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What is the measure, in degrees, of one interior angle of a regular hexagon?	What is the measure, in degrees, of one interior angle of a regular hexagon?
Let $a$ and $b$ be complex numbers such that the product $(z-a)(z-b)$ is a real monic quadratic polynomial for which $z=0$ is not a root. Determine the smallest possible value of $\|a\|+\|b\|$ when the graph of this quadratic polynomial is tangent to the unit circle.	Let $a$ and $b$ be complex numbers such that the product $(z-a)(z-b)$ is a real monic quadratic polynomial for which $z=0$ is not a root. Determine the smallest possible value of $\|a\|+\|b\|$ when the graph of this quadratic polynomial is tangent to the unit circle.
When rolling a fair 6-sided die, what is the expected value of the sum of the faces if it is rolled twice?	When rolling a fair 6-sided die, what is the expected value of the sum of the faces if it is rolled twice?
One hundred people were surveyed. Of these, $87$ indicated they liked Mozart and $70$ indicated they liked Bach. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach?	One hundred people were surveyed. Of these, $87$ indicated they liked Mozart and $70$ indicated they liked Bach. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach?
A certain type of tile is used to create patterns on a wall. Each tile has a square shape with an area of 1 square foot. The pattern requires that the tiles be arranged in a way that each row has the same number of tiles, and each column has the same number of tiles. In how many ways can a rectangular wall with an area...	A certain type of tile is used to create patterns on a wall. Each tile has a square shape with an area of 1 square foot. The pattern requires that the tiles be arranged in a way that each row has the same number of tiles, and each column has the same number of tiles. In how many ways can a rectangular wall with an area...
On a campus, 30 students are members of either the sports team or the debate club, with 10 students participating in both activities. If the total number of students on campus is 50, determine the minimum number of students who are neither part of the sports team nor the debate club.	On a campus, 30 students are members of either the sports team or the debate club, with 10 students participating in both activities. If the total number of students on campus is 50, determine the minimum number of students who are neither part of the sports team nor the debate club.
Let $x$ be the number of positive multiples of $12$ that are less than $40$. Let $y$ be the number of positive integers that are less than $40$, and a multiple of $4$ and a multiple of $3$. Compute $(x - y)^3$.	Let $x$ be the number of positive multiples of $12$ that are less than $40$. Let $y$ be the number of positive integers that are less than $40$, and a multiple of $4$ and a multiple of $3$. Compute $(x - y)^3$.
Problem: A fair six-sided die is rolled three times. What is the probability that the sum of the three numbers rolled is a multiple of 3?	Problem: A fair six-sided die is rolled three times. What is the probability that the sum of the three numbers rolled is a multiple of 3?
What is the value of $n$ in the equation $n + (n + 1) + (n + 2) = 9$?	What is the value of $n$ in the equation $n + (n + 1) + (n + 2) = 9$?
What is the smallest positive integer with exactly 14 positive divisors?	What is the smallest positive integer with exactly 14 positive divisors?
Problem: The number $e$ can be expressed as an infinite series: $$e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots$$ Find the value of $e$ rounded to three decimal places by summing up the first 10 terms of this infinite series. Solution: This problem requires the stud...	Problem: The number $e$ can be expressed as an infinite series: $$e = \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots$$ Find the value of $e$ rounded to three decimal places by summing up the first 10 terms of this infinite series. Solution: This problem requires the stud...
Let $x = \sqrt{a} + \sqrt{b}$ and $y = \sqrt{a} - \sqrt{b}$, where $a$ and $b$ are positive integers. If $x^2 + y^2 = 32$ and $xy = 12$, find the value of $a$ and $b$.	Let $x = \sqrt{a} + \sqrt{b}$ and $y = \sqrt{a} - \sqrt{b}$, where $a$ and $b$ are positive integers. If $x^2 + y^2 = 32$ and $xy = 12$, find the value of $a$ and $b$.
A bookshelf has 5 shelves, and each shelf can hold a certain number of books. If the total number of books that the bookshelf can hold is 140, and each shelf can hold the same number of books, how many books can each shelf hold?	A bookshelf has 5 shelves, and each shelf can hold a certain number of books. If the total number of books that the bookshelf can hold is 140, and each shelf can hold the same number of books, how many books can each shelf hold?
A car travels a certain distance at an average speed of 60 km/h. If the car's engine rotates at an average of 2500 revolutions per minute, how many revolutions does the engine make during the entire trip if the car's fuel efficiency is 10 km/liter and it consumes 15 liters of fuel during the trip?	A car travels a certain distance at an average speed of 60 km/h. If the car's engine rotates at an average of 2500 revolutions per minute, how many revolutions does the engine make during the entire trip if the car's fuel efficiency is 10 km/liter and it consumes 15 liters of fuel during the trip?
A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from the six digits 2, 2, 2, 3, 7, and 9, and the rock's age begins with an odd digit. How many possibilities are there for the rock's age?	A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from the six digits 2, 2, 2, 3, 7, and 9, and the rock's age begins with an odd digit. How many possibilities are there for the rock's age?
What is the remainder when 1,493,824 is divided by 4?	What is the remainder when 1,493,824 is divided by 4?
Let $f$ be a function defined on the set of real numbers, and let $g$ be a function defined on the interval $[0, 2\pi]$. If $f^{-1}(g(x)) = 2\sin x + 1$ and $g$ has an inverse, find $g^{-1}(f(1))$.	Let $f$ be a function defined on the set of real numbers, and let $g$ be a function defined on the interval $[0, 2\pi]$. If $f^{-1}(g(x)) = 2\sin x + 1$ and $g$ has an inverse, find $g^{-1}(f(1))$.
In a triangle $ABC$, the median $AM$ divides the triangle into two smaller triangles, $ABM$ and $ACM$. If the length of $AB$ is 7, the length of $AC$ is 9, and the length of $BC$ is 11, find the length of $AM$. Note: The solution requires a different approach, using the properties of medians in a triangle, rather than...	In a triangle $ABC$, the median $AM$ divides the triangle into two smaller triangles, $ABM$ and $ACM$. If the length of $AB$ is 7, the length of $AC$ is 9, and the length of $BC$ is 11, find the length of $AM$. Note: The solution requires a different approach, using the properties of medians in a triangle, rather than...
What is the greatest integer less than the cube root of 800?	What is the greatest integer less than the cube root of 800?
Let $\alpha$ be the smallest positive angle for which $\tan \alpha,$ $\tan 2 \alpha,$ $\tan 3 \alpha$ form a geometric progression, in some order. Find $\sin \alpha$.	Let $\alpha$ be the smallest positive angle for which $\tan \alpha,$ $\tan 2 \alpha,$ $\tan 3 \alpha$ form a geometric progression, in some order. Find $\sin \alpha$.
In the Smallest Positive Integer Game, Alice says, "My number is 24." Bob says, "What kind of silly smallest number is that? Every prime factor of your number is also a prime factor of my number." The smallest possible number that Bob could have is 12. Another problem inspired by this one: In the Smallest Positive I...	In the Smallest Positive Integer Game, Alice says, "My number is 24." Bob says, "What kind of silly smallest number is that? Every prime factor of your number is also a prime factor of my number." The smallest possible number that Bob could have is 12. Another problem inspired by this one: In the Smallest Positive I...
Let $\mathbf{A}$ be a $3 \times 3$ matrix such that \[\mathbf{A} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}, \quad \mathbf{A} \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad \text{and} \quad \mathbf{A} \begin{pmatrix} 1 \\ 1 \\ 0 \...	Let $\mathbf{A}$ be a $3 \times 3$ matrix such that \[\mathbf{A} \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \\ 1 \end{pmatrix}, \quad \mathbf{A} \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad \text{and} \quad \mathbf{A} \begin{pmatrix} 1 \\ 1 \\ 0 \...
Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers.	Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers.
A five-digit number $abcde$ is divisible by 99 if and only if the sum of its digits $a+b+c+d+e$ is divisible by 9 and the number formed by the last two digits $de$ is divisible by 11. How many different five-digit numbers $abcde$ can be formed such that $a+b+c+d+e=27$ and $de$ is divisible by 11?	A five-digit number $abcde$ is divisible by 99 if and only if the sum of its digits $a+b+c+d+e$ is divisible by 9 and the number formed by the last two digits $de$ is divisible by 11. How many different five-digit numbers $abcde$ can be formed such that $a+b+c+d+e=27$ and $de$ is divisible by 11?
Find the value of $k$ such that the graph of the equation $x^2 + y^2 = k^2$ passes through the point $(3, 4)$ and is tangent to the line $x + y = 5$.	Find the value of $k$ such that the graph of the equation $x^2 + y^2 = k^2$ passes through the point $(3, 4)$ and is tangent to the line $x + y = 5$.
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(1) = 1$ and $g(x^2 + y^2) = (x + y) (g(x) - g(y))$ for all real numbers $x$ and $y.$ Find the smallest possible value of $g(5).$	Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(1) = 1$ and $g(x^2 + y^2) = (x + y) (g(x) - g(y))$ for all real numbers $x$ and $y.$ Find the smallest possible value of $g(5).$
Let $f(x)$ be a function defined as $f(x) = \left \lceil x \right\rceil - \left \lfloor x \right \rfloor$. Determine the value of $\lim_{n\to\infty} f\left(\frac{1}{n}\right)$ if it exists.	Let $f(x)$ be a function defined as $f(x) = \left \lceil x \right\rceil - \left \lfloor x \right \rfloor$. Determine the value of $\lim_{n\to\infty} f\left(\frac{1}{n}\right)$ if it exists.
Consider a matrix $\mathbf{M}$ that represents a reflection about a line passing through the origin and making an angle of $45^\circ$ with the positive $x$-axis. Find the determinant of $\mathbf{M}$.	Consider a matrix $\mathbf{M}$ that represents a reflection about a line passing through the origin and making an angle of $45^\circ$ with the positive $x$-axis. Find the determinant of $\mathbf{M}$.
A pizza parlor offers three different types of pizza: plain, vegetarian, and meat-lovers. The plain pizza costs $5, the vegetarian pizza costs $2 more than the plain pizza, and the meat-lovers pizza costs $4 more than the vegetarian pizza. If you order 2 plain pizzas, 3 vegetarian pizzas, and 1 meat-lovers pizza, what ...	A pizza parlor offers three different types of pizza: plain, vegetarian, and meat-lovers. The plain pizza costs $5, the vegetarian pizza costs $2 more than the plain pizza, and the meat-lovers pizza costs $4 more than the vegetarian pizza. If you order 2 plain pizzas, 3 vegetarian pizzas, and 1 meat-lovers pizza, what ...
A chef has a secret recipe for a salad that requires a mixture of spinach, carrots, and sprouts. The ratio of the weights of spinach to carrots to sprouts is 2:5:7. If the chef needs 3.5 kilograms of carrots for a large batch of salad, how many kilograms of sprouts are needed?	A chef has a secret recipe for a salad that requires a mixture of spinach, carrots, and sprouts. The ratio of the weights of spinach to carrots to sprouts is 2:5:7. If the chef needs 3.5 kilograms of carrots for a large batch of salad, how many kilograms of sprouts are needed?
The equation $x^2+12x=73$ has two solutions. The positive solution has the form $\sqrt{a}-b$ for positive natural numbers $a$ and $b$. What is $a+b$?	The equation $x^2+12x=73$ has two solutions. The positive solution has the form $\sqrt{a}-b$ for positive natural numbers $a$ and $b$. What is $a+b$?
The difference when the product of 4 and a number is subtracted from 16 is greater than 10. How many positive integers satisfy the given conditions?	The difference when the product of 4 and a number is subtracted from 16 is greater than 10. How many positive integers satisfy the given conditions?
Let $a,$ $b,$ $c$ be real numbers such that $0 < a,$ $b,$ $c < 2.$ Find the maximum value of \[\frac{1}{(2 - a)(2 - b)(2 - c)} + \frac{1}{(a)(b)(c)}.\]	Let $a,$ $b,$ $c$ be real numbers such that $0 < a,$ $b,$ $c < 2.$ Find the maximum value of \[\frac{1}{(2 - a)(2 - b)(2 - c)} + \frac{1}{(a)(b)(c)}.\]
In right triangle $ABC$ with $\angle A = 90^\circ$, we have $AB =16$ and $BC = 24$. Find $\sin A$.	In right triangle $ABC$ with $\angle A = 90^\circ$, we have $AB =16$ and $BC = 24$. Find $\sin A$.
Consider a sequence of 10 digits, where each digit can be either 0 or 1. The sequence must start with 1, and no two consecutive digits can be the same. How many such sequences are possible?	Consider a sequence of 10 digits, where each digit can be either 0 or 1. The sequence must start with 1, and no two consecutive digits can be the same. How many such sequences are possible?
A bacteria culture starts with 1000 bacteria and triples every hour. However, due to a limitation in nutrients, the bacteria start to die at a rate of 10% per hour after the first 5 hours. How many bacteria will be present in the culture after 10 hours?	A bacteria culture starts with 1000 bacteria and triples every hour. However, due to a limitation in nutrients, the bacteria start to die at a rate of 10% per hour after the first 5 hours. How many bacteria will be present in the culture after 10 hours?
Let $a_n$ denote the number of sets of positive integers $x,y,z$, where $x<y<z$, which satisfy $xyz = n!$. For example, $a_3 = 2$, since the only possible sets are $\{1,2,6\}$ and $\{1,3,4\}$. Find $a_{100}$.	Let $a_n$ denote the number of sets of positive integers $x,y,z$, where $x<y<z$, which satisfy $xyz = n!$. For example, $a_3 = 2$, since the only possible sets are $\{1,2,6\}$ and $\{1,3,4\}$. Find $a_{100}$.
Jo adds up all the positive integers from 1 to 50. Kate does a similar thing with the first 50 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 50 values. What is the positive difference between Jo's sum and Kate's sum?	Jo adds up all the positive integers from 1 to 50. Kate does a similar thing with the first 50 positive integers; however, she first rounds every integer to its nearest multiple of 10 (rounding 5s up) and then adds the 50 values. What is the positive difference between Jo's sum and Kate's sum?
What is the coefficient of $x^3$ in the expansion of $(2x+1)^6$?	What is the coefficient of $x^3$ in the expansion of $(2x+1)^6$?
Factor the expression $x^2 + 5x + 6$ by finding the two numbers whose product is 6 and whose sum is 5.	Factor the expression $x^2 + 5x + 6$ by finding the two numbers whose product is 6 and whose sum is 5.