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The prime factorization of a number is 2^3 \cdot 3 \cdot 5^2. What is the arithmetic mean of all the positive integer divisors of this number that are multiples of 15? | nvidia |
The sum of the cubes of two positive integers is 945. The difference of the two integers is 3. What is the sum of the two integers? | nvidia |
Given that $a$, $b$, and $c$ are real numbers satisfying $a^2 + b^2 = c^2$, calculate the value of $(a+b)^2 - (a-b)^2$. | nvidia |
Given the ellipse defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a > b$, and the line $y = kx$, find the value of $k$ for which the tangent point of the line and ellipse coincides with the point on the ellipse furthest from the origin. | nvidia |
Let $P(x)$ be a polynomial such that $P(x^3) + P(-x^3) = x^6$ for all real numbers $x.$ Find $P(11).$ | nvidia |
In a triangle $ABC$, the side $BC$ is a diameter of a circle with radius $2\sqrt{10}$. The point $A$ lies on the circle. A line passing through the center of the circle intersects the line segment $BC$ at the point $D$ and intersects the circle at the point $E$. If the ratio of the area of triangle $ABD$ to the area of... | nvidia |
Find the sum of all positive integers less than 1000 that are not divisible by 2, 3, or 5. | nvidia |
In the equation
$$\frac{x}{x^2-4}=\frac{1}{x+3}$$
determine $x$ such that $x$ is a real number and $x\neq -3$. | nvidia |
The real and imaginary parts of a complex number z are both positive integers. If the sum of these two parts is 7, find the smallest possible value of |z|. | nvidia |
What is the positive difference between the sum of the first 20 positive even integers and the sum of the first 15 positive odd integers? | nvidia |
Find the smallest value of $x$ that satisfies the equation $|2x-5|=3x+1$. | nvidia |
Determine the value of $x$ such that the expression $\frac{x_{11}}{55_{6}}+\frac{33_{4}}{22_{5}}$ is equal to $\frac{2}{3}$ when expressed in base 10. | nvidia |
Consider the function $f(n)$ that represents the sum of all positive divisors of $n$.
Now, determine the smallest positive integer $n$ such that $f(f(n))=n^2$. | nvidia |
Let $a$, $b$, and $c$ be complex numbers such that
$a+b+c = i^4 + i^8 + i^{12}$,
$ab+ac+bc=i^8+i^{20}+i^{12}$,
and
$abc=i^{12}+i^{24}+i^{36}$.
Determine the value of $|a+b+c|^2$, knowing that
$|a|=|b|=|c|$ and that $a$, $b$, and $c$ form the vertices of an equilateral triangle on the complex plane. | nvidia |
Let $m$ be a positive integer such that $2^m\equiv 1\pmod{7}$. Determine $2^{m+1}+2^{m-1}\pmod{7}$. | nvidia |
Find the coordinates of the point that is one-third of the way from the point $(2,4)$ to the point $(8,10)$. | nvidia |
Find the largest integer $n$ for which $12^n$ evenly divides $20!$. | nvidia |
Given that $\triangle ABC$ and $\triangle A'B'C'$ are similar triangles with $A$ and $A'$ being the centers of two concentric circles, and $B$, $C$, $B'$, and $C'$ lying on the circles, find the ratio of the area of the smaller circle to the area of the larger circle if the ratio of the areas of the triangles is $1:4$ ... | nvidia |
Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $a^2+b^2+c^2=50$. Let $m=\min\{a,b,c\}$. Find the largest possible value of $m$. | nvidia |
Find the number of positive integers less than 50 that are not multiples of 3, 5, or 7. | nvidia |
Let $x$ be a positive real number such that the fractional part of $x$ is $\frac{1}{x}$. Find $x$. | nvidia |
A circle is inscribed inside a regular hexagon, such that the diameter of the circle is equal to the side length of the hexagon. The area of the hexagon is 864. Find the radius of the circle. | nvidia |
When four standard dice are tossed, the numbers $a,b,c,d$ are obtained. Find the probability that $(a-2)(b-2)(c-2)(d-2) \neq 0$ | nvidia |
The regular polygon $A_1A_2A_3A_4A_5...A_{100}$ is inscribed in a circle of radius $1.$ Point $P$ lies inside the circle so that the distance from $P$ to the center of the circle is $r$, where $0<r<1.$ The rays $\overrightarrow{PA_i}$ divide the circle into $100$ sectors. Determine the maximum number of sectors that ca... | nvidia |
It is known that, for all positive integers $k$,
$1^2+2^2+3^2+\ldots+k^{2}=\frac{k(k+1)(2k+1)}6$.
Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $200$.
| nvidia |
For how many four-digit positive integers is the sum of the digits equal to 9? | nvidia |
Determine the number of positive integers $n$ that satisfy the inequality $3^{n-1} < 2^n < 3^{n+1}$ | nvidia |
The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits $2, 5$, and $1$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible.
| nvidia |
Let $M$ be the midpoint of the segment $\overline{AP}$. $N$ is a point on the extension of the segment $\overline{MP}$ such that $MN = \frac{1}{3}MP$. A circle passes through the points $M$, $N$, and $P$, and intersects the segment $\overline{AP}$ at the point $Q$ such that $A$ and $Q$ are on opposite sides of the poi... | nvidia |
Find the real number $x$ such that $\lfloor x \rfloor \cdot x = x^2 - 1$. Express $x$ as a decimal. | nvidia |
Find the greatest integer that can be formed by the digits of the number 123456789, such that it is divisible by 9 and the sum of its digits is equal to 45. | nvidia |
Given the system of equations
$$x+y+z=6,$$
$$x^2+y^2+z^2=14,$$
find the largest whole number $N$ satisfying the inequality $|xy+yz+zx|<N$. | nvidia |
Given positive integers $a$ and $b$, where $b>a$, find the smallest possible value of $b$ such that the difference between the sum of divisors of $b$ and the sum of divisors of $a$ is 231, considering that the prime factorization of $a$ contains only one distinct prime. | nvidia |
Find the largest possible value of the expression $$\frac{a^3+b^3+c^3}{(a+b+c)^2},$$ where $a$, $b$, and $c$ are positive real numbers, and $a+b+c \neq 0.$ | nvidia |
Find the number of positive integers less than or equal to 1000 that have at least one digit repeated. | nvidia |
Determine the sum of all values of $s$ such that $s^3 = 8s$. | nvidia |
Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ | nvidia |
Let $g(x)$ be a function defined for all positive integers satisfying the conditions $g(x) > 0$ for all $x > 0$ and
\[g(x + y) = g(xy) + g(x)g(y)\]for all $x, y > 0.$ Determine $g(2010).$ | nvidia |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/29 of the original integer. | nvidia |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | nvidia |
Determine the smallest composite number that can be written as the difference of two powers of a prime number. | nvidia |
What is the minimum number of sides a convex polygon can have such that the number of its diagonals is greater than the number of its sides? | nvidia |
Let $a$ be a real number randomly selected from the interval $[-12,12]$. Then, $p$ and $q$ are two relatively prime positive integers such that $p/q$ is the probability that the equation $x^4+16a^2=(2a^2-6a)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $p+q$. | nvidia |
Find the value(s) of $x$ such that $8xy-12y+2x-3=0$ is true for all values of $y$. | nvidia |
Let $a_n$ denote the sum of the reciprocals of all the positive divisors of $n$. The sum $a_4a_5 \dotsm a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? | nvidia |
Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(m) \equiv 0 \pmod{m}$ for all positive integers $m$. Find the smallest possible degree of $f(x)$. | nvidia |
An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon? | nvidia |
The equation $x^2 + kx + m = 0$ has two distinct solutions $x_1$ and $x_2$ such that the sum of their squares is equal to twice the product of $x_1$ and $x_2$. Find the value of $k$ in terms of $m$. | nvidia |
What is the smallest positive integer $k$ for which there exists a positive integer $n$ such that $k^n$ can be expressed as the sum of exactly two perfect squares? | nvidia |
Find the product of all positive odd factors of $432$ that are not perfect squares. | nvidia |
Find a polynomial of degree 6, in x, with integer coefficients such that cos(π/7) is a root of the polynomial. | nvidia |
The sum of the areas of all the triangles formed by the diagonals of a regular heptagon is $S$. Find the largest integer less than $\frac{7S}{100}$. | nvidia |
For positive integers $n$, let $f(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point. How many positive integer divisors does $f(2010)$ have? | nvidia |
Find the smallest value of $x$ that satisfies the equation $|3x+7|=26$. | nvidia |
Find the largest integer $n$ such that $3^n$ is less than or equal to $80$. | nvidia |
Determine the largest positive integer $m$ such that the repeating decimal representation of $\frac{1}{m}$ has a period of at most 4 digits. | nvidia |
For all composite integers $n$, what is the largest integer that always divides into the difference between $n$ and the square of $n$? | nvidia |
Let $\alpha$ and $\beta$ be two positive real numbers such that $\alpha + \beta = 1$. Find the value of $x$ in the equation $\alpha^{\alpha} \beta^{\beta} + \alpha^{\beta} \beta^{\alpha} = \frac{1}{2^x}$. | nvidia |
A game show contestant chooses three distinct numbers at random from the set {1, 2,..., 6}. What is the probability that the sum of the three numbers is an even number? | nvidia |
What is the smallest positive integer $n$ such that $n\%$ of $n$ is greater than or equal to $120$? | nvidia |
Given that the determinant of a 2x2 matrix is 14, find the determinant of the matrix that results when the rows and columns of the original matrix are swapped, and then the matrix is multiplied by a scalar value of 3. | nvidia |
Let $a$ and $b$ be positive integers such that $ab+4a+4b=37$ and $a>b$. Determine $a$. | nvidia |
The product of the positive integer divisors of a positive integer $n$ is $2^7$. Determine $n$. | nvidia |
What is the smallest positive integer $n$ such that $\frac{n}{n+101}$ is equal to a terminating decimal? | nvidia |
Consider the set of all polynomials of degree 3 with real coefficients that have all roots equal to -1 or 1 and have the leading coefficient equal to 1.
Find the number of polynomials in this set, which have at least one root equal to -1. | nvidia |
In a regular polygon, the diagonals are the line segments that connect non-adjacent vertices. What is the minimum number of diagonals that can be drawn in a regular polygon with 12 sides such that every diagonal intersects at least two other diagonals? | nvidia |
A point $P(x,y)$ is such that its distance from point $A(-6,-10)$ is half its distance from point $B(2,16)$. Determine the $x$-coordinate of point $P$. | nvidia |
What is the sum of the squares of the positive integers from 1 to 100? | nvidia |
In quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $E$, with $AE=4$, $CE=3$, and $BE=5$. Find the length of $DE$, given that the area of triangle $AED$ is 17.5 and the area of triangle $BEC$ is 12.5. | nvidia |
Let a and b be positive real numbers. Find the maximum value of
$$\lfloor{\sqrt{a}}\rfloor\lfloor{\sqrt{b}}\rfloor,$$
given that a+b=9. | nvidia |
Find the sum of the $y$-coordinates of the points of intersection of the parabola $x^2-4x+4=y^2$ and the line $x-y-2=0$. | nvidia |
A family of circles is defined by the equation $(x-a)^2 + (y-a)^2 = a^2$, where $a$ is a parameter that varies over all real numbers. Find the curve that passes through all the centers of these circles. | nvidia |
Find the smallest positive integer value of $y$ such that $y + 1234$ results in a number that is a palindrome. | nvidia |
Find all pairs of three-digit numbers whose sum of squares is a perfect square. | nvidia |
Let $A = (a_{ij})_{3 \times 3}$ be a matrix such that for any $3 \times 3$ matrix $B$, the sum of the elements in the main diagonal of the matrix product $AB$ equals the sum of all elements of $B$. If the sum of the elements of $A$ is $99$, what is the sum of the elements in the main diagonal of $A$? | nvidia |
Find all integers $n$ such that $n^2+n+1$ is a perfect square. What is the sum of the squares of these integers? | nvidia |
A combinatorial equation $f(n)=\sum_{i=0}^{k}(-1)^{i}\binom{n}{i}$ is given, where $n$ and $k$ are positive integers and $f(n)$ denotes the number of subsets of a set $\{1, 2, \dots, n\}$ that contain at most $k$ elements.
Determine the largest possible value of $n$ such that $f(n) < 10^5$. | nvidia |
Find the smallest positive integer n such that 2^n - 1 is divisible by 13. | nvidia |
What is the largest even number that can be expressed as the product of two distinct prime numbers and is also a perfect square? | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a^2b^3c^5$ is a perfect cube. Find the smallest positive integer $n$ such that $n^3a^2b^3c^5$ is a perfect fifth power. | nvidia |
Determine the largest positive integer $k$ such that $x^k - 1$ is a factor of $x^{1001} - 1$ and $x^{1012} - 1$, where $x$ is an integer greater than 1. | nvidia |
Find the sum of the series:
$$1+2(0.7)+3(0.7)^2+4(0.7)^3+5(0.7)^4+...$$ | 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 |
Find the smallest positive integer $x$ that satisfies the congruence $37x \equiv 1 \pmod{19}$. | nvidia |
For each positive integer $n$, let $T(n)$ denote the product of the digits of $n$. For how many values of $n$ is $n+T(n)+T(T(n))=2023$? | nvidia |
Let $p(x)$ be a polynomial such that $p(x^2 + 1) = x^4 + 4x^2$ for all real numbers $x.$ Find $p(18).$ | nvidia |
Find the least positive value of $x$ such that
\[\cos (\arccos x), \ \cos (2\arccos x), \ \cos (3\arccos x), \ \cos (4\arccos x)\]is an arithmetic progression for some $x$ with $0 < x < 1$. | nvidia |
Find the number of positive integers $n \le 1000$ such that $n$ can be expressed as the difference of two perfect squares. | nvidia |
Given positive integers $m$ and $n$ such that $\frac{1}{m} + \frac{1}{3n} = \frac{1}{4}$, what is the least possible value of $mn$? | nvidia |
Find $b$ such that the lines given by the equations
$$y=3x+7\text{ and }y=bx+8$$
are perpendicular to each other. | nvidia |
Let $a,$ $b,$ and $c$ be constants such that the system of equations
\begin{align*}
x+y+z&=a,\\
x^2+y^2+z^2&=b,\\
x^3+y^3+z^3&=c,
\end{align*}has a unique solution $(x,y,z)$ where $x,$ $y,$ and $z$ are distinct positive integers. Find the smallest possible value of $a+b+c.$ | nvidia |
Find the sum of the solutions of: $|x+1|+|x-1|=3|x|$. | nvidia |
A circle with center $O$ is tangent to a triangle $ABC$ at points $D$, $E$, and $F$, where $D$, $E$, and $F$ are the midpoints of the sides of the triangle. If the radius of the circle is $r$, the perimeter of the triangle is $p$, and the area of the triangle is $a$, find the value of the sum $p + a + r$. | nvidia |
The geometric mean of an even number of consecutive positive integers is $x$. Find the product of the smallest and largest of the integers in terms of $x$. | nvidia |
Three integers $x$, $y$, and $z$ are chosen uniformly at random from the set of positive integers less than or equal to $11$, such that $x\neq y$, $y\neq z$, and $z\neq x$. Let $p(x,y,z)$ be the probability that $x$, $y$, and $z$ are the lengths of the sides of a triangle. Compute $p(3,5,7)$. | nvidia |
Let $S$ be the set of all integer sequences $(a_1,a_2,\ldots)$ with the following property: for all natural numbers $m$ and $n$, the sequence
\[(a_{m+1},a_{m+2},\ldots,a_{m+n})\]is a permutation of the sequence
\[(a_{n+1},a_{n+2},\ldots,a_{n+m}).\]For how many values of $k$ does there exist such a sequence with $a_1 = ... | nvidia |
Find the sum of all coefficients in the expansion of $(x+y+z)^5$. | nvidia |
Find the smallest positive integer $b$ such that $b^{-1}$ is undefined $\pmod{40}$ and $b^{-1}$ is also undefined $\pmod{50}$. | nvidia |
Find the largest value of $\frac{1}{x^2}$ if $x$ is a real number satisfying \[\sqrt{\frac{\log_{10}x^3}{\log_{10}x}}+\sqrt{\frac{\log_{10}x^2}{\log_{10}x}}=2.\] | nvidia |
What is the smallest positive integer that can be expressed as the sum of two consecutive odd numbers and is also divisible by 8 and 15? | nvidia |
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