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Let $\S$ be an operation that takes two integers as input and returns an integer. For any two integers $a$ and $b$, $a\,\S\,b$ is defined as the sum of all integers in the interval $[a,b]$, inclusive. If $7\,\S\,2=27$, what is the value of $5\,\S\,3$? | nvidia |
We define a function $f(x)$ such that $f(11)=34$, and if there exists an integer $a$ such that $f(a)=b$, then $f(b)$ is defined and
$f(b)=3b+1$ if $b$ is odd
$f(b)=\frac{b}{2}$ if $b$ is even.
What is the smallest possible number of integers in the domain of $f$? | nvidia |
The sum of two positive integers is 24 and their ratio is 3:5. What is the value of the larger of the two integers? | nvidia |
Find the sum of all real numbers $x$ such that $f(f(x))=x$, where $f(x)$ is a real-valued function satisfying $f(x+y)+f(x-y)=2f(x)f(y)$ for all real numbers $x$ and $y$. | nvidia |
The sum of the squares of the sides of a triangle is 54 cm^2.
Find the maximum area of the triangle. | nvidia |
Find the number of odd factors of $3^6 \cdot 5^8$ that are multiples of $3^4$. | nvidia |
Determine the number of divisors of $4^3 \cdot 5^4 \cdot 6^2$ that are perfect squares. | nvidia |
Find the sum of the values of $x$ which satisfy $x^2 +1992x = 1993$. | nvidia |
The problem is:
The pages of a book are numbered from 1 to n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice?
Another problem inspired by this one:
A teacher has a set of n d... | nvidia |
Find the number of ways to represent the number 2010 as a sum of four integers, where each integer is a product of a distinct prime number and a perfect square. | nvidia |
Given a sequence $a_n$ such that $a_1=i, a_2=-i, a_3=-1, a_4=1$, and for $n>4$, $a_n=a_{n-4}$, find the sum of the first 10000 terms of the sequence. | nvidia |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the maximum value of $x^2y + y^2z + z^2x$. | nvidia |
Find the largest real number $x$ such that
\[\frac{\lfloor x \rfloor}{x} = \frac{11}{12}.\] | nvidia |
Determine the number of integers between 100 and 900 such that their square roots are integers themselves. | nvidia |
How many three-digit numbers are there such that the product of the tens and units digits is a perfect square? | nvidia |
On a 5x5 grid, find the number of paths from the top-left corner to the bottom-right corner that pass through at least two consecutive points on the same row or column. | nvidia |
Find the number of pairs of positive integers $(a, b)$ that satisfy the equation $a^2 - b^2 = 56$. | nvidia |
For how many positive integers $n$ does $1+3+5+\cdots+(2n-1)$ evenly divide $15n$? | nvidia |
In a regular dodecahedron, the vertices are labeled with positive integers from 1 to 20. A dodecahedron is said to be "balanced" if the sum of the numbers on each pair of opposite faces is equal. The dodecahedron is placed on a table, and a series of rotations is performed, where each rotation consists of rotating the ... | nvidia |
Let $f(z) = \frac{1 - iz}{1 + iz}$ for all complex numbers $z.$ Find the smallest positive real number $k$ such that for any complex number $w$ with $|w| \le 1,$ the complex number $f(w)$ lies on a circle whose radius is at most $k.$ | nvidia |
Find the positive value of $n$ such that the equation $9x^2+nx+1=0$ has exactly one solution in $x$. | nvidia |
Given that $f(x) = x^2-4x+2$, determine the value of $x$ such that $f(f(x)) = f(3)$. | 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 |
Find the area of the triangle with vertices at (-3, 4), (0, 0), and (5, -2). | nvidia |
The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Let \a=|x^2+2bx+c|\le k. Determine all real numbers c, for any b,k that are given, such that the inequality has exactly one solution for x. | nvidia |
How many different 5-digit positive integers exist such that the first digit is not 0 and the last digit is even? | nvidia |
Find the integer $n$, $0 \le n \le 11$, such that \[n \equiv 10389 \pmod{12}.\] | nvidia |
In the triangle $ABC$, $A=(0,0)$, $B=(6,0)$, and $C=(x,y)$.
Find $x$ and $y$ such that
$AC$ is perpendicular to $BC$,
and
the midpoint of $BC$ lies on the line $3x+5y=20$. | nvidia |
Find all values of $t$ such that the equation $\frac{3t^2+2t-1}{2t^2-t-1} > 5$ holds true. Express the solution in terms of interval notation. | nvidia |
Let $a$ and $b$ be positive integers such that $a > b$.
What is the smallest possible value of $a$ for which the expression $a^{a+1} \div (a-b)^b$ is an integer? | nvidia |
Let's consider a grid of size $n \times m$ filled with numbers from 1 to $n \cdot m$ in a specific way. Each number can only be placed in a cell that is adjacent to a cell already filled with a number that is one less than the number to be placed. For example, the number 3 can be placed in a cell adjacent to a cell fil... | nvidia |
Let $z$ be a complex number such that $|z|=1$ and $|z^2+\frac{1}{z^2}|=1$. Find the minimum possible value of $|z+\frac{1}{z}|$. | nvidia |
Find the sum of all composite integers between 1 and 20, that can be expressed as a product of two distinct primes. | nvidia |
Find all nonzero polynomials $p(x)$ for which there exists a nonzero polynomial $q(x)$ such that
\[\frac{d}{dx} \left( p(x) q(x) \right) = p(x) q(x).\]
Enter all the solutions, separated by commas. | nvidia |
Find the number of positive integers less than or equal to 1000 that are divisible by both 3 and 7. | nvidia |
How many four-digit positive integers have at least one digit that is a 2 or a 3? | nvidia |
Find all real numbers $x$ such that the expression $(x + 1)^2 \sin(x) - (x + 2)^2 \cos(x)$ is equal to zero for all real values of $x$. | nvidia |
A triangle with sides of length $a$, $b$, and $c$ is called a Heronian triangle if $a$, $b$, and $c$ are all integers and the area of the triangle is also an integer. Determine the number of Heronian triangles with at least one side of length 17. | nvidia |
Compute the number of ways to arrange 5 red balls and 11 blue balls in a row such that no two red balls are adjacent. | nvidia |
Compute the sum of all possible products of three different numbers from the set {1, 2, 3, 4, 5}, divided by the square of the sum of the same numbers. | nvidia |
Determine the smallest positive integer $n$ such that $5^n\equiv n^5\pmod 3$. | nvidia |
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$
| nvidia |
Find the largest positive integer that is a perfect cube and is also the sum of two distinct prime numbers. | nvidia |
Given an equilateral triangle with a side length of 2 units, inscribe it inside a square such that one of the triangle's vertices lies on the square's corner and the other two vertices lie on the square's sides.
One of the square's diagonals is parallel to the triangle's base.
Determine the area of the square. | nvidia |
Consider the equation
\[\sqrt[3]{(1 + i)^x} + \sqrt[3]{(1 - i)^x} = 2,\]
where $i$ is the imaginary unit, such that $i^2 = -1$. Find all the real values of $x$ that satisfy the equation. | nvidia |
Let $a$, $b$ and $c$ be positive integers such that $a>1$. Find the smallest possible value of $a+b+c$ such that the domain of the function $f(x) = \sqrt{a^x}-\frac{b}{\arctan x}+\frac{c}{x^2+1}$ is a closed interval of length $1$. | nvidia |
Determine all real values of $a$ such that $ax^2+3x-2$ divides $x^5+x^3+3x^2+ax+2$. | nvidia |
Let $f(x)$ be a polynomial such that
$$f(x^2)=x^2(x^2+1)(x^2+2),$$
for all real numbers $x$. Compute the value of $f(4)$. | nvidia |
A point $P$ is inside a regular hexagon. Three lines pass through $P$, each line passing through two vertices of the hexagon. The distance between $P$ and one of the vertices is given as $d$. Determine the sum of all possible distances between $P$ and the vertices of the hexagon. | nvidia |
Determine the smallest positive number $a$ such that the function $y = \sin (ax) + \sin (x/a)$ is periodic with period $2\pi$. | nvidia |
The graph of a function $y = f(x)$ is shown below. The graph consists of two line segments, one connecting the points $(0,0)$ and $(2,2)$, and the other connecting the points $(2,2)$ and $(4,0)$.
What is the value of the integral $\int_{0}^{4} f(x) dx$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklen... | nvidia |
Let x_n=1+\frac{1}{2}+\cdots+\frac{1}{n} , i.e., denote by x_n the sum of the first n terms of a harmonic series. Prove that for any positive integer n we have
$$\lfloor x_n \rfloor - \lfloor x_{n-1} \rfloor = \left\{ \begin{array}{ll}
1 & \mbox{ if } n \mbox{ is a power of } 2 \\
0 & \mbox{ otherwise }
\end{array} \ri... | nvidia |
What is the smallest positive integer $n$ such that $n\cdot \frac{5}{8}$ is an integer? | nvidia |
A $\frac 1q$ -triangle is a structured, infinite, collection of numbers. For example, a $\frac 15$ -triangle is constructed as follows:
\begin{align*} 1 \qquad \frac 15\,\ \qquad \frac 1{25}\,\ \qquad \frac 1{125} \qquad &\cdots\\ \frac 1{10} \qquad \frac 1{50}\,\ \qquad \frac{1}{250} \qquad &\cdots\\ \frac 1{100} \qqu... | nvidia |
Find the number of distinct real solutions of the equation $x^2 + ax + b = 0$, where $a$ and $b$ are real numbers such that $a + b = 1$ and $a - b = -3$. | nvidia |
Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. | nvidia |
Find the maximum number of consecutive positive odd integers that can be multiplied together before the product exceeds 1000. | nvidia |
The expression $x^3 - 3x^2 + 3x - 1$ represents the volume of a rectangular prism with edge lengths $x$, $x-1$, and $x-2$. Find the value of $x$ such that the volume is equal to the sum of the volumes of three identical cubes with edge lengths $x-1$. | nvidia |
What is the sum of all positive integer divisors of 77? | nvidia |
On a circle with center $O$ and radius $r$, two tangents are drawn from an external point $P$. The tangents touch the circle at points $A$ and $B$. A third tangent is drawn from a point $Q$ on the line segment $OP$, intersecting the circle at point $C$. Given that $\angle AOB = 90^\circ$ and $OC \perp PQ$, find the are... | nvidia |
Find the largest positive integer $x$ such that $\gcd(x+7,x+11)=1$ and $x<1000$. | nvidia |
Let $S$ be a subset of the set $\{1, 2, \ldots, 10\}$. Compute the number of $S$ such that $|S|=8$ and $|S\cap\{1, 2, \ldots, 5\}|=3$. | nvidia |
Two circles, each with a radius of 4, are drawn with centers at (10, 40) and (15, 25). A line passing through (12, 30) divides the circles such that the combined area of the parts on one side of the line equals the combined area of the parts on the other side. Determine the absolute value of the slope of this line. | nvidia |
Find the smallest positive integer $y$ that satisfies the congruence $13y^2+5y\equiv 11\pmod {17}$. | nvidia |
Find the smallest positive integer n, such that the polynomial $\cos ^n(x) + \sin ^n(x)$ can be written in the form $P(\cos x)$, where P(t) is a polynomial with real coefficients, and determine this polynomial. | nvidia |
The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7? | nvidia |
Find the number of rational numbers $p/q$, where $1 \le p,q \le 100$, such that $p/q$ is a repeating decimal and $p+q=100$. | nvidia |
Given the equation of a parabola in the form of $x^2+Ay^2+Bx+Cy+D=0$, determine the value of A such that the parabola has only one real focus. | nvidia |
Find the value of $a$ for which the function $f(x)=x^2-4x+a$ has exactly one real root. | nvidia |
Find the minimum positive difference between two composite numbers whose sum is 95. | nvidia |
A number is divisible by 9 if the sum of its digits is divisible by 9.
Find the largest digit $d$ such that the sum of the digits in the number $23d456$ is divisible by 9, and $d$ is the units digit of the sum of the other digits. | nvidia |
The product of two positive integers is 60, and their difference is 5. Find the sum of their squares. | nvidia |
Let $c$ be a real number randomly selected from the interval $[-12,12]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation $x^4+16c^2=(3c^2-8c)x^2$ has $\textit{exactly}$ one real solution. Find the value of $m+n$. | nvidia |
Find the greatest value of $b$ such that $-b^2+7b-10 \ge 0$. | nvidia |
Find all real values of $x$ that satisfy $x + \frac{12}{x-2} = -3.$ Enter all the solutions, separated by commas. | nvidia |
The graphs of two quadratic functions $f(x)$ and $g(x)$ intersect at four different points. Prove that if the functions intersect at five different points then they must be identical, otherwise find the condition on the coefficients so that the parabolas do not have any common point if they already do not intersect in ... | nvidia |
Let $a$ and $b$ be two positive integers such that the highest power of $11$ that divides $a!$ is equal to the highest power of $11$ that divides $b!$. What is the minimum value of $|a-b|$? | nvidia |
A set of $n$ points in a three-dimensional space has the following properties: the distance between each pair of points is at least $5$ units, and the sum of the squares of the distances between all pairs of points is equal to $n^2(n-1)$. Find the smallest possible value of $n$. | nvidia |
Given a sequence of matrices $A_1, A_2, \dots, A_n$ such that each $A_i$ represents a rotation by $10^\circ$ in the plane, find the smallest positive integer $m$ such that the product $A_1A_2 \dots A_m$ represents a full rotation by $360^\circ$. | nvidia |
Let $S$ be the set of integers $n > 1$ for which $\tfrac1n = 0.d_1d_2d_3d_4\ldots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$?
Write another problem inspired by this one:
Consider the set $T$ of rati... | nvidia |
Let $c^2=\frac{25}{36}$ and $d^2=\frac{(3+\sqrt{7})^2}{12}$, where $c$ is a negative real number and $d$ is a positive real number. If $(c+d)^4$ can be expressed in the simplified form $\frac{m\sqrt{n}}{p}$ where $m$, $n$, and $p$ are positive integers, what is the value of the sum $m+n+p$? | nvidia |
Find the smallest positive integer $m$ such that $m^3 - m^2$ is divisible by some but not all integer values of $j$ when $1\leq j \leq m$. | nvidia |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)-... | nvidia |
The three sides of a triangle are $a+1$, $b+3$, and $c+2$, where $a$, $b$, and $c$ are positive integers. The perimeter of the triangle is 23 and the sum of the squares of the lengths of the three sides is 207. What is the length of the side opposite the largest angle of the triangle? | nvidia |
How many positive three-digit integers less than 500 have at least two digits that are the same? | nvidia |
Let $A$ be the set of all real numbers except $0$, $1$ and $-1$. Find the value of $n \in A$ that maximizes the sum
\[S_n = \sum_{k = 0}^{n} \frac{1}{\binom{n}{k}} + \sum_{k = 0}^{n} \frac{k}{\binom{n}{k}}.\] | nvidia |
How many positive three-digit integers less than 500 have at least two digits that are the same? | nvidia |
A very large integer $n$ has the property that when the digits of $n$ are reversed, the result is a number that is divisible by $n$. If $n$ is in decimal form and has exactly 2023 digits, what is the smallest possible value of the sum of the digits of $n$? | nvidia |
Let $a$, $b$, $c$ be three positive integers such that $a-b$, $b-c$, $c-a$ are three distinct prime numbers. Determine all possible triplets $(a, b, c)$ such that
$$\begin{cases}
a^2 \equiv 1 \pmod{b} \\
b^2 \equiv 1 \pmod{c} \\
c^2 \equiv 1 \pmod{a} \\
\end{cases}$$ | nvidia |
Find the sum of all values of $x$ such that $x^3 = 64$. | nvidia |
Compute the difference of the squares of two consecutive integers whose product is 238. | nvidia |
Determine the set of all real numbers $x$ that satisfy the inequality $|x^2 - 4| < |x - 2| < 3$. | nvidia |
An ellipse can be inscribed inside a square so that it touches the square at four points: the center of each side of the square. If the square's side length is 1, what is the area of the ellipse that touches the square in the way described above? | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
How many four-digit positive integers have at least one digit that is a 2 or a 3? | nvidia |
How many positive integer divisors of $2205^{2205}$ are divisible by exactly 1001 positive integers? | nvidia |
Determine all values of $x$ that satisfy the equation $|x + 2| = \sqrt{3x + 5}$. | nvidia |
In the three-digit number $ABC$, the hundreds and tens digits are consecutive integers, while the ones digit is twice as large as the sum of the other two digits. Determine the largest possible value of the three-digit number such that when it is divided by the sum of its digits, the remainder is 1. | nvidia |
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