF\) is constructed through \(B\) so that \(EF\) is perpendicular to \(DB\), and \(A\) and \(C\) lie on \(DE\) and \(DF\), respectively. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is $1:16$ PM the clock incorrectly shows $9:96$ PM. Find the fraction of the day during which the clock shows the correct time. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A convex polyhedron $Q$ has vertices $V_1,V_2,\cdots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\cdots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges does $R$ have?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Parallelogram $ABCD$ has area $1,000,000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every \(90\) seconds, and Robert runs clockwise and completes a lap every \(80\) seconds. Both start from the same line at the same time. At some random time between \(10\) minutes and \(11\) minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m, where k and m are integers in the simplest form of the fraction.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. If the answer is $x^\circ$, what is the value of $x$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? Provide your answer as a simplified fraction in the form \frac{k}{m}, and find the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Halfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ points, with other possible scores being $8, 4, 2$, and $0$ points. Chelsea always scores at least $4$ points on each shot. If Chelsea's next $n$ shots are bullseyes, she will be guaranteed victory. What is the minimum value for $n$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers \( x \) and \( x+32 \) are three-digit and four-digit palindromes, respectively. What is the sum of the digits of \( x \)?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let \( n \) be the smallest positive integer such that \( n \) is divisible by 20, \( n^2 \) is a perfect cube, and \( n^3 \) is a perfect square. Find the number of digits of \( n \).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The solution of the equation \(7^{x+7} = 8^x\) can be expressed in the form \(x = \log_b 7^7\). The original answer is in the format \(\frac{k}{m}\). Please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A palindrome between $1000$ and $10,000$ is chosen at random. Find the probability that it is divisible by $7$. The original answer is in \frac{k}{m} format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

On a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$. What is the maximum number of questions that Jesse could have answered correctly?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A solid cube has side length $3$ inches. A $2$-inch by $2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A $16$-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by $1$. Find how many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. Find the smallest value of $n$ for which $P(n) < \frac{1}{2010}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values of $x$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that

$P(1) = P(3) = P(5) = P(7) = a,$  and

$P(2) = P(4) = P(6) = P(8) = -a.$

What is the smallest possible value of $a$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let \(ABCD\) be a cyclic quadrilateral. The side lengths of \(ABCD\) are distinct integers less than \(15\) such that \(BC\cdot CD=AB\cdot DA\). Find the largest possible value of \(BD\) in the form \(\sqrt{\frac{x}{y}}\). Please provide the value of \(x + y\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \frac{1}{2010}$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. Find the sum of the minimum values of $P(x)$ and $Q(x)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$, then his sequence contains $5$ numbers:
$$\begin{array}{ccccc}
{}&{}&{}&{}&55\\
55&-&7^2&=&6\\
6&-&2^2&=&2\\
2&-&1^2&=&1\\
1&-&1^2&=&0\\
\end{array}$$
Let $N$ be the smallest number for which Jim’s sequence has $8$ numbers. What is the units digit of $N$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that
$P(1) = P(3) = P(5) = P(7) = a$,  and
$P(2) = P(4) = P(6) = P(8) = -a$.
Find the smallest possible value of $a$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $32$? Provide your answer as an integer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $61$ points. How many free throws did they make?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $17.71$. What was the cost of a pencil in cents?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two real numbers are selected independently and at random from the interval \([-20,10]\). Find the probability that the product of those numbers is greater than zero. Provide your answer as a simplified fraction \(\frac{k}{m}\), and then give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of \(6\) meters, and it takes her \(36\) seconds longer to walk around the outside edge of the track than around the inside edge. The original answer for Keiko's speed is in the format \(\frac{k\pi}{m}\), where k and m are integers. Please find the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What is the percentage of the area of rectangle $AQRD$ that is shaded? If the answer is $x\%$, what is the value of $x$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. Find $\cos\left(\angle AVB\right)$. The original answer is in the form -\frac{k}{m}, where k and m are integers. Please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{\sqrt{k} - 1}{m}\). Please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Rhombus \(ABCD\) has side length \(2\) and \(\angle B = 120^\circ\). Region \(R\) consists of all points inside the rhombus that are closer to vertex \(B\) than any of the other three vertices. The area of \(R\) is given in the form \(\frac{k\sqrt{n}}{m}\). Please find the value of \(k + m + n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4 : 5$. What is the degree measure of angle $BCD$? If the answer is $x^\circ$, what is the value of $x$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \geq 2$. Find the sum of the digits of $h_{2011}(1)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let \(f_{1}(x)=\sqrt{1-x}\), and for integers \(n \geq 2\), let \(f_{n}(x)=f_{n-1}(\sqrt{n^2 - x})\). If \(N\) is the largest value of \(n\) for which the domain of \(f_{n}\) is nonempty, the domain of \(f_{N}\) is \([c]\). Find the value of \(N+c\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the hundreds digit of \(2011^{2011}\)?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. Find the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$. The original answer is in the form of $k\sqrt{m}$, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose that the euro is worth $1.3$ dollars. If Diana has $500$ dollars and Etienne has $400$ euros, by what percent is the value of Etienne's money greater than the value of Diana's money? Provide your answer as a percentage.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

An iterative average of the numbers \(1, 2, 3, 4\), and \(5\) is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. Find the difference between the largest and smallest possible values that can be obtained using this procedure. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A triangle has an area of $30$, one side of length $10$, and the median to that side has a length of $9$. Let $\theta$ be the acute angle formed by that side and the median. The original answer for $\sin{\theta}$ is in the form \frac{k}{m}. Please provide the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt{2}$, and $\angle BAC = 45^\circ$. Point $D$ is $20$ meters due north of point $C$. Calculate the distance $AD$ and determine the interval in which this distance falls, expressed as (a, b) where a and b are consecutive integers. Please find the value of a + b.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A square region \(ABCD\) is externally tangent to the circle with equation \(x^2+y^2=1\) at the point \((0,1)\) on the side \(CD\). Vertices \(A\) and \(B\) are on the circle with equation \(x^2+y^2=4\). The side length of this square can be expressed in the form \(\frac{k\sqrt{m}-n}{p}\), where \(\sqrt{m}\) is a square root of a prime number. Find the value of \(k + m + n + p\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many sequences of zeros and ones of length $20$ have all the zeros consecutive, or all the ones consecutive, or both?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Jesse cuts a circular disk of radius \(12\), along \(2\) radii to form \(2\) sectors, one with a central angle of \(120^\circ\). He makes two circular cones using each sector to form the lateral surface of each cone. The ratio of the volume of the smaller cone to the larger cone is in the form \(\frac{\sqrt{k}}{n}\). Please give the value of \(k + n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A unit cube has vertices \(P_1,P_2,P_3,P_4,P_1',P_2',P_3',\) and \(P_4'\). Vertices \(P_2\), \(P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\le i\le 4,\) vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2\), \(P_1P_3\), \(P_1P_4\), \(P_1'P_2'\), \(P_1'P_3'\), and \(P_1'P_4'\). The original answer is in the form \(\frac{k\sqrt{n}}{m}\). Please find the value of k + m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. Find the sum of all possible values of $n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A circle of radius \(2\) is cut into four congruent arcs. The four arcs are joined to form the star figure shown. Find the ratio of the area of the star figure to the area of the original circle. The original answer is in the form \(\frac{k-\pi}{m\pi}\). Please provide the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home. How old is Joey?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Six points are equally spaced around a circle of radius \(1\). Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle? The original answer is in \(\frac{\sqrt{k}}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Line \(l_1\) has equation \(3x - 2y = 1\) and goes through \(A = (-1, -2)\). Line \(l_2\) has equation \(y = 1\) and meets line \(l_1\) at point \(B\). Line \(l_3\) has positive slope, goes through point \(A\), and meets \(l_2\) at point \(C\). The area of \(\triangle ABC\) is \(3\). The original answer for the slope of \(l_3\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$A$ flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What is the percentage of the flowers that are carnations? Provide your answer as a percentage.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $S$ be the set of positive integers $n$ for which $\frac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. Find the sum of the elements of $S$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Ted's grandfather used his treadmill on $3$ days this week. He went $2$ miles each day. On Monday he jogged at a speed of $5$ miles per hour. He walked at the rate of $3$ miles per hour on Wednesday and at $4$ miles per hour on Friday. If Grandfather had always walked at $4$ miles per hour, he would have spent less time on the treadmill. How many minutes less did he spend?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. Find the perimeter of parallelogram $ADEF$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit $6$. For how many positive integers $b$ does the base-$b$-representation of $2013$ end in the digit $3$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The real numbers \(c, b, a\) form an arithmetic sequence with \(a \geq b \geq c \geq 0\). The quadratic \(ax^2+bx+c\) has exactly one root. The original answer is in the form \(-k+\sqrt{m}\). Please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The number $2013$ is expressed in the form
$2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,
where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. Find the value of $|a_1 - b_1|$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Samantha lives $2$ blocks west and $1$ block south of the southwest corner of City Park. Her school is $2$ blocks east and $2$ blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Six spheres of radius \(1\) are positioned so that their centers are at the vertices of a regular hexagon of side length \(2\). The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. The radius of this eighth sphere can be expressed in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest terms. Please find the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base $10$ with no leading zeros. A $6$-digit palindrome $n$ is chosen uniformly at random. Find the probability that \(\frac{n}{11}\) is also a palindrome. The original answer is in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Square $ABCD$ has side length $s$, and a circle centered at $E$ has radius $r$. Both $r$ and $s$ are rational numbers. The circle passes through $D$, and $D$ lies on $\overline{BE}$. Point $F$ lies on the circle, on the same side of $\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF = \sqrt{9 + 5\sqrt{2}}$. Find the ratio \(\frac{r}{s}\) in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In \(\triangle ABC\) shown in the figure, \(AB=7\), \(BC=8\), \(CA=9\), and \(\overline{AH}\) is an altitude. Points \(D\) and \(E\) lie on sides \(\overline{AC}\) and \(\overline{AB}\), respectively, so that \(\overline{BD}\) and \(\overline{CE}\) are angle bisectors, intersecting \(\overline{AH}\) at \(Q\) and \(P\), respectively. Find the length of \(PQ\). The answer is in the form \(\frac{k}{m}\sqrt{n}\), please provide the value of \(k + m + n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The sides of a right triangle are $a$ and $b$, and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, find the ratio of $r$ to $s$. The original answer is in the format of k:m, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The area of a circle inscribed in a regular hexagon is \(100\pi\). What is the area of the hexagon? The original answer is in the form of \(k\sqrt{m}\), please provide the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Mary is about to pay for five items at the grocery store. The prices of the items are $7.99, $4.99, $2.99, $1.99, and $0.99. Mary will pay with a twenty-dollar bill. What percentage of the $20.00 will she receive in change? Provide your answer as an integer percentage.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The graphs of $y = -|x - a| + b$ and $y = |x - c| + d$ intersect at points $(2,5)$ and $(8,3)$. Find the value of $a + c$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many unique $3$-letter sequences with no spaces can be made using the letters in "AUGUSTIN LOUIS CAUCHY", which contains $19$ letters? For example, "GAA" is one acceptable sequence, but "GGA" is not an acceptable sequence because there is only one G available. The original ordering of the letters does not have to be preserved.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

When \(p = \sum_{k=1}^{6} k \ln{k}\), the number \(e^p\) is an integer. What is the largest power of \(2\) that is a factor of \(e^p\)? If the answer is \(2^x\), what is the value of \(x\)?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The vertices of triangle \(PQR\) have coordinates as follows: \(P(0,a)\), \(Q(b,0)\), \(R(c,d)\), where \(a\), \(b\), \(c\), and \(d\) are positive. The origin and point \(R\) lie on opposite sides of \(PQ\). Find the expression for the area of triangle \(PQR\). The original answer is in the form \(\frac{kac + mbd - nab}{p}\), please give the value of k + m + n + p.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the adjoining figure, \( \triangle CDE \) is an equilateral triangle and \( \square ABCD \) and \( \square DEFG \) are squares. If the measure of \( \angle GDA \) is \( x^\circ \), what is the value of \( x \)?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If the angles $a, b, c, d,$ and $e$ are known, what is the expression for the angle $u$ in terms of $a$, $b$, $c$, $d$, and $e$? The original answer is in the form of m*a + n*c + p*d. Please provide the value of m + n + p.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The Tigers beat the Sharks $2$ out of the first $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017} = \log_b (a^{2017})$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the greatest integer $n$ such that $$n \leq 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{2014}}?$$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Pablo buys popsicles for his friends. The store sells single popsicles for $1$ each, 3-popsicle boxes for $2$, and 5-popsicle boxes for $3$. What is the greatest number of popsicles that Pablo can buy with $8$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many $5$-digit positive numbers which contain only odd numbers are there such that there is at least one pair of consecutive digits whose sum is $10$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The symbol $R_k$ stands for an integer whose base-ten representation is a sequence of $k$ ones. For example, $R_3=111$, $R_5=11111$, etc. When $R_{24}$ is divided by $R_4$, the quotient $Q=\frac{R_{24}}{R_4}$ is an integer whose base-ten representation is a sequence containing only ones and zeros. Find the number of zeros in $Q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Juan rolls a fair regular octahedral die marked with the numbers $1$ through $8$. Then Amal rolls a fair six-sided die. Find the probability that the product of the two rolls is a multiple of $3$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum
\begin{align*}
f\left(\frac{1}{2019}\right)-f\left(\frac{2}{2019}\right)+f\left(\frac{3}{2019}\right)-&f\left(\frac{4}{2019}\right)+\cdots\\
 &\,+f\left(\frac{2017}{2019}\right) - f\left(\frac{2018}{2019}\right)?
\end{align*}

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Beth adds the consecutive positive integers $a$, $b$, $c$, $d$, and $e$, and finds that the sum is a perfect square. She then adds $b$, $c$, and $d$ and finds that this sum is a perfect cube. What is the smallest possible value of $e$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. The distance $DE$ is in the form of k\sqrt{m}. Please find the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Represent the hypotenuse of a right triangle by $c$ and the area by $A$. What is the altitude on the hypotenuse? The original answer is in the form of $\frac{mA}{nc}$, where $m$ and $n$ are integers. Please give the value of $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A square with an area of \(2\) is inscribed within a larger square with an area of \(3\), forming four congruent triangles. Determine the ratio of the shorter leg to the longer leg in the shaded right triangle. The original answer is in the form \(k - \sqrt{m}\). Please provide the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many positive integers less than $1000$ are there such that they cannot be written as the sum of $2$ or more successive positive integers? Fill in the blank with the correct number.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If the points $(1, y_1)$ and $(-1, y_2)$ lie on the graph of $y = ax^2 + bx + c$, and $y_1 - y_2 = -6$, the original answer is in the form of $k$. Please find the value of $k$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Bertha has $6$ daughters and no sons. Some of her daughters have $6$ daughters, and the rest have none. Bertha has a total of $30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no children? Fill in the blank with the correct number.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let \(ABCD\) be an isosceles trapezoid with parallel bases \(\overline{AB}\) and \(\overline{CD}\), where \(AB > CD\). Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles with areas \(2\), \(3\), \(4\), and \(5\), starting with the triangle with base \(\overline{CD}\) and moving clockwise. The original answer for the ratio \(\frac{AB}{CD}\) is in the form \(k + \sqrt{m}\). Please calculate the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $0.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3$? The original answer is in the form \(\frac{k}{m}\), please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $K$ be the point of intersection of $AB$ and the line touching the circumcircle of $\triangle ABC$ at $C$ where $m(\widehat {A}) > m(\widehat {B})$. Let $L$ be a point on $[BC]$ such that $m(\widehat{ALB})=m(\widehat{CAK})$, $5|LC|=4|BL|$, and $|KC|=12$. Find the length of $|AK|$ in centimeters, denoted as x. Please provide the value of x.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A man with mass \(m\) jumps off a high bridge with a bungee cord attached to his ankles. The man falls through a maximum distance \(H\), at which point the bungee cord brings him to a momentary rest before he bounces back up. The bungee cord is perfectly elastic, obeying Hooke's force law with a spring constant \(k\), and stretches from an original length of \(L_0\) to a final length \(L = L_0 + h\). The maximum tension in the bungee cord is four times the weight of the man.

Determine the spring constant \(k\). The original answer is in the form \(\frac{kmg}{nH}\). Please provide the value of \(k+n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Liquid X does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$ cm thick. A rectangular box measuring $6$ cm by $3$ cm by $12$ cm is filled with liquid X. Its contents are poured onto a large body of water. The radius of the resulting circular film is in the form \(\sqrt{\frac{a}{\pi}}\). Please find the value of a + b + c, where a, b, and c are constants such that the expression under the square root is in the form \(\frac{a}{\pi}\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $U = 2 \times 2004^{2005}$, $V = 2004^{2005}$, $W = 2003 \times 2004^{2004}$, $X = 2 \times 2004^{2004}$, $Y = 2004^{2004}$, and $Z = 2004^{2003}$. Find the largest value among the expressions $U - V$, $V - W$, $W - X$, $X - Y$, and $Y - Z$. The answer should be given in the form of k - m, where k and m are placeholders for the original expressions. Provide the sum of the alphabetical positions of k and m as the final answer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A line passes through $A(1,1)$ and $B(100,1000)$. Find the number of other points with integer coordinates that are on the line and strictly between $A$ and $B$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In our number system, the base is ten. If the base were changed to four, you would count as follows: $1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, \ldots$ What would be the twentieth number in this system?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For a rhombus with a side length of 5, the length of one of its diagonals is not larger than $6$, and the length of the other diagonal is not smaller than $6$. Determine the maximum value of the sum of the two diagonals.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A plane intersects a right circular cylinder of radius $1$ forming an ellipse. If the major axis of the ellipse is $50\%$ longer than the minor axis, what is the length of the major axis?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For the integer numbers $i, j, k$ satisfying the condition $i^2 + j^2 + k^2 = 2011$, find the largest value of $i + j + k$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A sector with an acute central angle \(\theta\) is cut from a circle of radius 6. What is the radius of the circle circumscribed about the sector in terms of \(\theta\)? The original answer is in the form \(k \sec \frac{m}{n} \theta\). Please find the value of k + m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. The area of the sheet of wrapping paper can be expressed in the form $k(w+h)^m$. Please find the value of $k + m$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Assuming \(a\neq3\), \(b\neq4\), and \(c\neq5\), find the integer value of the following expression:
\[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\]
Please provide your answer as an integer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of the number of girls to the number of boys in her class? Provide your answer in the form of $a : b$, and then find the value of $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the guests to the rooms?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In rectangle \(ABCD\), \(DC = 2 \cdot CB\) and points \(E\) and \(F\) lie on \(\overline{AB}\) so that \(\overline{ED}\) and \(\overline{FD}\) trisect \(\angle ADC\) as shown. Find the ratio of the area of \(\triangle DEF\) to the area of rectangle \(ABCD\). The original answer is in the format \(\frac{\sqrt{k}}{m}\), please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? If the answer is $z$ degrees, what is the value of $z$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is the value of $k$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A \(4\times 4\times h\) rectangular box contains a sphere of radius \(2\) and eight smaller spheres of radius \(1\). The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. The original answer is in the form of \(k + m\sqrt{n}\). Please find the value of \(k + m + n\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. Find the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone. The original answer is in \(\frac{k+\sqrt{m}}{n}\) format, please give the value of k + m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? The answer should be in the form \(\frac{k}{m}\), where the fraction is fully simplified. Please find the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. Find the minimum possible value of $BE+DE+CD$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In rectangle $ABCD$, $\overline{AB}=20$ and $\overline{BC}=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. Find the length of $\overline{AE}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. Find how many pairs of integers $(m,n)$ there are such that $1\leq m\leq 2012$ and
$$5^n<2^m<2^{m+2}<5^{n+1}.$$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? Provide your answer as a simplified expression in the form of \(\frac{a+b\sqrt{c}}{d}\). Please find the value of a + b + c + d.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The number \(5^{867}\) is between \(2^{2013}\) and \(2^{2014}\). Find how many pairs of integers \((m,n)\) there are such that \(1\leq m\leq 2012\) and
\[5^n<2^m<2^{m+2}<5^{n+1}.\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a small pond there are eleven lily pads in a row labeled $0$ through $10.$ A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability \(\frac{N}{10}\) and to pad \(N+1\) with probability \(1-\frac{N}{10}\). Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Calculate the value of \(k+m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the sum of all the positive solutions of
\[2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1\]
If the answer is \(x\pi\), what is the value of \(x\)?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at $1:00$ PM and finishes the second task at $2:40$ PM. If she finishes the third task at $x$ PM, express the time $x$ in the format HH:MM PM, where HH and MM are two-digit numbers (possibly with leading zeros), and find the value of HH + MM.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The Tigers beat the Sharks $2$ out of the $3$ times they played. They then played $N$ more times, and the Sharks ended up winning at least $95\%$ of all the games played. What is the minimum possible value for $N$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Back in $1930$, Tillie had to memorize her multiplication facts from $0 \times 0$ to $12 \times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd? If the answer is $x$, what is the value of $100x$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$. Find the value of $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$. The original answer is in the form -\frac{m}{n}, where m and n are integers. Please provide the value of m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$? Fill in the blank with the correct number.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last $2$ marbles. What is the probability that Cheryl gets $2$ marbles of the same color? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a$, $b$, and $c$ be three distinct one-digit numbers. Find the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$. If the answer is $y$, what is the value of $10y$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have? Provide your answer as an integer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The line \(12x+5y=60\) forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle? The original answer is in the format \(\frac{k}{m}\), please give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$ so that the mean (average) of the remaining numbers is $6$? Provide the number of such subsets.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$? If the answer is $y$, what is the value of $10y$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A regular hexagon with sides of length $6$ has an isosceles triangle attached to each side. Each of these triangles has two sides of length $8$. The isosceles triangles are folded to make a pyramid with the hexagon as the base of the pyramid. What is the volume of the pyramid? If the answer is in the form $a\sqrt{b}$, what is the value of $a$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\frac{1}{3}$ of all the ninth graders are paired with $\frac{2}{5}$ of all the sixth graders, find the fraction of the total number of sixth and ninth graders that have a buddy. The original answer is in \frac{k}{m} format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Hexadecimal (base-$16$) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. Find the sum of the digits of $n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The isosceles right triangle \(ABC\) has a right angle at \(C\) and an area of \(12.5\). The rays trisecting \(\angle ACB\) intersect \(AB\) at \(D\) and \(E\). Find the area of \(\bigtriangleup CDE\). The original answer is in the form \(\frac{k - m\sqrt{n}}{p}\). Please provide the value of \(k + m + n + p\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. Find the perimeter of the triangle. The original answer is in the form $k + m\sqrt{n}$, please give the value of $k + m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For every positive integer $n$, let ${mod}_5 (n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f: \{0,1,2,3,\cdots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively as follows:
$$f(i,j)=\begin{cases}\operatorname{mod}_5(j+1)&\text{if}i=0\text{and}0\le j\le4,\\f(i-1,1)&\text{if}i\ge1\text{and}j=0\text{,and}\\f(i-1,f(i,j-1))&\text{if}i\ge1\text{and}1\le j\le4.\end{cases}$$
Find the value of $f(2015,2)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. Find the value of $FG + JH + CD$. The original answer is in the form k + \sqrt{m}, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. Find the coordinates of $p_{2015}$ in the form $(x, y)$, and provide the sum $x + y$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. Find the value of $\overline{AR}+\overline{BR}+\overline{CR}+\overline{DR}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. Find the number of ordered triples $(a,b,c)$ that are possible.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A collection of circles in the upper half-plane, all tangent to the \(x\)-axis, is constructed in layers as follows. Layer \(L_0\) consists of two circles of radii \(70^2\) and \(73^2\) that are externally tangent. For \(k \ge 1\), the circles in \(\bigcup\limits_{j=0}^{k-1}L_j\) are ordered according to their points of tangency with the \(x\)-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer \(L_k\) consists of the \(2^{k-1}\) circles constructed in this way. Let \(S=\bigcup\limits_{j=0}^{6}L_j\), and for every circle \(C\) denote by \(r(C)\) its radius. Find \(\sum\limits_{C\in S} \frac{1}{\sqrt{r(C)}}\). The original answer is in the form of \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

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 $a+b+c+d$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A rectangular box has integer side lengths in the ratio $1: 3: 4$. What could be the volume of the box?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Which of the following numbers is not a perfect square: $1^{2016}$, $2^{2017}$, $3^{2018}$, $4^{2019}$, or $5^{2020}$? The original answer is in the form $x^{y}$, please give the value of $x + y$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the tens digit of $2015^{2016}-2017?$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $M=\left\{(x, y) \mid x^{2}+2 y^{2}=3\right\}, N=\{(x, y) \mid y=m x+b\}$ 。若对于所有的 $m \in \mathbf{R}$, 均有 $M \cap N \neq \varnothing$, 求 $b$ 的取值范围。答案的形式为\left[-\frac{\sqrt{c}}{d}, \frac{\sqrt{c}}{d}\right]，请给出c和d的绝对值之和。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is the value of $a+b$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area of Carl’s garden in square yards?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$, and the line $x=5.1$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. Find the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two different numbers are randomly selected from the set \(\{ -2, -1, 0, 3, 4, 5\}\) and multiplied together. What is the probability that the product is \(0\)? Express your answer as a fraction in simplest form, \(\frac{k}{m}\), and give the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Circles with centers \(P, Q\) and \(R\), having radii \(1, 2\) and \(3\), respectively, lie on the same side of line \(l\) and are tangent to \(l\) at \(P', Q'\) and \(R'\), respectively, with \(Q'\) between \(P'\) and \(R'\). The circle with center \(Q\) is externally tangent to each of the other two circles. Find the area of triangle \(PQR\). The original answer is in the form \(\sqrt{a} - \sqrt{b}\), please give the value of a + b.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? Provide your answer as an integer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the area of the region enclosed by the graph of the equation \(x^2+y^2=|x|+|y|?\) Express your answer in the form \(a\pi + b\), where \(a\) and \(b\) are constants. Please provide the value of \(a + b\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s,$ and $t$ is $1.$ What is the value of $r+s+t$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Rectangle \(ABCD\) has \(AB=5\) and \(BC=4\). Point \(E\) lies on \(\overline{AB}\) so that \(EB=1\), point \(G\) lies on \(\overline{BC}\) so that \(CG=1\), and point \(F\) lies on \(\overline{CD}\) so that \(DF=2\). Segments \(\overline{AG}\) and \(\overline{AC}\) intersect \(\overline{EF}\) at \(Q\) and \(P\), respectively. The value of \(\frac{PQ}{EF}\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is the value of $a + b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q.$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let \(ABCD\) be a unit square. Let \(Q_1\) be the midpoint of \(\overline{CD}\). For \(i=1,2,\dots,\) let \(P_i\) be the intersection of \(\overline{AQ_i}\) and \(\overline{BD}\), and let \(Q_{i+1}\) be the foot of the perpendicular from \(P_i\) to \(\overline{CD}\). Find the value of
\[\sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \,.\]
The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Three numbers in the interval \([0,1]\) are chosen independently and at random. Find the probability that the chosen numbers are the side lengths of a triangle with positive area. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$? Please provide your answer as an integer without any commas.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's original number?

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