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 polynomial \((x+y)^9\) is expanded in decreasing powers of \(x\). The second and third terms have equal values when evaluated at \(x=p\) and \(y=q\), where \(p\) and \(q\) are positive numbers whose sum is one. 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.

The number $n$ is called a composite number if it can be written in the form $n = a \times b$, where $a, b$ are positive integers greater than $1$. Write the number $2013$ as a sum of $m$ composite numbers. What is the largest possible value of $m$? Please provide the value of 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.

Triangle $ABC$ has side-lengths $AB=12$, $BC=24$, and $AC=18$. The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. Find the perimeter of $\triangle AMN$.

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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 the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. If the area 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.

What is the average length in letters of a word in this question? 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.

A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, find the smallest distance that the grasshopper can be away from the point $2010$.

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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.

Right triangle \(ACD\) with a right angle at \(C\) is constructed outwards on the hypotenuse \(\overline{AC}\) of isosceles right triangle \(ABC\) with leg length \(1\), as shown, so that the two triangles have equal perimeters. Find the value of \(\sin(2\angle BAD)\). 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.

Line \(\ell_1\) has equation \(3x-2y=1\) and goes through \(A=(-1,-2)\). Line \(\ell_2\) has equation \(y=1\) and meets line \(\ell_1\) at point \(B\). Line \(\ell_3\) has positive slope, goes through point \(A\), and meets \(\ell_2\) at point \(C\). The area of \(\triangle ABC\) is \(3\). The slope of \(\ell_3\) is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. 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.

Let \(ABCDA'B'C'D'\) be a rectangular prism with \(|AB|=2|BC|\). \(E\) is a point on the edge \([BB']\) satisfying \(|EB'|=6|EB|\). Let \(F\) and \(F'\) be the feet of the perpendiculars from \(E\) at \(\triangle AEC\) and \(\triangle A'EC'\), respectively. If \(m(\widehat{FEF'})=60^{\circ}\), find the value of \(|BC|/|BE|\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\). 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.

Find the number of triples \((a,b,c)\) of positive integers which satisfy the simultaneous equations:
\[ ab + bc = 44, \]
\[ ac + bc = 23. \]

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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 the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ are there that do not contain 4 consecutive integers?

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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.

After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The original answer is in the format k: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.

A student is asked to measure $30.0 \, \text{g}$ of methanol $(d=0.7914 \, \text{g/mL \text{ at } 25^{\circ}\text{C}})$ but has only a graduated cylinder with which to measure it. Calculate the volume of methanol, in mL, that the student should use to obtain the required $30.0 \, \text{g}$, and then multiply this volume by 10. What is the resulting value?

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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.

Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's by $5070$. What was the sum of their original numbers?

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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 number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$. 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.

If the following instructions are carried out by a computer, what is the value of $X$ that will be printed because of instruction 5?

1. Start $X$ at $3$ and $S$ at $0$.
2. Increase the value of $X$ by $2$.
3. Increase the value of $S$ by the value of $X$.
4. If $S$ is at least $10000$, then go to instruction 5; otherwise, go to instruction 2 and proceed from there.
5. Print the value of $X$.
6. Stop.

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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 pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. Find the volume of this cube. The original answer is in the form \(k\sqrt{2} + m\), 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.

Jerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis.

Dennis then announces, "Aaron's number is at least three times Jerry's number."

Aaron says, "I still don't know Jerry's number."

Jerry then replies, "Oh, now I know Aaron's number."

What is the sum of their numbers?

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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.

Five unit squares are arranged in the coordinate plane with the lower left corner at the origin. The slanted line, extending from \((a,0)\) to \((3,3)\), divides the entire region into two regions of equal area. 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.

LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip, it turned out that LeRoy had paid \(A\) dollars and Bernardo had paid \(B\) dollars, where \(A < B\). How many dollars must LeRoy give to Bernardo so that they share the costs equally? Provide your answer in the form of \(\frac{k(B-A)}{m}\), and then 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.

\(y\) varies inversely as the square of \(x\). When \(y=16\), \(x=1\). Find the value of \(y\) when \(x=8\). 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.

A parking lot has $16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires $2$ adjacent spaces. What is the probability that she is able to park? Express your answer in the form \(\frac{k}{m}\), where the fraction is simplified, and 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.

The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. Find the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is 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 $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|BD|=2$ and $|DC|=6$. If $|AB|=4$ and $m(\widehat{ACB})=20^\circ$, then what is the value of $m(\widehat{BAD})$ in degrees?

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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 circle with radius $4$ passing through points $A$ and $B$ is tangent to another circle with radius $3$ passing through points $A$ and $C$, where the distance $|AB| = 2$. If the line $BC$ is tangent to the second circle, what is the value of $|BC|^2$?

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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.

Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. How many distinct lines pass through at least two members 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 circle $C_1$ with radius $6$ and the circle $C_2$ with radius $8$ are externally tangent to each other at $A$. The circle $C_3$, which is externally tangent to both $C_1$ and $C_2$, has a radius of $21$. The common tangent of $C_1$ and $C_2$ that passes through $A$ meets $C_3$ at points $B$ and $C$. The original answer is in the form $k\sqrt{m}$, 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.

What is the area enclosed by the graph of $|3x| + |4y| = 12$?

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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 class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$ correct answers, at least $2$ of any $40$ gave exactly $2$ correct answers, and at least $3$ of any $40$ gave exactly $1$ correct answer. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers?

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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 non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ such that the sum of the three interior angles at $A$ is $360^\circ$. Thus, the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can 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 regular polygon with $n$ sides is inscribed in a circle with radius $R$. The area of the polygon is $3R^2$. Determine the 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 sides of a triangle are \(25, 39,\) and \(40\). What is the diameter of the circumscribed circle? 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.

If the tangents of all interior angles of a triangle are integers, what is the sum of these integers?

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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.

At a dance party, a group of boys and girls exchange dances as follows: one boy dances with $5$ girls, a second boy dances with $6$ girls, and so on, with the last boy dancing with all the girls. If $b$ represents the number of boys and $g$ the number of girls, find the relation between $b$ and $g$. The original answer is in the form $b = mg + n$. 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.

Let \(ABC\) be a triangle with \(m(\widehat{A})=90^\circ\), \(|AB|=4\), and \(|AC|=3\). Let \(D\) be the foot of the perpendicular from \(A\) to \([BC]\). If \(P\) is a point on \([BD]\) such that \(5|AP|=13|PD|\), find the length of \(|CP|\). 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.

In triangle $ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Let $D$ denote the midpoint of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the bisector of angle $BAC$. Find the area of the triangle $ADE$ that is closest to 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 sequence $a_1$, $a_2$, $a_3$, $\dots$ satisfies $a_1 = 19$, $a_9 = 99$, and, for all $n \ge 3$, $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find the value of $a_2$.

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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.

Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB}) = 11^{\circ}$, and $m(\widehat{EBC}) = 71^{\circ}$. If the answer is $x^{\circ}$, find the value of $x$ for $m(\widehat{DCE})$.

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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 circle with center at \(O\) and radius \(r\), chord \(AB\) is drawn with length equal to \(r\) (units). From \(O\), a perpendicular to \(AB\) meets \(AB\) at \(M\). From \(M\), a perpendicular to \(OA\) meets \(OA\) at \(D\). In terms of \(r\), find the area of triangle \(MDA\) in appropriate square units. The original answer is in the form \(\frac{k r^2 \sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers. 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.

If the sum of digits in the decimal representation of a positive integer $n$ is $111$ and the sum of digits in the decimal representation of $7002n$ is $990$, what is the sum of digits in the decimal representation of $2003n$?

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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 how many primes $p$ less than $15$, there exist integer triples $(m,n,k)$ such that:
\[
\begin{array}{rcl}
m+n+k &\equiv& 0 \pmod{p} \\
mn+mk+nk &\equiv& 1 \pmod{p} \\
mnk &\equiv& 2 \pmod{p}.
\end{array}
\]
Find the number of such primes $p$.

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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.

Let $ABCD$ be a convex quadrilateral such that $m (\widehat{ADB}) = 15^{\circ}$ and $m (\widehat{BCD}) = 90^{\circ}$. The diagonals of the quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC| = 4$, $|EA| = 8$, and $|EP| = 2$. 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.

The acronym AMC is shown in the rectangular grid below with grid lines spaced \(1\) unit apart. In units, find the sum of the lengths of the line segments that form the acronym AMC. The original answer is in the form \(a + b\sqrt{c}\), please provide the value of \(a + b + c\).

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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.

Let $C$ and $D$ be points on the circle with center $O$ and diameter $[AB]$, where $C$ and $D$ are on different semicircles with diameter $[AB]$. Let $H$ be the foot of the perpendicular from $B$ to $[CD]$. If $|AO| = 13$, $|AC| = 24$, and $|HD| = 12$, what is $\widehat{DCB}$ in degrees? 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.

Larry always orders pizza with exactly two of his three favorite toppings: pepperoni, bacon, and sausage. If he has ordered a total of $600$ pizzas and has had each topping equally often, how many pizzas has he ordered with pepperoni? 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.

Find the value of \( \frac {1}{2 - \frac {1}{2 - \frac {1}{2 - \frac12}}} \). The original answer 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.

A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile, the speed is constant. If the second mile is traversed in $2$ hours, find the time, in hours, needed to traverse the $n$th mile. The original answer is in the form $k(n-m)$, 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 point object of mass \(m\) is connected to a cylinder of radius \(R\) via a massless rope. At time \(t = 0\), the object is moving with an initial velocity \(v_0\) perpendicular to the rope. The rope has a length \(L_0\) and a non-zero tension. All motion occurs on a horizontal frictionless surface. The cylinder remains stationary and does not rotate. The object moves in such a way that the rope slowly winds up around the cylinder. The rope will break when the tension exceeds \(T_{max}\). Express your answers in terms of \(T_{max}\), \(m\), \(L_0\), \(R\), and \(v_0\). The original answer is in the form \(\frac{k m^2 v_0^3}{n T_{max}}\). Please calculate 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.

Let \(S_1\) be the area of the regular pentagon \(ABCDE\). And let \(S_2\) be the area of the regular pentagon whose sides lie on the lines \(AC, CE, EB, BD, DA\). The original answer is in the form \(\frac{k+m\sqrt{5}}{n}\). 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 circle passes through the vertices of a triangle with side-lengths of \(7\frac{1}{2}, 10, 12\frac{1}{2}\). The radius of the circle can be expressed in the form \(\frac{k}{m}\). 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.

If \(V = gt + V_0\) and \(S = \frac{1}{2}gt^2 + V_0t\), find the expression for \(t\) in terms of \(S\), \(V\), and \(V_0\). The original answer is in the form \(\frac{kS}{m(V + V_0)}\), 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.

We draw a circle with radius $5$ on a gridded paper where the grid consists of squares with sides of length $1$. The center of the circle is placed in the middle of one of the squares. Find the number of squares through which the circle passes. 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.

How many pairs of integers \((x, y)\) are there such that \(2x + 5y = xy - 1\)?

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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.

Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. Find the probability that Flora will eventually land at $10$. 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.

For how many positive integer values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities \[ \begin{cases} 2x > 3x - 3 \\ 3x - a > -6 \end{cases} \]?

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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 $78$ is divided into three parts which are proportional to $1, \frac{1}{3}, \frac{1}{6}$, find the value of the middle part in its simplest fractional form. 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.

In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box? 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.

How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?

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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 all positive integers $n$, the function $f$ satisfies the following conditions:
- $f(1) = 1$
- $f(2n + 1) = 2f(n)$
- $f(2n) = 3f(n) + 2$

Determine how many positive integers $x \leq 100$ make the value of $f(x)$ odd.

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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 perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \, \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \, \text{cm}$. The square has a perimeter greater than 0. How many positive integers are NOT possible values for $d$?

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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 quadrilateral has consecutive sides of lengths $70$, $90$, $130$, and $110$. It is both inscribed in a circle and has a circle inscribed within it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find 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.

In the diagram below, for each row except the bottom row, the number in each cell is determined by the sum of the two numbers beneath it. Find the sum of all cells denoted with a question mark.
```latex
\begin{asy}
unitsize(2cm);
path box = (-0.5,-0.2)--(-0.5,0.2)--(0.5,0.2)--(0.5,-0.2)--cycle;
draw(box); label("$2$",(0,0));
draw(shift(1,0)*box); label("$?$",(1,0));
draw(shift(2,0)*box); label("$?$",(2,0));
draw(shift(3,0)*box); label("$?$",(3,0));
draw(shift(0.5,0.4)*box); label("$4$",(0.5,0.4));
draw(shift(1.5,0.4)*box); label("$?$",(1.5,0.4));
draw(shift(2.5,0.4)*box); label("$?$",(2.5,0.4));
draw(shift(1,0.8)*box); label("$5$",(1,0.8));
draw(shift(2,0.8)*box); label("$?$",(2,0.8));
draw(shift(1.5,1.2)*box); label("$9$",(1.5,1.2));
\end{asy}
```

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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 perimeter of the right triangle whose exradius of the hypotenuse is $30$?

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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 \( y = x + \frac{1}{x} \), then transform the equation \( x^4 + x^3 - 4x^2 + x + 1 = 0 \) into the form \( x^2(y^2 + y - k) = 0 \). 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.

$ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Find the value of $\sin \theta$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

```asy
\draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
\draw((0,0)--(2,1));
\draw((0,0)--(1,2));
\label("A", (0,0), SW);
\label("B", (0,2), NW);
\label("C", (2,2), NE);
\label("D", (2,0), SE);
\label("M", (1,2), N);
\label("N", (2,1), E);
\label("$\theta$", (.5,.5), SW);
```

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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 $\triangle ABC$, we have $AC = BC = 7$ and $AB = 2$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 8$. What is the length of $BD$?

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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 \(E\), \(F\), \(G\) be points on sides \([AB]\), \([BC]\), \([CD]\) of the rectangle \(ABCD\), respectively, such that \(|BF|=|FQ|\), \(m(\widehat{FGE})=90^\circ\), \(|BC|=\frac{4\sqrt{3}}{5}\), and \(|EF|=\sqrt{5}\). The value of \(|BF|\) is in the form \(\frac{\sqrt{m} - \sqrt{n}}{p}\). Please find 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.

Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. The probability of obtaining a six on any toss is \(\frac{1}{6}\), independent of the outcome of any other toss. 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.

In right triangle \(ABC\), \(BC = 5\), \(AC = 12\), and \(AM = x\); \(\overline{MN} \perp \overline{AC}\), \(\overline{NP} \perp \overline{BC}\); \(N\) is on \(AB\). If \(y = MN + NP\), one-half the perimeter of rectangle \(MCPN\), find the value of \(y\). The original answer is in the form \(\frac{k - mx}{n}\). 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.

If the function \( f \) satisfies the equation \( f(x) + f\left( \frac{1}{\sqrt[3]{1-x^3}}\right) = x^3 \) for every real \( x \neq 1 \), find the value of \( f(-1) \). The original answer is in the form \(\frac{k}{m}\), where 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 square has sides of length $10$, and a circle centered at one of its vertices has radius $10$. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in the form \(a + b\pi\), where \(a\) and \(b\) are integers. 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.

Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping squares, respectively. If the pattern were continued, find the number of nonoverlapping squares in figure $100$.

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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.

Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1$? Provide your answer in the form of \frac{k}{m}, and 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.

By inserting parentheses, it is possible to give the expression $2 \times 3 + 4 \times 5$ several values. How many different values can be obtained?

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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 coin is altered so that the probability that it lands on heads is less than \(\frac{1}{2}\). When the coin is flipped four times, the probability of getting an equal number of heads and tails is \(\frac{1}{6}\). What is the probability that the coin lands on heads? Express your answer in the form \(\frac{k - \sqrt{m}}{n}\), where the fraction is in its simplest form. Please calculate 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.

Adams plans a profit of $10\%$ on the selling price of an article and his expenses are $15\%$ of sales. If the rate of markup on an article that sells for $\$5.00$ is $x\%$, find the value of $3x$. Provide your answer as an integer without the percentage symbol.

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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 sequence $2001, 2002, 2003, \ldots$, each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$. What is the value of the $2004^\text{th}$ term in this sequence?

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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 graphs $y = Ax^2$ and $y^2 + 3 = x^2 + 4y$, where $A$ is a positive constant and $x$ and $y$ are real variables. Determine the number of points at which the two graphs intersect.

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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.

At a party with $n \geq 4$ people, if every 3 people have exactly 1 common friend, find the number of different values that $n$ can 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.

Find the integer that does not divide $n^{2225} - n^{2005}$ for every integer 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.

Find the number of primes $p$, such that $x^{3} - 5x^{2} - 22x + 56 \equiv 0 \pmod{p}$ has no three distinct integer roots in $[0, p)$.

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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.

Let $E$ be a point inside the rhombus $ABCD$ such that $|AE| = |EB|$, $m(\widehat{EAB}) = 12^\circ$, and $m(\widehat{DAE}) = 72^\circ$. What is the value of $m(\widehat{CDE})$ in degrees?

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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 how many integers \( n \) between 1 and 100 does \( x^2 + x - n \) factor into the product of two linear factors with integer coefficients?

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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 $k$ which makes the statement "When we take any $6$ subsets with $5$ elements of the set \( \{1,2,\dots, 9\} \), there exist $k$ of them having at least one common element." true?

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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 $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. Find the number of different possible denominators.

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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 six-sided dice are fair in the sense that each face is equally likely to turn up. However, one of the dice has the $4$ replaced by $3$ and the other die has the $3$ replaced by $4$. When these dice are rolled, find the probability that the sum is an odd number. 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 the roots of the polynomial \(x^3 + 4x^2 - 7x - 1\). Find the polynomial that has roots \(ab\), \(bc\), and \(ac\). The original answer is in the form \(mx^3 + nx^2 + px + q\). Please give the value of \(m + n + p + q\).

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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.

Let $ABCD$ be a convex quadrilateral with $\angle C = \angle D = 90^\circ$. The circle $K$ passing through $A$ and $B$ is tangent to $CD$ at $C$. Let $E$ be the intersection of $K$ and $[AD]$. If $|BC| = 20$, $|AD| = 16$, then find the length of $|CE|$. 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.

Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?

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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 circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 30^\circ$, and $\angle DOB = 45^\circ$. Find the ratio of the area of the smaller sector $COD$ to the area of the circle. 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 $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P \in [AC]$, $Q \in [BC]$, $R \in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K \in [BC]$, $M \in [AB]$, and $L \in [AC]$. Find $|AB|+|AC|$.

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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 \(D\) be a point on side \([BC]\) of \(\triangle ABC\) such that \(|AB|=3\), \(|CD|=1\), and \(|AC|=|BD|=\sqrt{5}\). If the \(B\)-altitude of \(\triangle ABC\) meets \(AD\) at \(E\), find the length of \(|CE|\). 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.

When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. What is the smallest possible value 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.

Given three squares as in the figure, where the largest square has area 1, and the area $A$ is known. Find the area $B$ of the smallest square in terms of $A$. The original answer is in the form \frac{(k - A)^m}{n}, where k, m, and n are integers. Please calculate 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.

If $a$ and $b$ are the roots of the equation $x^2 - 2cx - 5d = 0$, and $c$ and $d$ are the roots of the equation $x^2 - 2ax - 5b = 0$, where $a, b, c, d$ are distinct real numbers, 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.

If \(x \neq 0\) or \(4\) and \(y \neq 0\) or \(6\), then \(\frac{2}{x}+\frac{3}{y}=\frac{1}{2}\) is equivalent to the equation \(\frac{ay}{y-b}=x\). Please determine the sum of 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.

Two medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \sqrt{15}$ square inches. Find the length of the third median in inches. 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.

For every $n \ge 2$, $a_n = \frac{\sqrt[3]{n^3 + n^2 - n - 1}}{n}$. Find the least value of positive integer $k$ satisfying $a_2a_3\cdots a_k > 3$.

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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 pattern of figures $\triangle$ $\bullet$ $\square$ $\blacktriangle$ $\circ$ is repeated in the sequence:
\[
\triangle, \bullet, \square, \blacktriangle, \circ, \triangle, \bullet, \square, \blacktriangle, \circ, \ldots
\]
Determine the 214th figure in the sequence. Write the position of the figure in the pattern as a number from 1 to 5.

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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 sum of distinct real numbers \(x\) such that \((2x^2+5x+9)^2=56(x^3+1)\)? 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 ordered triples of integers $(a, b, c)$ satisfy the following equations?
\[ |a + b| + c = 19 \]
\[ ab + |c| = 97 \]
Provide the number of such ordered triples.

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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 integers \( a \), \( b \), and \( c \), each greater than 1, such that \( \sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}} \) for all \( N > 1 \). Find the value of \( b \).

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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.

What is the sum of the real roots of the equation $(x\sqrt{x})^x = x^{x\sqrt{x}}$? Provide your answer in the form of 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.

Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have an area of $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? 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.

Emily has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any previous boxes. How many balls in total are in the boxes as a result of Emily's $2017$th 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.

Let $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\sqrt{6}$, $m(\widehat{ABD})=60^\circ$, $m(\widehat{BDC})=45^\circ$, and $|BD|=13+3\sqrt{3}$. Find the length of $|AC|$.

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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 zeroes of the function $f(x) = x^2 - ax + 2a$ are integers. Find the sum of all possible values 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 $f(x) = x - \frac{1}{x}$. How many different solutions are there to the equation $f(f(f(x))) = 1$? 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.

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0)$, $(2020, 0)$, $(2020, 2020)$, and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) If the answer is $d$ to the nearest tenth, what is the value of $10d$?

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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.

Lines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1)$. Find the equation of one of these lines. The equation is in the form mx + ny + p = 0. Please give 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.

Let \[x = .123456789101112\ldots998999,\] where the digits are obtained by writing the integers 1 through 999 in order. What is the 1983rd digit to the right of the decimal point?

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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 value of \(\dfrac{3\times 5}{9\times 11}\times \dfrac{7\times 9\times 11}{3\times 5\times 7}\).

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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 whole numbers between $100$ and $400$ contain the digit $2$? Fill in the blank with the number of such whole numbers.

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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 grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ 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 large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. Find the total number of smaller cubes that must have at least one face shaded.

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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 $A$ and $B$ are nonzero digits, then find the number of digits (not necessarily different) in the sum of the three whole numbers:
$$\begin{array}{cccc}9&8&7&6\\&{A}&3&2\\&{B}&1&\\\hline\end{array}$$

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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 rose bushes are spaced about 1 foot apart, approximately how many bushes are needed to surround a circular patio whose radius is 12 feet? 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.

A fifth number, $n$, is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median. Find the number of 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.

When placing each of the digits $2,4,5,6,9$ in exactly one of the boxes of this subtraction problem, find the smallest possible difference.

$$\begin{array}{rrrr}&&\square&\square&\square\\&-&&\square&\square\\\hline\end{array}$$

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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 $47$ be written as the sum of two primes?

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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.

Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even? Express your answer as a fraction in simplest form. 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.

What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?

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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 decimal number $0.12345$, which digit, when changed to $9$, gives the largest possible number? Indicate the original position of this 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.

If an integer of two digits is \(k\) times the sum of its digits, find the expression that represents the number formed by interchanging the digits as the sum of the digits multiplied by \(x\). The original answer is in the form \(m - k\). 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.

Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then how many students are at the table?

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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 different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different. Find the number of such different patterns.

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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 largest quotient that can be formed using two numbers chosen from the set $\{ -24, -3, -2, 1, 2, 8 \}$?

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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.

A cube of edge $3 \text{ cm}$ is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, find the 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 digit-sum of $998$ is $9+9+8=26$. Find the number of $3$-digit whole numbers, whose digit-sum is $26$, that are even.

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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 distance between the $5^\text{th}$ and $26^\text{th}$ exits on an interstate highway is $118$ miles. If any two consecutive exits are at least $5$ miles apart, find the largest number of miles there can be between two consecutive exits that are between the $5^\text{th}$ and $26^\text{th}$ exits.

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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 rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. Find the area of quadrilateral $ABDF$.

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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 value of $(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193)$.

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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 $10^{93}-93$ is expressed as a single whole number, what is the sum of the digits?

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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 children at a time can play pairball. For $90$ minutes, with only two children playing at a time, five children take turns so that each one plays the same amount of time. How many minutes does each child play?

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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 \(W, X, Y,\) and \(Z\) be four different digits selected from the set

\(\{ 1,2,3,4,5,6,7,8,9\}.\)

Find the smallest possible value of the sum \(\frac{W}{X} + \frac{Y}{Z}\). The original answer is in the format \(\frac{m}{n}\). 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.

A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green squares. 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.

Find the sum of the digits in the answer to

$\underbrace{9999\cdots 99}_{94\mathrm{ nines}} \times \underbrace{4444\cdots 44}_{94\mathrm{ fours}}$

where a string of $94$ nines is multiplied by a string of $94$ fours. What is the sum of the digits in the result?

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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 operation has the same effect on a number as multiplying by \(\dfrac{3}{4}\) and then dividing by \(\dfrac{3}{5}\)? Provide your answer in the form of a fraction \(\frac{k}{m}\), 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.

Six different digits from the set
$\{ 1,2,3,4,5,6,7,8,9\}$
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is $23$ and the sum of the entries in the horizontal row is $12$.
What is the sum of the six digits used?

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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 containing three different numbers can be selected from the set
$\{ 89,95,99,132,166,173 \}$
so that the sum of the three numbers is even?

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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.

Ahn chooses a two-digit integer, subtracts it from $200$, and doubles the result. What is the largest number Ahn can get? 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.

Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. Find the ratio of the area of the inscribed square to the area of the large square. The original answer is in \(\dfrac{k}{m}\) 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.

There are positive integers that have these properties:

the sum of the squares of their digits is $50$, and
each digit is larger than the one to its left.
Find the product of the digits of the largest integer with both properties.

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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.

All of the even numbers from $2$ to $98$ inclusive, excluding those ending in $0,$ are multiplied together. What is the rightmost digit (the units digit) of the product?

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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 value of \(100 \times 19.98 \times 1.998 \times 1000\) and express your answer in the form of \(x^2\). 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.

Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.
Rule $1$: If the integer is less than $10$, multiply it by $9$.
Rule $2$: If the integer is even and greater than 9, divide it by $2$.
Rule $3$: If the integer is odd and greater than $9$, subtract $5$ from it.
$A$ sample sequence: $23, 18, 9, 81, 76, \cdots$.
Find the value of the $98^\text{th}$ term of the sequence that begins $98, 49, \cdots$.

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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 board of $8$ columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered $1$ through $8$, row two is $9$ through $16$, and so on. A student shades square $1$, then skips one square and shades square $3$, skips two squares and shades square $6$, skips $3$ squares and shades square $10$, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?

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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 the five numbers $1, 4, 7, 10$, and $13$ is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. What is the largest possible value for the horizontal or vertical sum?

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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 far-off land, three fish can be traded for two loaves of bread, and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? Provide your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers, 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.

Points $B$, $D$, and $J$ are midpoints of the sides of right triangle $ACG$. Points $K$, $E$, $I$ are midpoints of the sides of triangle $JDG$, etc. If the dividing and shading process is done 100 times (the first three are shown) and $AC=CG=6$, then find the total area of the shaded triangles rounded to the nearest 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.

In the year $2001$, the United States will host the International Mathematical Olympiad. Let $I, M,$ and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. What is the largest possible value of the sum $I + M + O$?

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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.

Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, find how many of the following four quantities change:

$(a)$ the length of the segment $MN$

$(b)$ the perimeter of $\triangle PAB$

$(c)$ the area of $\triangle PAB$

$(d)$ the area of trapezoid $ABNM$

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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 $|x - 2| = p$, where $x < 2$, then find the value of $x - p$. The original answer is in the form $a - bp$, 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 \(ABCD\), \(AD=1\), \(P\) is on \(\overline{AB}\), and \(\overline{DB}\) and \(\overline{DP}\) trisect \(\angle ADC\). The perimeter of \(\triangle BDP\) is in the form \(k+\frac{m\sqrt{n}}{3}\), where \(k\), \(m\), and \(n\) are integers. 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.

Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? 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.

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walters entered?

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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, M, and C be nonnegative integers such that A + M + C = 12. What is the maximum value of \( A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C \)?

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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.

Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$. The original answer is in the form -\frac{k}{m}, where the fraction is in simplest terms. 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 checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$, the second row $18,19,\ldots,34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$, the second column $14,15,\ldots,26$ and so on across the board, find the sum of the numbers in the squares that have the same numbers in both numbering systems.

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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 centered at \(O\) has radius \(1\) and contains the point \(A\). The segment \(AB\) is tangent to the circle at \(A\) and \(\angle AOB = \theta\). If point \(C\) lies on \(\overline{OA}\) and \(\overline{BC}\) bisects \(\angle ABO\), find the value of \(OC\). The original answer is in the form \(\frac{m}{n+\sin\theta}\). 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.

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is \(m\) times the area of the square. The original answer is in \(\frac{k}{nm}\) format, where \(m\) is given in the problem. Please find 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.

In triangle $ABC$, $AB = 13$, $BC = 14$, $AC = 15$. Let $D$ denote the midpoint of $\overline{BC}$ and let $E$ denote the intersection of $\overline{BC}$ with the bisector of angle $BAC$. Find the integer closest to the area of the triangle $ADE$.

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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 $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=10$. Find the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A$.

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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.

Keiko tosses one penny and Ephraim tosses two pennies. Find the probability that Ephraim gets the same number of heads that Keiko gets. Express your answer as a fraction in simplest form. The answer 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.

The area of rectangle $ABCD$ is $72$ units squared. If point $A$ and the midpoints of $\overline{BC}$ and $\overline{CD}$ are joined to form a triangle, what is the area of that triangle?

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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 of the twelve pentominoes pictured below have at least one line of reflectional symmetry?

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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 dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. The speed of sound is $1088$ feet per second and one mile is $5280$ feet. Estimate, to the nearest half-mile, how far Snoopy was from the flash of lightning. If the answer is $x$ miles, 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 charity sells $140$ benefit tickets for a total of $＄2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money, in dollars, is raised by the full-price tickets? 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.

How many positive integers not exceeding $2001$ are multiples of $3$ or $4$ but not $5$? 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.

A street has parallel curbs $40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $15$ feet and each stripe is $50$ feet long. Find the distance, in feet, between the stripes.

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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 piece of paper, \(4\) inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. Find the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle. 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 right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter \(10\) and altitude \(12\), and the axes of the cylinder and cone coincide. Find the radius of the cylinder. The original answer is in the form \(\frac{k}{m}\), where the fraction is in 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.

Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:

$$\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}$$

Find the value of $a-d$.

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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.

Points $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?

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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 $24$ four-digit whole numbers that use each of the four digits $2, 4, 5$ and $7$ exactly once. Only one of these four-digit numbers is a multiple of another one. Find this four-digit 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.

The function $f$ is given by the table
$$\begin{array}{|c||c|c|c|c|c|}\hline\text{x}&1&2&3&4&5\\\hline\text{f(x)}&4&1&3&5&2\\\hline\end{array}$$
If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \ge 0$, find the value of $u_{2002}$.

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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 arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. Find the digit that $M$ doesn't 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.

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. If the area of the shaded region is $x\pi$, find 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.

Circles of radius $2$ and $3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region. If the answer is $x\pi$, what is the value of $x$?

[asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4;  real r1=3; real r2=2; real r3=5; pair A=(-2,0), B=(3,0), C=(0,0); pair X=(1,0), Y=(5,0); path circleA=Circle(A,r1); path circleB=Circle(B,r2); path circleC=Circle(C,r3); fill(circleC,gray); fill(circleA,white); fill(circleB,white); draw(circleA); draw(circleB); draw(circleC); draw(A--X); draw(B--Y);  pair[] ps={A,B}; dot(ps);  label("$3$",midpoint(A--X),N); label("$2$",midpoint(B--Y),N); [/asy]

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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 \(i + 2i^2 + 3i^3 + \cdots + 2002i^{2002}\). The original answer is in the form \(a + bi\), where \(a\) and \(b\) are integers. 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.

Using the digits $1, 2, 3, 4, 5, 6, 7,$ and $9,$ form $4$ two-digit prime numbers, using each digit only once. What is the sum of the $4$ prime numbers?

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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 $1001$ red marbles and $1001$ black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ be the probability that they are different colors. Find the absolute difference \(|P_s - P_d|\). 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 $3\times3\times3$ cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?

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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 how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime 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.

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach? If the area can be expressed as $x\pi$, find 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.

Suppose that $\{a_n\}$ is an arithmetic sequence with
$$a_1+a_2+\cdots+a_{100}=100 ~\text{and}~ a_{101}+a_{102}+\cdots+a_{200}=200.$$
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.

Let $\triangle XOY$ be a right-angled triangle with $m\angle XOY = 90^{\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN = 19$ and $YM = 22$, find the length of $XY$.

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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 sequence of numbers: $4,7,1,8,9,7,6,\cdots$ For $n>2$, the $n$-th term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. Find the smallest value of $n$ for which $S_n>10,000$.

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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.

Under the new AMC $10, 12$ scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have $24$ correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are (three) scores that can be obtained in exactly three ways. If the sum of these scores is $x$, what is the value of $10x$?

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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 all integers $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find the integer value of $b - c$.

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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.

The equation \(z(z+i)(z+3i)=2002i\) has a zero of the form \(a+bi\), where \(a\) and \(b\) are positive real numbers. The original answer is in the form \(\sqrt{m}\). Please find the value of \(m\).

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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 \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(\overline{BC}\) and the median from \(A\) to \(\overline{BC}\) have the same length. If the length of \(BC\) is in the form \(\sqrt{k}\), 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.

In trapezoid $ABCD$ with bases $AB$ and $CD$, we have $AB = 52$, $BC = 12$, $CD = 39$, and $DA = 5$. Find the area of $ABCD$.

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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.

Loki, Moe, Nick, and Ott are good friends. Ott had no money, but the others did. Moe gave Ott one-fifth of his money, Loki gave Ott one-fourth of his money, and Nick gave Ott one-third of his money. Each gave Ott the same amount of money. What fractional part of the group's money does Ott now have? 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.

The sum of the two $5$-digit numbers $AMC10$ and $AMC12$ is $123422$. What is the value of $A+M+C$?

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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.

The symbolism $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. For example, $\lfloor 3 \rfloor = 3,$ and $\lfloor \frac{9}{2} \rfloor = 4$. Compute the sum
\[\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor.\]

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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.

Simplify \(\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}\) and express your answer in the form of \(\sqrt[k]{x^p}\). Please find the value of k + 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.

Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times has the number of possible license plates increased? Provide your answer as a fraction in the form \(\frac{a^2}{b}\). Please calculate 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.

Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of \(3:2:1\), respectively. Due to some confusion, they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed? 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.

Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

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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 octagon \(ABCDEFGH\) has an area of one square unit. What is the area of the rectangle \(ABEF\)? Provide your answer in the form \(\frac{k}{m}\) as a simplified fraction, 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.

A point \(P\) is chosen at random in the interior of equilateral triangle \(ABC\). What is the probability that \(\triangle ABP\) has a greater area than each of \(\triangle ACP\) and \(\triangle BCP\)? Express your answer in the form \(\frac{k}{m}\), where \(k\) and \(m\) are coprime positive integers. Find the value of \(k + m\).

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