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

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $75\%$ of the volume of the frozen ice cream. What is the ratio of the cone's height to its radius? Provide your answer in the form $x:1$, where $x$ is a positive integer. Please 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.

If \(\log (xy^3) = 1\) and \(\log (x^2y) = 1\), find the value of \(\log (xy)\). The answer should be in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) 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.

In rectangle \(ABCD\), \(AB=5\) and \(BC=3\). Points \(F\) and \(G\) are on \(\overline{CD}\) so that \(DF=1\) and \(GC=2\). Lines \(AF\) and \(BG\) intersect at \(E\). The area of \(\triangle AEB\) 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.

How many $15$-letter arrangements of $5$ A's, $5$ B's, and $5$ C's have no A's in the first $5$ letters, no B's in the next $5$ letters, and no C's in the last $5$ letters? Provide your answer as a sum of binomial coefficients in the form \(\sum\limits_{k=0}^{n}\binom{m}{k}^{p}\), and then calculate the value of m + n + p.

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

The area of trapezoid $ABCD$ is $164\ \text{cm}^2$. The altitude is $8\ \text{cm}$, $AB$ is $10\ \text{cm}$, and $CD$ is $17\ \text{cm}$. What is the length of $BC$, in centimeters?

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Solve the following math problem step 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 points are chosen randomly and independently on a circle. Find the probability that all three pairwise distances between the points are less than the radius of the circle. The probability is expressed in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).

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

How many two-digit positive integers have at least one $7$ as a digit? Fill in the blank with the correct number of such 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.

What is the value of \(x\) if \(|x-1|=|x-2|\)? Provide your answer in the form of \(\frac{k}{m}\), where \(k\) and \(m\) are integers. 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.

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

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

An integer \(x\), with \(10 \leq x \leq 99\), is to be chosen. If all choices are equally likely, what is the probability that at least one digit of \(x\) is a \(7\)? Express your answer as a fraction in simplest form \(\frac{a}{b}\), and 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.

A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively. How many rounds will there be in the game?

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

Minneapolis-St. Paul International Airport is $8$ miles southwest of downtown St. Paul and $10$ miles southeast of downtown Minneapolis. Find the number of miles between downtown St. Paul and downtown Minneapolis that is closest to the actual distance.

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Solve the following math problem step 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 $10$. Find the area of the union of the regions enclosed by the square and the circle. The original answer is in the form of k + m\pi, 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 $49$ consecutive integers is $7^5$. What is their median? Provide your answer in the form of $7^x$, where $x$ is an integer. Please give 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.

All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?

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

Henry's Hamburger Haven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties and any collection of condiments. How many different kinds of hamburgers can be ordered?

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Solve the following math problem step 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 United States, coins have the following thicknesses: penny, $1.55 \, \text{mm}$; nickel, $1.95 \, \text{mm}$; dime, $1.35 \, \text{mm}$; quarter, $1.75 \, \text{mm}$. If a stack of these coins is exactly $14 \, \text{mm}$ high, how many coins are in the stack?

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

Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?
Provide your answer as an integer.

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

The $5\times 5$ grid shown contains a collection of squares with sizes from $1\times 1$ to $5\times 5$. How many of these squares contain the black center square?

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

Three circles of radius \(1\) are externally tangent to each other and internally tangent to a larger circle. The radius of the large circle can be expressed in the form \(\frac{k + 2\sqrt{m}}{n}\), where the expression is in its simplest form. Find the value of \(k + m + n\).

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

In the sequence $2001$, $2002$, $2003$, $\cdots$, 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 $2004^{th}$ 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.

In \(\triangle ABC\), points \(D\) and \(E\) lie on \(BC\) and \(AC\), respectively. If \(AD\) and \(BE\) intersect at \(T\) so that \(\frac{AT}{DT}=3\) and \(\frac{BT}{ET}=4\), find the value of \(\frac{CD}{BD}\). 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.

Select numbers $a$ and $b$ between $0$ and $1$ independently and at random, and let $c$ be their sum. Let $A, B$ and $C$ be the results when $a, b$ and $c$, respectively, are rounded to the nearest integer. Find the probability that $A + B = C$. 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.

Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. What is the length of $\overline{CE}$? 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.

The square
$$
\begin{array}{|c|c|c|}\hline50&b&c\\\hline d&e&f\\\hline g&h&2\\\hline\end{array}
$$
is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, find the sum of the possible values of $g$.

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Solve the following math problem step 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 \(AB=7\), \(AC=8\), \(BC=9\). Point \(D\) is on the circumscribed circle of the triangle so that \(AD\) bisects angle \(BAC\). The value of \(\frac{AD}{CD}\) is given in the form \(\frac{k}{m}\), where k and m are coprime integers. Please find the value of \(k + m\).

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

Two \(4 \times 4\) squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. Find the area of the shaded region created by removing the circle from the squares. The answer is in the form \(k - m\pi\), please provide the value of \(k + m\).

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

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. Find the ratio of the area of the smallest circle to the area of the largest square. The original answer is in \(\frac{k\pi}{m}\) 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.

Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\sqrt{50}$ and $BE = 1$. What is the area of the inner square $EFGH$?

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Solve the following math problem step 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 $|3x|+|4y|=12$? 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.

Three tiles are marked \(X\) and two other tiles are marked \(O\). The five tiles are randomly arranged in a row. Find the probability that the arrangement reads \(XOXOX\). 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.

One fair die has faces \(1, 1, 2, 2, 3, 3\) and another has faces \(4, 4, 5, 5, 6, 6\). The dice are rolled and the numbers on the top faces are added. Find the probability that the sum will be odd. 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 sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds $6%$ sales tax, then subtracts $20%$ from this total. Jill rings up $90.00$, subtracts $20%$ of the price, then adds $6%$ of the discounted price for sales tax. What is the value of Jack's total minus Jill's total?

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Solve the following math problem step 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 five-sided star shown, the letters $A$, $B$, $C$, $D$ and $E$ are replaced by the numbers $3, 5, 6, 7$ and $9,$ although not necessarily in that order. The sums of the numbers at the ends of the line segments $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, $\overline{DE}$, and $\overline{EA}$ form an arithmetic sequence, although not necessarily in that order. What is the middle term of the arithmetic 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.

On a standard die, one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? Express your answer as a fraction in simplest 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 Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled in total?

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

An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $20$ or more? Express your answer in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) 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.

The sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Determine which digit is not included among the eight 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.

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure $1$. The cube is then cut in the same manner along the dashed lines shown in Figure $2$. This creates nine pieces. Find the volume of the piece that contains vertex $W$. 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.

All of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David 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.

On a certain math exam, $10\%$ of the students got $70$ points, $25\%$ got $80$ points, $20\%$ got $85$ points, $15\%$ got $90$ points, and the rest got $95$ points. What is the difference between the mean and the median score on this exam?

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Solve the following math problem step 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 faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?

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Solve the following math problem step 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 $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set $\{-7,-5,-3,-2,2,4,6,13\}.$ Find the minimum possible value of $(a+b+c+d)^{2}+(e+f+g+h)^{2}.$

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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.

A sequence of complex numbers $z_{0}, z_{1}, z_{2}, \ldots$ is defined by the rule

$$z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},$$

where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. Find the number of possible values for $z_{0}$. The original answer is in the form $x^{y}$, please provide the value of x + y.

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

Two distinct numbers \(a\) and \(b\) are chosen randomly from the set \(\{2, 2^2, 2^3, \cdots, 2^{25}\}\). Find the probability that \(\log_a b\) is an integer. The original answer is in \(\frac{k}{m}\) format, where the fraction is in 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.

For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. Find the number of positive integers $n$ for which both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$ hold 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.

Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. The probability that no two ants arrive at the same vertex is given in the form \(\frac{k}{m}\). Please find the value of k + m.

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

Elisa swims laps in the pool. When she first started, she completed $10$ laps in $25$ minutes. Now, she can finish $12$ laps in $24$ minutes. By how many minutes has she improved her lap time? Provide your answer in the form of a simplified fraction \(\frac{k}{m}\), 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.

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

What non-zero real value for \(x\) satisfies \((7x)^{14}=(14x)^7\)? Provide your answer in the form of \(\frac{a}{b}\), where \(a\) and \(b\) are integers. Please find the value of \(a + b\).

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

How many sets of two or more consecutive positive integers have a sum of $15$? 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 player pays $\textdollar 5$ to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case, the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game, the probability of winning times the amount won is what the player should pay.) If the answer is $\textdollar x$, what is the value of $x$?

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

Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet? Provide your answer in the format of AA:BB, where AA is the hour and BB is the minutes. Calculate the sum of AA and BB.

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Solve the following math problem step 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 \(a\) and \(b\) be the roots of the equation \(x^2-mx+2=0\). Suppose that \(a+\frac{1}{b}\) and \(b+\frac{1}{a}\) are the roots of the equation \(x^2-px+q=0\). The original answer for the value of \(q\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

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

Two farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

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Solve the following math problem step 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 \(\cos x=0\) and \(\cos (x+z)=\frac{1}{2}\). What is the smallest possible positive value of \(z\) in terms of \(\pi\)? The answer should be in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of \(k + m\).

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

Circles with centers \(A\) and \(B\) have radius \(3\) and \(8\), respectively. A common internal tangent intersects the circles at \(C\) and \(D\), respectively. Lines \(AB\) and \(CD\) intersect at \(E\), and \(AE=5\). Find the length of \(CD\). 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 circle of radius \(2\) is centered at \(O\). Square \(OABC\) has side length \(1\). Sides \(AB\) and \(CB\) are extended past \(B\) to meet the circle at \(D\) and \(E\), respectively. Find the area of the shaded region in the figure, which is bounded by \(BD\), \(BE\), and the minor arc connecting \(D\) and \(E\). The original answer is in the form \(\frac{k\pi}{m} + n - \sqrt{p}\). Please give the value of \(k + m + n + p\).

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

In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$? 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.

Circles with centers \(A\) and \(B\) have radius 3 and 8, respectively. A common internal tangent intersects the circles at \(C\) and \(D\), respectively. Lines \(AB\) and \(CD\) intersect at \(E\), and \(AE=5\). Find the length of \(CD\). 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.

Circles with centers \( O \) and \( P \) have radii \( 2 \) and \( 4 \), respectively, and are externally tangent. Points \( A \) and \( B \) on the circle with center \( O \) and points \( C \) and \( D \) on the circle with center \( P \) are such that \( AD \) and \( BC \) are common external tangents to the circles. Find the area of the concave hexagon \( AOBCPD \). The original answer is in the form \( k\sqrt{m} \). Please provide the value of \( k + m \).

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

Let \(S\) be the set of all points \((x,y)\) in the coordinate plane such that \(0 \le x \le \frac{\pi}{2}\) and \(0 \le y \le \frac{\pi}{2}\). Find the area of the subset of \(S\) for which
\[
\sin^2x-\sin x \sin y + \sin^2y \le \frac{3}{4}.
\]
The answer is in the form \(\frac{k\pi^2}{m}\). Please calculate the value of \(k + m\).

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

How many non-empty subsets $S$ of $\{1,2,3,\ldots ,15\}$ have the following two properties?

$(1)$ No two consecutive integers belong to $S$.

$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.

Provide the number of such subsets.

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

A traffic light runs repeatedly through the following cycle: green for \(30\) seconds, then yellow for \(3\) seconds, and then red for \(30\) seconds. Leah picks a random three-second time interval to watch the light. What is the probability, expressed as a fraction in simplest form \(\frac{k}{m}\), that the color changes while she is watching? 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.

Four circles of radius \(1\) are each tangent to two sides of a square and externally tangent to a circle of radius \(2\), as shown. The area of the square is expressed in the form \(k + m\sqrt{2}\), where \(k\) and \(m\) are integers. Find the value of \(k + m\).

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

The geometric series \(a+ar+ar^2+\ldots\) has a sum of \(7\), and the terms involving odd powers of \(r\) have a sum of \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

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

A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. 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.

Let $a$, $b$, and $c$ be digits with $a\ne 0$. The three-digit integer $abc$ lies one third of the way from the square of a positive integer to the square of the next larger integer. The integer $acb$ lies two thirds of the way between the same two squares. Find 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.

A set of 25 square blocks is arranged into a \(5 \times 5\) square. Find the number of different combinations of 3 blocks that can be selected from that set so that no two are in the same row or column.

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Solve the following math problem step 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 parallelogram bounded by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$, and $y=bx+d$ has area $18$. The parallelogram bounded by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$, and $y=bx-d$ has area $72$. Given that $a$, $b$, $c$, and $d$ are positive integers, find the smallest possible value of $a+b+c+d$.

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

A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of these four distances in meters?

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Solve the following math problem step 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 with a square base is cut by a plane that is parallel to its base and $2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. The original answer is in the form k + m\sqrt{2}. Please provide the value of k + m, where k and m are 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.

What is the area of the shaded pinwheel shown in the $5 \times 5$ grid? 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 positive integers $(a,b)$ are there such that $a$ and $b$ have no common factors greater than $1$ and:
$\frac{a}{b} + \frac{14b}{9a}$
is an integer? Provide the number of such pairs.

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

Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\cdots,12\},$ including the empty set, are spacy?

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Solve the following math problem step 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 bakery owner turns on his doughnut machine at $8:30$ AM. At $11:10$ AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the entire job? Write your answer in the format "HH:MM AM/PM", and then provide the sum of HH and MM.

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

A $4\times 4$ block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?

$\begin{array}{|c|c|c|c|}\hline1&2&3&4\\\hline8&9&10&11\\\hline15&16&17&18\\\hline22&23&24&25\\\hline\end{array}$

Find the positive difference between the two diagonal sums.

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Solve the following math problem step 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 ten-letter code $BEST ~OF ~LUCK$ represents the ten digits $0-9$, in order. What $4$-digit number is represented by the code word $CLUE$?

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Solve the following math problem step 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 volume of a cube whose surface area is twice that of a cube with volume $1$? Provide your answer in the form of $a\sqrt{b}$ where $a$ and $b$ are integers, and 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.

Older television screens have an aspect ratio of $4: 3$. That is, the ratio of the width to the height is $4: 3$. The aspect ratio of many movies is not $4: 3$, so they are sometimes shown on a television screen by "letterboxing" - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $2: 1$ and is shown on an older television screen with a $27$-inch diagonal. If the height of each darkened strip is $x$ inches, 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.

Points \(A\) and \(B\) are on a circle of radius \(5\) and \(AB = 6\). Point \(C\) is the midpoint of the minor arc \(AB\). What is the length of the line segment \(AC\)? The answer 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.

A ball is dropped from a height of $3$ meters. On its first bounce, it rises to a height of $2$ meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of $0.5$ meters? Provide your answer as an integer.

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

What is the area of the region defined by the inequality $|3x-18|+|2y+7|\le3$?

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Solve the following math problem step 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 Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ games was also an integer. Find the product of the number of points she scored in the ninth and tenth games.

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Solve the following math problem step 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 floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a,b)$?

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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.

The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{th}$ term of the sequence is $\log{b^n}$. What is 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.

Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$?

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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.

Triangle \(ABC\), with sides of length \(5\), \(6\), and \(7\), has one vertex on the positive \(x\)-axis, one on the positive \(y\)-axis, and one on the positive \(z\)-axis. Let \(O\) be the origin. Find the volume of tetrahedron \(OABC\). The original answer is in the form \(\sqrt{m}\). 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.

Rectangle \(PQRS\) lies in a plane with \(PQ=RS=2\) and \(QR=SP=6\). The rectangle is rotated \(90^\circ\) clockwise about \(R\), then rotated \(90^\circ\) clockwise about the point \(S\) moved to after the first rotation. If the length of the path traveled by point \(P\) is \(x\pi\), the original answer is in the form of \(k+\sqrt{m}\). Please find the value of \(k+m\).

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

In the expansion of
\[\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,\]
find the coefficient of $x^{28}$.

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Solve the following math problem step 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 function \(f\) is defined by \(f(z) = (4 + i) z^2 + \alpha z + \gamma\) for all complex numbers \(z\), where \(\alpha\) and \(\gamma\) are complex numbers and \(i^2 = -1\). Suppose that \(f(1)\) and \(f(i)\) are both real. The original answer is in the form \(\sqrt{m}\). Find the smallest possible 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.

Trapezoid $ABCD$ has bases $\overline{AB}$ and $\overline{CD}$ and diagonals intersecting at $K.$ Suppose that $AB = 9$, $DC = 12$, and the area of $\triangle AKD$ is $24.$ Find the area of trapezoid $ABCD$.

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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.

Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet?

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Solve the following math problem step 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 circles of radius \(1\) are to be constructed as follows. The center of circle \(A\) is chosen uniformly and at random from the line segment joining \((0,0)\) and \((2,0)\). The center of circle \(B\) is chosen uniformly and at random, and independently of the first choice, from the line segment joining \((0,1)\) to \((2,1)\). Find the probability that circles \(A\) and \(B\) intersect. The original answer is in the form \(\frac{k \sqrt{m} - n}{p}\). Please find the value of \(k + m + n + p\).

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

Jacob uses the following procedure to write down a sequence of numbers. First, he chooses the first term to be \(6\). To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts \(1\). If it comes up tails, he takes half of the previous term and subtracts \(1\). What is the probability that the fourth term in Jacob's sequence is an integer? 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 $n$ are both $\frac{n}{3}$ and $3n$ three-digit 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.

Let $ABCD$ be a trapezoid with $AB||CD, AB=11, BC=5, CD=19,$ and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. If the area of hexagon $ABQCDP$ is $x\sqrt{3}$, 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 carton contains milk that is \(2\%\) fat, an amount that is \(40\%\) less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? Provide your answer as a fraction in the form \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.

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

Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

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

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

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

How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$?

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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.

On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? If Monday is day 1, what is the number of the day?

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

Points \(A\) and \(C\) lie on a circle centered at \(O\), each of \(\overline{BA}\) and \(\overline{BC}\) are tangent to the circle, and \(\triangle ABC\) is equilateral. The circle intersects \(\overline{BO}\) at \(D\). The original answer is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

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