instruction_seed stringlengths 13 12.7k | response_seed stringlengths 13 12.7k |
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Determine the number of complex solutions of the equation
\[ \frac{z^2 + 1}{z^2 - z + 1} = \left( \frac{z + 1}{z - 2} \right)^2 \]
for $z \in \mathbb{C}$. | Determine the number of complex solutions of the equation
\[ \frac{z^2 + 1}{z^2 - z + 1} = \left( \frac{z + 1}{z - 2} \right)^2 \]
for $z \in \mathbb{C}$. |
The number $0.\overline{36}$ can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers with no common factors other than 1. Determine the value of $a^2 + b^2$. | The number $0.\overline{36}$ can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers with no common factors other than 1. Determine the value of $a^2 + b^2$. |
Let $\tau (n)$ denote the number of positive integer divisors of $n$ (including $1$ and $n$). Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.
| Let $\tau (n)$ denote the number of positive integer divisors of $n$ (including $1$ and $n$). Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.
|
A basketball tournament is held among eight teams, each playing the other exactly once. No draws are possible, each team has an equal chance of winning each match, and the outcomes of the games are independent. In each game, the winner earns one point, and the loser gets zero points. The accumulated points determine th... | A basketball tournament is held among eight teams, each playing the other exactly once. No draws are possible, each team has an equal chance of winning each match, and the outcomes of the games are independent. In each game, the winner earns one point, and the loser gets zero points. The accumulated points determine th... |
Find the quadratic polynomial, with real coefficients, which has $-2 - 3i$ as a root, and where the coefficient of $x$ is $-4$. | Find the quadratic polynomial, with real coefficients, which has $-2 - 3i$ as a root, and where the coefficient of $x$ is $-4$. |
At the mall's food court, Crystal wants to buy a meal consisting of one entree, one drink and one dessert. The table below lists Crystal's favorite foods in the food court. How many distinct possible meals can she buy from these options?
\begin{tabular}{ |c | c | c | }
\hline \textbf{Entrees} & \textbf{Drinks}&\textbf... | At the mall's food court, Crystal wants to buy a meal consisting of one entree, one drink and one dessert. The table below lists Crystal's favorite foods in the food court. How many distinct possible meals can she buy from these options?
\begin{tabular}{ |c | c | c | }
\hline \textbf{Entrees} & \textbf{Drinks}&\textbf... |
How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit? | How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit? |
For any real number $k,$ the graph of
\[y = 7x^2 + kx - 4k\]passes through a fixed point $(a,b).$ Find $(a,b).$ | For any real number $k,$ the graph of
\[y = 7x^2 + kx - 4k\]passes through a fixed point $(a,b).$ Find $(a,b).$ |
Given that the sum of the squares of the roots of a quadratic equation is 52 and the sum of the roots is -8, find the product of the roots. | Given that the sum of the squares of the roots of a quadratic equation is 52 and the sum of the roots is -8, find the product of the roots. |
How many of the natural numbers from 1 to 1000, inclusive, have exactly three digits, none of which is 0? | How many of the natural numbers from 1 to 1000, inclusive, have exactly three digits, none of which is 0? |
Let $x$ and $y$ be integers satisfying the system of congruences:
\begin{align*}
7x&\equiv 11\pmod{15},\\
4y&\equiv 13\pmod{15}.
\end{align*}
Determine the remainder when $xy$ is divided by 15. | Let $x$ and $y$ be integers satisfying the system of congruences:
\begin{align*}
7x&\equiv 11\pmod{15},\\
4y&\equiv 13\pmod{15}.
\end{align*}
Determine the remainder when $xy$ is divided by 15. |
The blue parabola shown is the graph of the equation $y = ax^2 + bx + c$. Find $c$. Assume that the graph has the standard unit scale.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tick... | The blue parabola shown is the graph of the equation $y = ax^2 + bx + c$. Find $c$. Assume that the graph has the standard unit scale.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tick... |
A random variable $X$ takes values in the set $\{1,2,3\}$ with equal probabilities. Another random variable $Y$ takes values in the set $\{1,2,3,4,5,6\}$. The probability that $Y$ takes the value $i$ is $\frac{i}{21}$ for $i=1,2,3,4,5,6$. What is the probability that the product of $X$ and $Y$ is a multiple of 3? | A random variable $X$ takes values in the set $\{1,2,3\}$ with equal probabilities. Another random variable $Y$ takes values in the set $\{1,2,3,4,5,6\}$. The probability that $Y$ takes the value $i$ is $\frac{i}{21}$ for $i=1,2,3,4,5,6$. What is the probability that the product of $X$ and $Y$ is a multiple of 3? |
Find the equation of the directrix of the parabola $y = 5x^2 - 1$. | Find the equation of the directrix of the parabola $y = 5x^2 - 1$. |
Find the values of x that satisfy the equation $\sqrt[3]{x^2+7x-12}=-1$ and $|x|<3$.
Note: This problem requires a different approach than the original one, as it involves a cubic root and an absolute value inequality, which requires a different method to solve. | Find the values of x that satisfy the equation $\sqrt[3]{x^2+7x-12}=-1$ and $|x|<3$.
Note: This problem requires a different approach than the original one, as it involves a cubic root and an absolute value inequality, which requires a different method to solve. |
How many positive four-digit integers are divisible by $8$? | How many positive four-digit integers are divisible by $8$? |
Find the product of the greatest common divisor and the least common multiple of 24 and 36. | Find the product of the greatest common divisor and the least common multiple of 24 and 36. |
Maria collects coins. She puts the same number of coins in each pouch and then stores each pouch in one of her two treasure chests. One of her treasure chests has a total of 630 coins. The other has 900. What is the largest number of coins that Maria could be putting in each pouch? | Maria collects coins. She puts the same number of coins in each pouch and then stores each pouch in one of her two treasure chests. One of her treasure chests has a total of 630 coins. The other has 900. What is the largest number of coins that Maria could be putting in each pouch? |
What is the greatest integer less than $\frac{1000}{\sqrt{15} + 1}$? | What is the greatest integer less than $\frac{1000}{\sqrt{15} + 1}$? |
A water tank can be filled by two pipes, A and B. Pipe A fills the tank at a rate of 2 cubic meters per minute, while pipe B fills it at a rate of 6x cubic meters per minute. If the tank is initially empty and both pipes are opened simultaneously, how many minutes will it take to fill a tank that can hold 54x^3 cubic m... | A water tank can be filled by two pipes, A and B. Pipe A fills the tank at a rate of 2 cubic meters per minute, while pipe B fills it at a rate of 6x cubic meters per minute. If the tank is initially empty and both pipes are opened simultaneously, how many minutes will it take to fill a tank that can hold 54x^3 cubic m... |
Find the base six equivalent of $213_{10}$. | Find the base six equivalent of $213_{10}$. |
A recipe for making a certain type of cake calls for a ratio of 2:5 of sugar to flour. If a baker has 1200 grams of flour and wants to use up all of it, but only has a 200-gram bag of sugar, how much more sugar does the baker need to buy in order to make the cake according to the recipe? | A recipe for making a certain type of cake calls for a ratio of 2:5 of sugar to flour. If a baker has 1200 grams of flour and wants to use up all of it, but only has a 200-gram bag of sugar, how much more sugar does the baker need to buy in order to make the cake according to the recipe? |
Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as the first one). What is the probability that Bob's password consists of an odd single-digit number followed by a letter and a positive single-digit number? | Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as the first one). What is the probability that Bob's password consists of an odd single-digit number followed by a letter and a positive single-digit number? |
Find the sum of all real solutions to the equation:
$$\frac{x-1}{x^2+3x+2} = \frac{x-4}{x^2-7x+1}$$ | Find the sum of all real solutions to the equation:
$$\frac{x-1}{x^2+3x+2} = \frac{x-4}{x^2-7x+1}$$ |
Determine all real numbers $x$ such that $\sin (2x+30^\circ) = \cos (3x-20^\circ)$. | Determine all real numbers $x$ such that $\sin (2x+30^\circ) = \cos (3x-20^\circ)$. |
A rectangular swimming pool is enlarged so that the new length is three times the old length, while the new width is only twice the old width. What is the ratio of the original volume to the enlarged volume if the depth of the pool remains the same? Express your answer as a common fraction. | A rectangular swimming pool is enlarged so that the new length is three times the old length, while the new width is only twice the old width. What is the ratio of the original volume to the enlarged volume if the depth of the pool remains the same? Express your answer as a common fraction. |
A store is selling a certain product at a price that increases by $1 every day. On the nth day, the total revenue collected from the sales of the product is equal to n(n+1)/2 dollars. What is the price of the product on the 2008th day? | A store is selling a certain product at a price that increases by $1 every day. On the nth day, the total revenue collected from the sales of the product is equal to n(n+1)/2 dollars. What is the price of the product on the 2008th day? |
In quadrilateral $ABCD$, the diagonals are perpendicular. If $AC=12$, $BD=8$, and $BC=5$, then what is the area of quadrilateral $ABCD$? | In quadrilateral $ABCD$, the diagonals are perpendicular. If $AC=12$, $BD=8$, and $BC=5$, then what is the area of quadrilateral $ABCD$? |
In a right triangle, two lines are drawn parallel to the legs of the triangle from a point on the hypotenuse, resulting in a rectangle and two smaller right triangles. The area of one of the smaller right triangles is $n$ times the area of the rectangle. Find the ratio of the area of the other small right triangle to t... | In a right triangle, two lines are drawn parallel to the legs of the triangle from a point on the hypotenuse, resulting in a rectangle and two smaller right triangles. The area of one of the smaller right triangles is $n$ times the area of the rectangle. Find the ratio of the area of the other small right triangle to t... |
A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form. | A sphere is inscribed in a cube with edge length 9 inches. Then a smaller cube is inscribed in the sphere. How many cubic inches are in the volume of the inscribed cube? Express your answer in simplest radical form. |
If the sum of the cubes of nonnegative real numbers $x, y,$ and $z$ is $198$, and $xy + yz + zx = 45$, then what is the sum of $x, y,$ and $z$? | If the sum of the cubes of nonnegative real numbers $x, y,$ and $z$ is $198$, and $xy + yz + zx = 45$, then what is the sum of $x, y,$ and $z$? |
A triangular prism has a base with 12 stones on one edge, and each successive edge has one fewer stone, ending with three stones on the smallest edge. If the prism has a height of 5 layers, how many stones are in the prism? | A triangular prism has a base with 12 stones on one edge, and each successive edge has one fewer stone, ending with three stones on the smallest edge. If the prism has a height of 5 layers, how many stones are in the prism? |
Find the number of positive three-digit integers that can be formed using only the digits 1, 2, 3, 5, and 7, such that the sum of the digits is a multiple of 5. | Find the number of positive three-digit integers that can be formed using only the digits 1, 2, 3, 5, and 7, such that the sum of the digits is a multiple of 5. |
The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $\frac{-b}{a}$. If the sum of the roots of the equation $2x^2 - 6x + 3 = 0$ is equal to the sum of the roots of the equation $x^2 - 4x + 2 = 0$, find the value of $a$ in the equation $ax^2 - 4ax + 2 = 0$. | The sum of the roots of a quadratic equation $ax^2 + bx + c = 0$ is given by $\frac{-b}{a}$. If the sum of the roots of the equation $2x^2 - 6x + 3 = 0$ is equal to the sum of the roots of the equation $x^2 - 4x + 2 = 0$, find the value of $a$ in the equation $ax^2 - 4ax + 2 = 0$. |
Find $x$ if $\log_x32 = \dfrac{5}{2}$. | Find $x$ if $\log_x32 = \dfrac{5}{2}$. |
Simplify the expression
$$\frac{2}{3} \cdot \frac{3}{5} \cdot \frac{5}{6} \cdot \frac{6}{7} \cdot \frac{7}{8}.$$ | Simplify the expression
$$\frac{2}{3} \cdot \frac{3}{5} \cdot \frac{5}{6} \cdot \frac{6}{7} \cdot \frac{7}{8}.$$ |
A library has a unique system for lending books to its patrons. For the first five books borrowed, the loan period is one week; for the next five books, the loan period is two weeks; and so on, increasing by one week for every five books borrowed. However, if a patron borrows a book that is in high demand, the loan per... | A library has a unique system for lending books to its patrons. For the first five books borrowed, the loan period is one week; for the next five books, the loan period is two weeks; and so on, increasing by one week for every five books borrowed. However, if a patron borrows a book that is in high demand, the loan per... |
Find $\cos \left( \frac{3\pi}{2} \right)$. | Find $\cos \left( \frac{3\pi}{2} \right)$. |
For how many non-negative real values of $x$ is $\sqrt{144-\sqrt[3]{x}}$ an integer? | For how many non-negative real values of $x$ is $\sqrt{144-\sqrt[3]{x}}$ an integer? |
What is the greatest integer $n$ such that the number $n!$ contains the same number of zeros at the end as it contains nonzero digits? | What is the greatest integer $n$ such that the number $n!$ contains the same number of zeros at the end as it contains nonzero digits? |
Compute $\binom{17}{9}$. You are told that $\binom{15}{6} = 5005$ and $\binom{15}{8} = 6435$. | Compute $\binom{17}{9}$. You are told that $\binom{15}{6} = 5005$ and $\binom{15}{8} = 6435$. |
Let $f(x)$ be a polynomial with integer coefficients. If $f(17)=13$, $f(13)=17$ and for every positive integer $k$ not equal to $13$ or $17$, there is a positive integer $n$ such that $n$ and $k+n$ are roots of $f(x)$, find $f(x)$. | Let $f(x)$ be a polynomial with integer coefficients. If $f(17)=13$, $f(13)=17$ and for every positive integer $k$ not equal to $13$ or $17$, there is a positive integer $n$ such that $n$ and $k+n$ are roots of $f(x)$, find $f(x)$. |
What is the sum of the first 100 terms of the sequence defined by $a_n = n^2 - 2n + 1$, where $n$ starts from 1? | What is the sum of the first 100 terms of the sequence defined by $a_n = n^2 - 2n + 1$, where $n$ starts from 1? |
Let $k$ be a real number such that $k > 1$ and
\[\sum_{n=1}^{\infty} \frac{5n-1}{k^n} = \frac{13}{4}.\]Find $k.$ | Let $k$ be a real number such that $k > 1$ and
\[\sum_{n=1}^{\infty} \frac{5n-1}{k^n} = \frac{13}{4}.\]Find $k.$ |
A bookshelf has five shelves, and the average number of books on each shelf is 8. If the top shelf has 3 more books than the bottom shelf, and the middle three shelves have the same number of books, what is the total number of books on the bookshelf? | A bookshelf has five shelves, and the average number of books on each shelf is 8. If the top shelf has 3 more books than the bottom shelf, and the middle three shelves have the same number of books, what is the total number of books on the bookshelf? |
Five standard dice are rolled. Let $A$ be the product of the five numbers shown, where a blank is considered as a 6. Let $B$ be the product of the non-blank numbers shown. Determine all possible values for $B-A$, where $B>A$. | Five standard dice are rolled. Let $A$ be the product of the five numbers shown, where a blank is considered as a 6. Let $B$ be the product of the non-blank numbers shown. Determine all possible values for $B-A$, where $B>A$. |
The employees of a company participate in a charity run. Their finishing times (in minutes) are recorded as follows:
| Name | Finishing Time |
| --- | --- |
| John | 42 |
| Emma | 35 |
| David | 50 |
| Sarah | 38 |
| Michael | 45 |
If the median finishing time is 3 minutes more than the mode (the most frequently occu... | The employees of a company participate in a charity run. Their finishing times (in minutes) are recorded as follows:
| Name | Finishing Time |
| --- | --- |
| John | 42 |
| Emma | 35 |
| David | 50 |
| Sarah | 38 |
| Michael | 45 |
If the median finishing time is 3 minutes more than the mode (the most frequently occu... |
A digit is written to the right of the units digit of $757$. If the resulting four-digit number is divisible by $3$, how many possibilities are there for the digit that was written? | A digit is written to the right of the units digit of $757$. If the resulting four-digit number is divisible by $3$, how many possibilities are there for the digit that was written? |
Find the difference between $1000_7$ and $666_7$ in base $7$. | Find the difference between $1000_7$ and $666_7$ in base $7$. |
Let $x^2 + ax + b = 0$ be a quadratic whose roots are the squares of the roots of the quadratic $x^2 - 3x + 2 = 0$. Find $b$. | Let $x^2 + ax + b = 0$ be a quadratic whose roots are the squares of the roots of the quadratic $x^2 - 3x + 2 = 0$. Find $b$. |
The figure shows a regular hexagon $ABCDEF$ divided into six congruent triangles of side length 1. Let $\overline{AE}$ and $\overline{BD}$ intersect at point $O$. The area of quadrilateral $ABOD$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
defaultpen(linewidth(0.7));
... | The figure shows a regular hexagon $ABCDEF$ divided into six congruent triangles of side length 1. Let $\overline{AE}$ and $\overline{BD}$ intersect at point $O$. The area of quadrilateral $ABOD$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[asy]
defaultpen(linewidth(0.7));
... |
Consider the operation $a\oplus b = a + b^{\lfloor a\rfloor}$, where $\lfloor a\rfloor$ denotes the greatest integer less than or equal to $a$. Find the value of $(3\oplus 2) \oplus 4$. | Consider the operation $a\oplus b = a + b^{\lfloor a\rfloor}$, where $\lfloor a\rfloor$ denotes the greatest integer less than or equal to $a$. Find the value of $(3\oplus 2) \oplus 4$. |
Let $a$ and $b$ be two distinct numbers chosen randomly from the set $\{1, 2, 3, \dots, 99\}$. What is the probability that $a^3 + b^3 + a + b$ is divisible by $a + b$? | Let $a$ and $b$ be two distinct numbers chosen randomly from the set $\{1, 2, 3, \dots, 99\}$. What is the probability that $a^3 + b^3 + a + b$ is divisible by $a + b$? |
What is the largest two-digit multiple of 17? | What is the largest two-digit multiple of 17? |
How many positive, four-digit integers have at least one repeated digit and do not contain the digit 0? | How many positive, four-digit integers have at least one repeated digit and do not contain the digit 0? |
Find the equation of the hyperbola with center at the origin, asymptotes $y=\pm \frac{x}{3}$, and vertices at $(\pm 2,0)$. | Find the equation of the hyperbola with center at the origin, asymptotes $y=\pm \frac{x}{3}$, and vertices at $(\pm 2,0)$. |
The parabola $y^2 = 8x$ and the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at two points $A$ and $B.$ Find the distance $AB.$
New problem:
In the triangle $ABC$, the parabola $y^2 = 8x$ and the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at points $A$ and $B$. The point $C$ is the center of the circle. Find the lengt... | The parabola $y^2 = 8x$ and the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at two points $A$ and $B.$ Find the distance $AB.$
New problem:
In the triangle $ABC$, the parabola $y^2 = 8x$ and the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at points $A$ and $B$. The point $C$ is the center of the circle. Find the lengt... |
A small pizza shop has 15 employees, and the average monthly salary of all employees is $2500. If 5 new employees are hired at a monthly salary of $2000 each, what is the average monthly salary of all 20 employees? | A small pizza shop has 15 employees, and the average monthly salary of all employees is $2500. If 5 new employees are hired at a monthly salary of $2000 each, what is the average monthly salary of all 20 employees? |
A $\textit{composite number}$ is a number that has two or more prime factors. The number 145 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers? | A $\textit{composite number}$ is a number that has two or more prime factors. The number 145 can be expressed as the sum of two composite numbers in many ways. What is the minimum positive difference between two such numbers? |
Two concentric spheres have radii of 4 and 7 units. A point $T$ lies on the surface of the larger sphere, and its coordinates are $(3, 4, 0)$. Another point $U$ lies on the surface of the smaller sphere, and its coordinates are $(0, 0, a)$. If the distance between $T$ and $U$ is 6 units, find the value of $a$. | Two concentric spheres have radii of 4 and 7 units. A point $T$ lies on the surface of the larger sphere, and its coordinates are $(3, 4, 0)$. Another point $U$ lies on the surface of the smaller sphere, and its coordinates are $(0, 0, a)$. If the distance between $T$ and $U$ is 6 units, find the value of $a$. |
Two numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors? | Two numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors? |
A certain medical test for a rare disease is accurate 95% of the time. If 2% of the population actually has the disease, what is the probability that a person who tests positive actually has the disease? | A certain medical test for a rare disease is accurate 95% of the time. If 2% of the population actually has the disease, what is the probability that a person who tests positive actually has the disease? |
If $\left|\frac{12}{x}+3\right|=2$, find the product of all possible values of $x$. Express your answer as an improper fraction. | If $\left|\frac{12}{x}+3\right|=2$, find the product of all possible values of $x$. Express your answer as an improper fraction. |
Two parallel conducting wires are held at a constant distance of 5 cm apart. A current flows through the first wire with a magnitude of 20 A, and the resulting magnetic field induces a current in the second wire with a magnitude of 0.4 A. If the current in the first wire is doubled, what is the minimum distance the sec... | Two parallel conducting wires are held at a constant distance of 5 cm apart. A current flows through the first wire with a magnitude of 20 A, and the resulting magnetic field induces a current in the second wire with a magnitude of 0.4 A. If the current in the first wire is doubled, what is the minimum distance the sec... |
Triangle $AHI$ is equilateral. We know $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction.
[asy]
unitsize(0.2inch);
defaultpen(linewid... | Triangle $AHI$ is equilateral. We know $\overline{BC}$, $\overline{DE}$ and $\overline{FG}$ are all parallel to $\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction.
[asy]
unitsize(0.2inch);
defaultpen(linewid... |
Let's start with n equally sized squares, where n can be any positive integer. Each square has side length s. The total perimeter is 10s if we line up all these squares end to end along one edge, say, horizontally (it makes no difference whether they're lined up vertically). This perimeter must be less than or equal to... | Let's start with n equally sized squares, where n can be any positive integer. Each square has side length s. The total perimeter is 10s if we line up all these squares end to end along one edge, say, horizontally (it makes no difference whether they're lined up vertically). This perimeter must be less than or equal to... |
Find all solutions to the inequality \[\frac{x^2}{(x-3)^2} \ge 0.\](Give your answer in interval notation.) | Find all solutions to the inequality \[\frac{x^2}{(x-3)^2} \ge 0.\](Give your answer in interval notation.) |
If $(x + y)^3 = 27$ and $x^2y + xy^2 = 9$, what is the value of $x^3 + y^3$? | If $(x + y)^3 = 27$ and $x^2y + xy^2 = 9$, what is the value of $x^3 + y^3$? |
Convert the point $(\rho,\theta,\phi) = \left( 2, \pi, \frac{\pi}{4} \right)$ in spherical coordinates to rectangular coordinates. | Convert the point $(\rho,\theta,\phi) = \left( 2, \pi, \frac{\pi}{4} \right)$ in spherical coordinates to rectangular coordinates. |
The 5-digit number $52\,28\square$ is a multiple of 6. Which digit is represented by $\square$? | The 5-digit number $52\,28\square$ is a multiple of 6. Which digit is represented by $\square$? |
Let $f(n)$ be the largest integer $m$ such that $m^3\le n$. Evaluate $f(30)+f(300)+f(3000)$. | Let $f(n)$ be the largest integer $m$ such that $m^3\le n$. Evaluate $f(30)+f(300)+f(3000)$. |
An equilateral triangle is circumscribed around a circle. One of the sides of the triangle is $12$, and the circle passes through the midpoint of this side. Find the area of the triangle. | An equilateral triangle is circumscribed around a circle. One of the sides of the triangle is $12$, and the circle passes through the midpoint of this side. Find the area of the triangle. |
The vertices of a convex decagon are connected by segments in such a way that no three vertices are collinear. How many distinct triangles can be formed using these segments as sides? | The vertices of a convex decagon are connected by segments in such a way that no three vertices are collinear. How many distinct triangles can be formed using these segments as sides? |
As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three people to do it, but there are five other p... | As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three people to do it, but there are five other p... |
Find the value of $y$ that will give the minimum value for $2y^2 + 5y - 8$. | Find the value of $y$ that will give the minimum value for $2y^2 + 5y - 8$. |
In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=24$. [asy]
pair A, B, C, D, E;
A=(0,20.785);
B=(0,0);
C=(9,-5.196);
D=(13.5,-2.598);
E=(12,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B",... | In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=24$. [asy]
pair A, B, C, D, E;
A=(0,20.785);
B=(0,0);
C=(9,-5.196);
D=(13.5,-2.598);
E=(12,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B",... |
Find the product of the complex numbers $e^{i\theta}$, where $\theta$ ranges over all solutions to the equation $\sin \theta = \frac{1}{\sqrt{2}}$ for $0 \le \theta < 2\pi$. | Find the product of the complex numbers $e^{i\theta}$, where $\theta$ ranges over all solutions to the equation $\sin \theta = \frac{1}{\sqrt{2}}$ for $0 \le \theta < 2\pi$. |
Find the value of
\[\frac{\frac{2025}{1} + \frac{2024}{2} + \frac{2023}{3} + \dots + \frac{1}{2025}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2026}}.\] | Find the value of
\[\frac{\frac{2025}{1} + \frac{2024}{2} + \frac{2023}{3} + \dots + \frac{1}{2025}}{\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots + \frac{1}{2026}}.\] |
What is the positive difference between $\frac{6^2 + 6^2}{6}$ and $\frac{6^2 \times 6^2}{6}$? | What is the positive difference between $\frac{6^2 + 6^2}{6}$ and $\frac{6^2 \times 6^2}{6}$? |
Solve the inequality $|2x - 1| - 3x \geq -2$ for $x$. | Solve the inequality $|2x - 1| - 3x \geq -2$ for $x$. |
What is the positive difference between the probability of a fair coin landing heads up exactly 2 times out of 3 flips and the probability of a fair coin landing heads up 3 times out of 3 flips? Express your answer as a common fraction. | What is the positive difference between the probability of a fair coin landing heads up exactly 2 times out of 3 flips and the probability of a fair coin landing heads up 3 times out of 3 flips? Express your answer as a common fraction. |
A band has 72 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 5 and 20 students per row, in how many possible row-lengths can the band be arranged? | A band has 72 members who will all be marching during halftime. They need to march in rows with the same number of students per row. If there must be between 5 and 20 students per row, in how many possible row-lengths can the band be arranged? |
A point $(x,y)$ is randomly selected such that $0<x<1$ and $0<y<1$. What is the probability that $x+y$ is NOT an integer? | A point $(x,y)$ is randomly selected such that $0<x<1$ and $0<y<1$. What is the probability that $x+y$ is NOT an integer? |
Find the sum of the coefficients in the polynomial $(x^3 - 2x^2 + x + 1)(2x^2 + 3x - 1)$ when it is fully expanded and simplified. | Find the sum of the coefficients in the polynomial $(x^3 - 2x^2 + x + 1)(2x^2 + 3x - 1)$ when it is fully expanded and simplified. |
Find the shortest distance between the point $(3,0)$ and the curve given by the equation $y = x^3 + x.$ | Find the shortest distance between the point $(3,0)$ and the curve given by the equation $y = x^3 + x.$ |
The value of $z$ varies inversely as the cube root of $t$ and when $t=27$, $z=2$. What is $t$ when $z=6$? | The value of $z$ varies inversely as the cube root of $t$ and when $t=27$, $z=2$. What is $t$ when $z=6$? |
In a right triangle $ABC$, the length of the hypotenuse $AC$ is 20 units and the length of the altitude $AD$ is 8 units. Determine the area of $\triangle ABC$. | In a right triangle $ABC$, the length of the hypotenuse $AC$ is 20 units and the length of the altitude $AD$ is 8 units. Determine the area of $\triangle ABC$. |
In the pyramid $VABCD$, the base $ABCD$ is a square with side length 12. The vertex $V$ is directly above the center of the base, and the altitude from $V$ to the base has a length of 6. The four triangular faces of the pyramid are congruent, and each face has a height of 3. The pyramid is then cut by a plane that inte... | In the pyramid $VABCD$, the base $ABCD$ is a square with side length 12. The vertex $V$ is directly above the center of the base, and the altitude from $V$ to the base has a length of 6. The four triangular faces of the pyramid are congruent, and each face has a height of 3. The pyramid is then cut by a plane that inte... |
Ming and Catherine walked to the store together to buy some pencils. Ming bought $40$ pencils and Catherine bought $24$. If each package of pencils sold at the store contain the same number of pencils, what is the largest possible number of pencils in a package? | Ming and Catherine walked to the store together to buy some pencils. Ming bought $40$ pencils and Catherine bought $24$. If each package of pencils sold at the store contain the same number of pencils, what is the largest possible number of pencils in a package? |
When the graph of $y = 2x^2 - x + 7$ is shifted four units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$. | When the graph of $y = 2x^2 - x + 7$ is shifted four units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$. |
Given the equation
$$2(x-1)+7=x+3(x+5)$$
find the value of $x^2+2x-5$. | Given the equation
$$2(x-1)+7=x+3(x+5)$$
find the value of $x^2+2x-5$. |
A set of 10 circles, each with a different radius between 1 and 10 units, is drawn on a plane such that no two circles intersect. A line is drawn on the plane such that it intersects 3 of the circles. How many distinct sets of 3 circles can be intersected by a single line? | A set of 10 circles, each with a different radius between 1 and 10 units, is drawn on a plane such that no two circles intersect. A line is drawn on the plane such that it intersects 3 of the circles. How many distinct sets of 3 circles can be intersected by a single line? |
Find the value of $(x^2 + 3x + 2)^2 - (2x + 4)^2$. | Find the value of $(x^2 + 3x + 2)^2 - (2x + 4)^2$. |
A regular hexagon and an equilateral triangle both have perimeters of 72 cm. What is the ratio of the length of the side of the hexagon to the length of the side of the triangle? Express your answer as a common fraction. | A regular hexagon and an equilateral triangle both have perimeters of 72 cm. What is the ratio of the length of the side of the hexagon to the length of the side of the triangle? Express your answer as a common fraction. |
Compute \[
\left\lfloor \frac{2007! + 2004!}{2006! + 2005!}\right\rfloor.
\]
Problem:
A company has 2007 boxes, each containing a different number of marbles. The total number of marbles in all the boxes is 2007! (2007 factorial). The company wants to pack the marbles into 2006 boxes, such that each box contains a dif... | Compute \[
\left\lfloor \frac{2007! + 2004!}{2006! + 2005!}\right\rfloor.
\]
Problem:
A company has 2007 boxes, each containing a different number of marbles. The total number of marbles in all the boxes is 2007! (2007 factorial). The company wants to pack the marbles into 2006 boxes, such that each box contains a dif... |
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square? | A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square? |
A die is rolled twice, and the results are recorded. What is the probability that at least one of the rolls results in an even number or a number greater than 4? | A die is rolled twice, and the results are recorded. What is the probability that at least one of the rolls results in an even number or a number greater than 4? |
Let $b_n$ be a sequence defined by the recursion
$$b_{n+2}=\frac{b_n^2+b_{n+1}}{b_n+1}$$for all $n\geq 1$. Suppose that $b_1$ and $b_2$ are both integers. Determine the smallest possible value of $|b_1-b_2|$. | Let $b_n$ be a sequence defined by the recursion
$$b_{n+2}=\frac{b_n^2+b_{n+1}}{b_n+1}$$for all $n\geq 1$. Suppose that $b_1$ and $b_2$ are both integers. Determine the smallest possible value of $|b_1-b_2|$. |
Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive intege... | Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let $d$ be Dick's present age. How many ordered pairs of positive intege... |
Evaluate $$\lfloor 1.00001 \rfloor + \lfloor 1.000011 \rfloor + \lfloor 1.000001111 \rfloor + \cdots + \lfloor 1.111111111111 \rfloor$$ where the number of 1's in the decimal part increases by 1 in each subsequent term, and $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x$. | Evaluate $$\lfloor 1.00001 \rfloor + \lfloor 1.000011 \rfloor + \lfloor 1.000001111 \rfloor + \cdots + \lfloor 1.111111111111 \rfloor$$ where the number of 1's in the decimal part increases by 1 in each subsequent term, and $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x$. |
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