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A rectangular prism has a total surface area of 648 square units. The area of the largest pair of parallel faces is twice the area of the smallest pair of parallel faces, while the area of the middle-sized pair of parallel faces is the average of the areas of the other two pairs. Determine the volume of the prism. | nvidia |
Find the least positive integer that is divisible by 4, 7, and 9. | nvidia |
What is the smallest positive integer that has exactly three positive divisors? | nvidia |
Given that $\alpha$ and $\beta$ are roots of $x^2-x-1=0$, and that $\gamma$ and $\delta$ are roots of $x^2+3x+1=0$, determine the value of $(\alpha+\beta)(\gamma+\delta)$ | nvidia |
Determine the number of positive integers $n$ such that the number $2022_n$ is a multiple of $5$ and the sum of its digits in base $n$ is a prime number. | nvidia |
Find the sum of all the odd numbers between 1 and 20 that are not divisible by 3. | nvidia |
Find a positive integer that is divisible by 17 and has a cube root between 5 and 5.2. | nvidia |
Find the number of integers $n$ that satisfy
\[10 < n^2 < 99.\] | nvidia |
Given a set of ten points $Q_1, Q_2, \dotsb, Q_{10}$ on the complex plane such that the points are evenly spaced on a circle centered at the origin, determine the value of the sum
\[\frac{1}{Q_1} + \frac{1}{Q_2} + \frac{1}{Q_3} + \dotsb + \frac{1}{Q_{10}}.\] | nvidia |
For a positive integer n, find the minimum value of $a_1 + a_2 +... + a_n$ given that $a_1, a_2,..., a_n$ are non-negative real numbers such that $\sqrt{a_1} + \sqrt{a_2} +... + \sqrt{a_n} = 4$ and $\sqrt{a_1}^2 + \sqrt{a_2}^2 +... + \sqrt{a_n}^2 = 6$. | nvidia |
In a 5-digit number, the sum of the digits is 30. If the digits are increased by 1, the resulting number is divisible by 5. If the digits are increased by 2, the resulting number is divisible by 4. Find the original 5-digit number. | nvidia |
Given a triangle $ABC$ with vertices $A(0,8)$, $B(2,0)$, and $C(8,0)$, a circle intersects the lines $AC$ and $BC$ at points $P$ and $Q$, respectively, such that the area of the circular sector $PCQ$ is 20. Determine the distance from point $C$ to the center of the circle. | nvidia |
Consider the sequence of numbers defined recursively by $s_1=2$ and for $n>1$ by $s_n=s_{n-1}+s_{n/2}$ when $n$ is even and by $s_n=s_{n-1}^2$ when $n$ is odd. Given that $s_n=2047$, find $n$. | nvidia |
Let $f(x)$ be a rational function of the form $\frac{ax^2+bx+c}{dx^2+ex+f}$. Find the sum of the roots of $f(x)$ if it is known that $f(x)$ has infinitely many solutions for $x$ when equated to a constant $k$, and the sum of the values of $x$ for which $f(x)$ is undefined is 10. | nvidia |
Determine the value of $a$ such that the quadratic equation $x^2+ax+4=0$ has roots whose product is twice their sum. | nvidia |
Find all real numbers $b$ such that the equation $\frac{x}{b} + \frac{b}{x} = \frac{5}{2}$ has exactly one solution for $x$. | nvidia |
Determine the smallest integer value of x for which the sum of the squares of the roots of the equation $x^2 + bx + c = 0$ is less than 2. | nvidia |
Determine the polynomial in two variables with rational coefficients and the smallest possible degree that has the root $(x,y)=(\sqrt[3]{2},\sqrt[3]{3})$. | nvidia |
Compute the number of ways to arrange the letters in the word "committee" such that the vowels are together. | nvidia |
In a parallelogram ABCD with side lengths 5 and 10, find the sum of the squares of the lengths of the diagonals AC and BD. | nvidia |
Find all $x$ such that $x^2+5x<6$. Express your answer in interval notation. | nvidia |
Given a positive real number $x$, find the maximum value of $y$ such that $2y^3 - 3xy^2 + x^3 \geq 0$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $\frac{1}{x} + \frac{1}{y} = 2$. Find the maximum value of $x + y$. | nvidia |
Find the number of distinct decimal numbers that can be formed by rounding to two decimal places the fractions $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots, \frac{1}{100}$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x+y) = g(x)g(y)+x+y\]
for all real numbers $x$ and $y.$
Determine all possible values of $g(2).$ | nvidia |
Determine the value of $x$ such that the area of the triangle formed by the points $(0, 0)$, $(x, 0)$, and $(x, x)$ is equal to $\frac{1}{2}$. | nvidia |
Find the minimum value of
\[x^2 + 2xy + 3y^2\]
over all real numbers $x$ and $y$. | nvidia |
Find the value of $x$ that satisfies the equation:
$$\frac{2^{2x}}{2^{x+1}} = 3^x \cdot 2^{-1}$$ | nvidia |
Find the number of positive integers $n,$ $1 \le n \le 1000,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\frac14$? | nvidia |
Determine the area of a rectangle with sides $2x$ and $3x$, where $x$ is a positive real number, given that the perimeter of the rectangle is equal to $20$. | nvidia |
What is the sum of the reciprocals of all the positive integer divisors of 24? | nvidia |
The function $g$ has the property that for each real number $x$ in its domain, $-x$ is also in its domain and \[
g(x) + g(-x) = x^2.
\]What is the largest set of real numbers that can be in the domain of $g$? | nvidia |
Find the number of positive integers less than 100 that are not divisible by 2, 3, or 5, but are divisible by 7. | nvidia |
Find all values of $x$ such that $\lceil x \rceil - \lfloor x \rfloor = 0.7$. | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
Find the number of whole number solutions for $n$ in the inequality $\frac{n+1}{2}<e<n-\frac{1}{2}$. | nvidia |
Determine the sum of the squares of the coefficients in the expansion of $(x+y)^7$. | nvidia |
Find the positive integer $M$ for which $$7^3 \times 14^3 = 2^3 \times M^3$$ | nvidia |
If $a$, $b$, and $c$ are positive integers such that $ab+bc+ca=98$, $a^2+b^2+c^2=128$, and $a^3+b^3+c^3=580$, what is the value of $a+b+c$? | nvidia |
Find the smallest positive integer that is not divisible by any prime number less than 10, but is divisible by at least two distinct primes greater than 10. | nvidia |
Find the smallest positive integer $n$ such that the equation $x^2 + nx + 1 = 0$ has at least one irrational root that can be expressed as a finite decimal. | nvidia |
Find the smallest positive integer $n$ such that $\frac{1}{n}$ is a repeating decimal and $n$ contains the digit 7. | nvidia |
Find all values of $a$ such that $a = 3 - \cos a$. | nvidia |
A circle of radius $3$ units is centered at the point $(h, 0)$. The circle intersects the parabola $y = x^2$ at two points. Find the sum of all possible values for $h$. | nvidia |
Let $x$ be a real number with $0 \le x < \pi$ that satisfies the equation $\tan x = \tan^2 x + \tan^3 x.$ Enter all the solutions, separated by commas. | nvidia |
Find the sum of all real values of $x$ that are not solutions to the equation
$$\frac{\sqrt{x+3}\cdot\sqrt[3]{x+7}}{x+10} = 1$$ | nvidia |
What is the largest five-digit number that can be formed using each of the digits 1, 2, 3, 4, and 5 exactly once, such that the sum of the first two digits is greater than the sum of the last two digits? | nvidia |
Determine the number of six-digit natural numbers divisible by 5 that can be formed using digits 1, 2, 3, 4, 5, 6, 7, 8, 9, such that each digit is used only once and the digits of each number are in increasing order. | nvidia |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$. | nvidia |
Consider the following five statements about real numbers $a$ and $b$:
$a < b$
$a^2 < b^2$
$\frac{1}{a} < \frac{1}{b}$
$a + b < 0$
$ab > 0$
What is the maximum number of these statements that can be true for any values of $a$ and $b$? | nvidia |
Find all positive integer values of $n$ such that $2^n$ is a factor of $3^n+1$. | nvidia |
Find the greatest integer value of $c$ for which the expression $\frac{2x^3+5x^2+7x+1}{x^2+cx+9}$ has a domain of all real numbers. | nvidia |
How many distinct ways can the seven letters of the word "Laughed" be arranged so that no two vowels are adjacent? | nvidia |
The degree measure of the sum of the interior angles of a convex polygon with $n$ sides is 1800. What is the degree measure of the sum of the interior angles of a convex polygon with $n + 2$ sides? | nvidia |
How many triangles with integer side lengths are there, such that one side is the geometric mean of the other two? | nvidia |
Find the ratio of the area of a regular hexagon inscribed in a circle to the area of a regular triangle inscribed in the same circle, given that the radius of the circle is 1 unit. | nvidia |
Determine the number of distinct four-digit integers that can be formed using the digits of the number 1333, if the resulting integer is divisible by 3. | nvidia |
Let $S$ be a set of distinct positive integers. Define a "cohesive" subset of $S$ to be a subset where every pair of elements in the subset has a common divisor greater than 1. For example, if $S = \{2,3,4,6,9\}$, then the cohesive subsets of $S$ are $\{2,4\}$, $\{2,6\}$, $\{3,6\}$, $\{3,9\}$, and $\{4,6\}$. Let $T$ be... | nvidia |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas. | nvidia |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | nvidia |
Find the number of positive integers less than or equal to 100 that are relatively prime to 15. | nvidia |
Find the sum of the values of $x$ that satisfy the equation $x^3-4x^2-17x+60=0$ and also make the expression $\sqrt{2x+5}$ an integer. | nvidia |
Find the sum of all the roots of the equation $x^{17}-x^{15}+x^{13}-x^{11}+\cdots +x^3-x = 0$. | nvidia |
What is the smallest integer value of $x$ for which the sum of the squares of the positive integer solutions to $x^2 - |y| = 1$ is less than 100? | nvidia |
Count the number of 6-digit positive integers with all digits being either 3 or 4, such that no two consecutive digits are the same. | nvidia |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [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 tickdownlength=-0.15inch;
real tickdownbase=0.3;
real who... | nvidia |
Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $x+y+z=12,$ $xy+yz+xz=47,$ and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1.$ Find $xyz.$ | nvidia |
If the graph of the function $f(x)$ contains the point $(a,b)$ and the graph of the function $g(x)$ contains the point $(b,a)$, what is the sum of the coordinates of the point that must be on the graph of the function $y = f(g(x))$? | nvidia |
What is the greatest positive integer that can be subtracted from 537 so that the resulting number is a multiple of 7? | nvidia |
In a triangle, the length of the hypotenuse is 15 meters, and the ratio of the other two sides is 2:5. Find the area of the triangle. | nvidia |
Given a cubic polynomial P(x) with real coefficients, find the number of distinct possible values of the real number 'a' such that P(x) = 0 and P(x+a) = 0 have exactly two common roots for any value of x. | nvidia |
Let $f(z)$ be a monic quartic polynomial with real coefficients such that two of its roots are $\frac{\sqrt{3}}{2}i-\frac{1}{2}$ and $\frac{1}{2}+\frac{\sqrt{3}}{2}i$. Suppose that for every root $r$ of $f(z)$, $|r|=1$, and that the four roots form a rectangle when plotted in the complex plane. Find $\sum_{n=1}^\infty ... | nvidia |
If $p$ is a non-real complex number such that $p^2+p+1=0$, determine the value of $|p^4|$. | nvidia |
The graph of the ellipse
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\]where $a > b,$ has the property that the area of the triangle formed by the two foci and the point $(0,b)$ is $\frac{1}{4}$ of the area of the triangle formed by the two foci and the point $(0,a)$. Find $\frac{a}{b}$. | nvidia |
Find the value of $x$ that satisfies the equation $\frac{x}{2} \cdot (x+1) \cdot (x+2) = 12 + \frac{12}{x}$. | nvidia |
Find the largest integer $x$ that satisfies the inequality:
$$x \leq \sqrt{5x+2} < x+1.$$ | nvidia |
Find the only positive real number $y$ for which $\displaystyle \frac{y-3}{7} = \frac{3}{y-7}$. | nvidia |
Let $a$ and $b$ be real numbers such that
\[\sin (a + b) = \sin a \cos b.\]
Find the minimum value of $\sin a.$ | nvidia |
Consider the triangular array of numbers with 0, 1, 2, 3, $\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & & & \\
& & 3 & & 4... | nvidia |
Find the probability that the product of the three numbers obtained by tossing three standard dice is a multiple of 3. | nvidia |
In $\triangle RED$, $\measuredangle DRE=75^{\circ}$ and $\measuredangle RED=45^{\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC}\perp\overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA=AR$. Then $AE=\fr... | nvidia |
Find the number of all distinct values of the sum of the squares of the roots of the equation $x^2+px+q=0$, where $p$ and $q$ are positive integers such that $p+q=2019$. | nvidia |
Let $F$ be a function that maps positive integers to real numbers such that
$$\frac{F(m)}{F(n)} = \frac{mF(n+1)}{nF(m+1)}$$for all positive integers $m$ and $n.$ Given that $F(1) = 1$ and $F(2) = 4,$ find $F(100).$ | nvidia |
Let $F(n)$ denote the number of distinct prime factors of a positive integer $n.$ For example, $F(1)=0,F(2)=1,$ and $F(60)=3.$ Define $T(n)$ by $T(n)=F(1)+F(2)+\cdots+F(n).$ Determine all positive integers $n$ such that $T(n)=T(2n)$. | nvidia |
Let $f$ be a function with the property that for each real number $x$ in its domain, $x^2$ is also in its domain and
\[
f(x) \cdot f(x^2) = x.
\]What is the largest set of real numbers that can be in the domain of $f$? | nvidia |
In a geometric sequence, the sum of the first 1234 terms is 150, and the sum of the first 2468 terms is 300. Find the sum of the first 3702 terms. | nvidia |
Find all real numbers $b$ such that the inequality $|x^3 - 3bx^2 + 2b^2x - b^3| \le 1$ has exactly two solutions in $x$. | nvidia |
Find the smallest three-digit number $n$ such that the last three digits of $13\cdot n$ are $123$. | nvidia |
Factor the expression $x^2+bx+c$, where $b$ and $c$ are integers. The equation $x^2+bx+c=0$ has real roots, and the discriminant of this equation is a perfect square. What can be said about the expression $c+2x+b^2$, when $x$ is an integer? | nvidia |
Find the greatest $b$ such that $\frac{3\sqrt{(3b)^2+(2)^2}-2b^2-4}{\sqrt{4+9b^2}+2}=1$. | nvidia |
Let $a$ be a real number such that the sequence $\{a_n\}_{n=0}^{2018}$, defined by $a_n = a^n$, has the property that the sum of any 19 consecutive terms of the sequence is divisible by 2019. Find the remainder when $a^{2018}$ is divided by 2019. | nvidia |
Let $N$ be the number of positive integers $x$ such that
$x(x+1)(x+2)(x+3)<17!$ and $\ds\frac{x+3}{x}>14$. Find $N$. | nvidia |
In triangle $ABC$, $AB=AC$, and point $P$ lies inside the triangle such that $\angle{APB}=\angle{APC}$. The perimeter of triangle $ABC$ is 24, and the area of triangle $APB$ is 8. Find the area of triangle $APC$. | nvidia |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | nvidia |
Find the number of real values of $k$ such that the equation $\left|\frac{k}{1+i}\right| = 2$ has a solution. | nvidia |
Find the sum of all positive integers less than 100 that are perfect cubes of perfect squares. | nvidia |
A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$.
| nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 6.\]Find the minimum value of $x^3 y^2 z.$ | nvidia |
Let $x,$ $y,$ $z$ be a three-term geometric series where all the terms are positive, such that $x+y+z = 60.$ Find the largest possible value of $y.$ | nvidia |
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