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Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^3 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite". | nvidia |
Find the remainder when the product of all positive integers from 1 to 25 that are not divisible by 3 is divided by 4. | nvidia |
What is the smallest positive integer that can be subtracted from 219 to yield a number that is a multiple of 5 and has exactly two distinct prime factors? | nvidia |
Let
$f(x)=ax^4+bx^3+cx^2+dx+e$
be a polynomial with integer coefficients such that
$f(x)=0$
for the values of $x$ equal to $1,-2,3,4$, and $-5$. Determine $f(0)$. | nvidia |
A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes? | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 2$. Find the set of all possible values of $\frac{1}{x} + \frac{1}{y}$. | nvidia |
In a quadrilateral, the perpendicular bisectors of the sides are tangent to a circle. Two opposite vertices of the quadrilateral are endpoints of a diameter of the circle, and the other two vertices are on the circle. Find the area of the quadrilateral in terms of the radius of the circle. | nvidia |
A bookshelf has two types of books: fiction and non-fiction. The number of fiction books, in boxes, is at least 8 boxes more than one-third the number of non-fiction books, and is no more than three times the number of non-fiction books. Find the least number of boxes of non-fiction books that can be placed on the book... | nvidia |
In the equation
$$5x^2-kx+1=0$$
determine $k$ such that the difference of the roots be equal to unity.
Write another problem inspired by this one:
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$.
To create a new problem inspired by this one, we can change th... | nvidia |
How many positive integers less than 100 are there such that their sum of digits is prime and none of the digits are zero? | nvidia |
Let $f(x)$ be a function defined on a subset of the real numbers, with the property that for every $x$ in its domain, the number $x - f(x)$ is also in its domain and $f(x - f(x)) = f(x).$ What is the largest set of real numbers that can be in the domain of $f$? | nvidia |
In a circle with radius 5 cm, two congruent squares are inscribed, such that their diagonals are diameters of the circle and intersect at the center of the circle. What is the side length of one of the squares? | nvidia |
Three natural numbers $a$, $b$, and $c$ are randomly chosen from the set $\{1,2,3,4,5,6,7,8,9\}$. What is the probability that the sum $a+b+c$ is divisible by $3$? | nvidia |
Find the sum of all even numbers between 20 and 50, inclusive, that are divisible by 3. | nvidia |
What is the maximum number of vertices of a convex pentagon that can be placed inside a circle with radius 5, such that each vertex of the pentagon lies on the circle and the lengths of all five sides of the pentagon are different integers? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) + y) = g(x) \cdot g(g(y) + g(-x)) + x^2\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the product of all possible values of $g(2).$ Find $m + t.$ | nvidia |
Find the equation of the line that passes through the points $(2,3)$ and $(4,5)$ and determine the product of its $x$-intercept and $y$-intercept. | nvidia |
Find the side length of the largest equilateral triangle that can fit inside a circle of radius 3. | nvidia |
Let f be a function such that for all positive integers x and y it holds that $f(xy)=f(x)+f(y)$. It is known that $f(2) = 2$ and $f(3)=3$. What is the value of $f(12)$? | nvidia |
A cube is made up of smaller unit cubes, with each edge consisting of $n$ of these unit cubes. Each of the six faces of the larger cube has a unique arrangement of the unit cubes, with some being black and the rest white. The number of white unit cubes on each face is equal to the sum of the number of white unit cubes ... | nvidia |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin^2 x + \cos^2 x = 1$ and $\sin x \cos x = 0.$ Enter all the solutions, separated by commas. | nvidia |
Find the smallest positive integer that is a multiple of 5 and has exactly 3 distinct prime factors. | nvidia |
Given a set of 10 distinct integers, find the number of ways to partition the set into two non-empty subsets such that the sum of the integers in each subset is equal. | nvidia |
A circle with center at $(2,3)$ and radius $4$ intersects the parabola $y=x^2$ at two points $A$ and $B$. Find the area of the portion of the circle enclosed by the arc $AB$ and the parabola $y=x^2$. | nvidia |
Find the smallest positive integer $n$ such that the sum of its divisors (excluding $n$ itself) is equal to the number of its divisors. | nvidia |
Find the sum of all constants $k$ such that the quadratic $x^2 - kx + 12$ can be factored in the form $(x+c)(x+d)$, where $c$ and $d$ are integers. | nvidia |
The sum of the areas of three congruent equilateral triangles is equal to the area of a square. The side length of the square is 12 cm. Find the length of the side of one of the equilateral triangles. | nvidia |
If $\sqrt{3x^2-2}=\sqrt{11}$, find the sum of all the possible values of $x$. | nvidia |
In the equation of the circle $(x-7)^2+(y+8)^2=65$, determine the coordinates of the point where the circle intersects with the line $x-y-4=0$ and is farthest from the center of the circle. | nvidia |
Given points $A=(0,0),$ $B=(2,1),$ $C=(4,3),$ and $D=(6,2),$ a triangle $ABC$ and a parallelogram $ACBD$ are created by connecting the points. A point $P=(p_1,p_2)$ in the coordinate plane can be moved to minimize the sum of the distances from $P$ to $A$, $P$ to $B$, and $P$ to $D$. Determine the coordinates of $P$ suc... | nvidia |
Let $x$ and $y$ be positive integers such that $x+y<10$. What is the sum of all possible integer values of $x-y$? | nvidia |
Find the positive real number $y$ such that $\lfloor y \rfloor \cdot y = 42$. Express $y$ as a decimal. | nvidia |
Let $Q$ be the product of the first $200$ positive even integers. Find the largest integer $m$ such that $Q$ is divisible by $2^m$. | nvidia |
Let $P$ be the product of the first $100$ positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k .$
| nvidia |
Given a positive integer $n$, find the smallest possible value of
\[\left\lceil{\frac{a_1+a_2}{a_3}}\right\rceil+\left\lceil{\frac{a_2+a_3}{a_4}}\right\rceil+\cdots+\left\lceil{\frac{a_{n-1}+a_n}{a_1}}\right\rceil,\]
where $a_1,a_2,\ldots,a_n$ are positive real numbers, and $\lceil{x}\rceil$ denotes the smallest intege... | 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 |
Find the smallest positive integer $n$ such that $12n$ and $20n$ have a common factor greater than $n$. | nvidia |
Let $$P(x)=2x^4+ax^3+bx^2+cx+d$$ be a monic quartic polynomial with real coefficients such that
$$P(1)=P(2)=P(3)=P(4)=P(5)=0.$$ Determine $a+b+c+d$. | nvidia |
An integer can be expressed as a sum of two distinct squares of integers in exactly three ways: $117=6^2+9^2$, $117=8^2+7^2$, and $117=4^2+11^2$. How many integers can be expressed as a sum of two distinct squares of integers in exactly two ways? | nvidia |
Determine the number of positive integers $b$ less than $18$ such that the congruence $bx\equiv 1\pmod{18}$ has a solution in $x$. | nvidia |
Let $g$ be a function from the set of positive integers to the set of positive integers, such that $g(a) \times g(b) = g(a+b) \times g(a \times b)$ for all positive integers $a$ and $b$. If $g(1)=1$, find the number of distinct possible values of $g(5)$. | nvidia |
The city has a grid of streets that form a 6x6 square. The city needs to place streetlights on some of the intersections of the streets so that at least one streetlight is visible from every intersection. How many different ways can the city place the streetlights if at least one streetlight is on each row and at least... | nvidia |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $x^2+y^2\le 25$? | nvidia |
If $m$ and $n$ are positive integers with $m+n=40$, what is the smallest possible value of $\frac{m}{n}+\frac{n}{m}$? | nvidia |
Find the value of the sum
\[\sum_{k=0}^{100} \frac{\sin((2k+1)\pi/101)}{2k+1}.\] | nvidia |
Find the smallest positive integer n such that it leaves a remainder of 1 when divided by each of the following integers: 2, 3, 4, 5, 6, 7, 8, and 9, and also satisfies the condition that the sum of its digits is a prime number. | nvidia |
Determine the value of $a$ such that the sum of the squares of the roots of the quadratic equation $ax^2 + 2x + 1 = 0$ is equal to 10. | nvidia |
Let ABC be a triangle with $\angle A=60^\circ.$ Find all points P in the plane such that
\[\frac{|PA|}{1}=\frac{|PB|}{2}=\frac{|PC|}{3}.\] For each point found, draw segments from the point to A, B and C, and label their intersections with the sides of the triangle, if any, with X, Y and Z, respectively. | nvidia |
Count the number of positive integers not exceeding 100 that are either a perfect square or a prime number, but not both. | nvidia |
Find the product of all the solutions of the equation $$\frac{5x^2-1}{3x^2+1}=\frac{3}{x^2}$$ | nvidia |
If the inverse tangent of the product of two positive numbers is equal to the sum of their inverse tangents, find the ratio of these two numbers. | nvidia |
For how many positive integral values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$
\begin{cases}
2x>3x-3\\
3x-a>-6
\end{cases}
$$ | nvidia |
The lines with the given equations are perpendicular. Find the value of $a$ so that the lines with the given equations are perpendicular.
Find the value of $a$ so that the lines with the given equations are perpendicular.
\begin{align*}
y &= 2x+13 \\
6y+ax &= 6.
\end{align*}
The lines with the given equations are pe... | nvidia |
Let $g(n)$ return the number of distinct triplets of positive integers $(a, b, c)$ such that for each triplet, $a^3 + b^3 + c^3 = n$. Find the smallest positive integer $n$ for which $g(n) = 5$. | nvidia |
A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit.
What is the value of $a+b+2c$?
[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=-... | nvidia |
Determine $a$ such that the difference of the roots of the equation
$$5x^2-kx+1=0$$
is equal to $2a$ and the product of the roots is equal to $5a$. | nvidia |
Find the largest integer that can be subtracted from 945 to result in a number that is divisible by 7. | nvidia |
A certain graph consists of $n$ vertices, and for every pair of vertices, there is an edge between them with probability $\frac{1}{3}$, independently of other pairs. The graph is said to be "triangle-friendly" if it contains at least one triangle (a set of three vertices connected by three edges). Find the smallest val... | nvidia |
Given that $a$, $b$, and $c$ are positive integers, find the smallest possible value of $c$ such that there exist solutions to the system of equations
$$x \equiv y \pmod a,$$
$$2x \equiv z \pmod b,$$
$$3y \equiv c \pmod {ab},$$
subject to the constraint that $x$, $y$, and $z$ are integers. | nvidia |
Find the smallest positive integer that can be expressed as the sum of two distinct perfect squares, each of which is the square of a positive integer. | nvidia |
Let $n$ be the smallest positive integer with 2015 digits that is divisible by 15. What is the largest positive integer $m$ such that $n^2 + m$ is divisible by 20? | nvidia |
A 57-gon Q1 is drawn in the Cartesian plane. The sum of the y-coordinates of the 57 vertices equals 171. The midpoints of the sides of Q1 form a second 57-gon, Q2. Finally, the midpoints of the sides of Q2 form a third 57-gon, Q3. Find the sum of the y-coordinates of the vertices of Q3. | nvidia |
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | nvidia |
What is the number of positive integer divisors of 540 that are less than or equal to its square root? | nvidia |
A number $x$ is such that $x^2=2^m\cdot3^n$, where $m$ and $n$ are positive integers. If the difference between $m$ and $n$ is 3, find the value of $x$ when $m+n=7$. | nvidia |
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.Find the sum of all integers $n$ such that $\dfrac{12}{n}$ is also an integer.
| nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 3.$ Find the minimum value of
\[\frac{4}{x} + \frac{9}{y} + \frac{16}{z}.\] | nvidia |
Let g(n) be the product of all positive divisors of n. Find the smallest integer k such that for every n > k, g(n) is a composite number. | nvidia |
Consider the polynomials \[f(x)=1-12x+3x^2-4x^3+5x^4\] and \[g(x)=3-2x-6x^3+9x^4.\] Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 3. | nvidia |
In an equilateral triangle of side length $8$ cm, a small equilateral triangle is formed by joining the midpoints of the sides. When this small triangle is folded along one of its sides so that the vertex opposite the fold touches the midpoint of the side of the larger triangle, the area of the folded triangle that ove... | nvidia |
Let $x,y,z$ be positive real numbers such that $x+y+z=12$ and $xy+yz+zx=36$. Let $n=\max\{xy,yz,zx\}$. Find the smallest possible value of $n$. | nvidia |
Find the sum of all the entries in the first ten rows of Pascal's Triangle that are not the first or last entry in their row, and that are divisible by 3. | nvidia |
Find the number of pairs of non-zero vectors on the three-dimensional lattice that are perpendicular to each other and have a magnitude of 2. | nvidia |
The sum of the squares of four consecutive positive integers is 1310. What is the smallest of the four integers? | nvidia |
Given a $3 \times 3$ matrix $\mathbf{A}$ with entries $a_{ij}$, where $a_{ii} = x$ for all $i$ and $a_{ij} = 1$ for all $i \neq j$, find the value of $x$ such that $\det(\mathbf{A}) = 0$. | nvidia |
Find the sum of all integers between 300 and 500 that are multiples of 5. | nvidia |
The vertices of an equilateral triangle are located on the circle with radius $\frac{16}{\sqrt{3}}$ cm, such that a square inscribed inside the triangle has the area of 256 cm^2. What is the area of a regular hexagon inscribed inside the same circle? | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x^3 + yg(z)) = x^2g(x) + z^2g(y)\]for all real numbers $x,$ $y,$ and $z.$
Let $m$ be the number of possible values of $g(7),$ and let $t$ be the product of all possible values of $g(7).$ Find $m + t.$ | nvidia |
A sequence of three numbers has the property that the sum of any two of the numbers is a square number. If the sum of the three numbers is 20, what is the largest possible value of the product of the three numbers? | nvidia |
The function $f$ has the property that $f(x+y) = f(x) + f(y)$ for all real $x$ and $y$, and $f(1) = 3$. Find the value of $f(2)$. | nvidia |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x + \cos x = \sqrt{2}.$ Enter all the solutions, separated by commas. | nvidia |
Let $g$ be a function taking the positive integers to the positive integers, such that
\[3g(abc) = g(a)g(b)g(c)\]for all positive integers $a,$ $b,$ and $c.$
Let $m$ be the number of possible values of $g(100),$ and let $t$ be the sum of the possible values of $g(100).$ Find $m \times t.$ | nvidia |
Let $m$ be the smallest positive integer that is divisible by 16 and is a perfect cube, and $n$ be the smallest positive integer that is divisible by 16 and is a perfect square. Cameron writes down all the positive integers less than $10^4$ that are divisible by 16 and have exactly three distinct positive divisors. How... | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(0) = 0$ and
\[g(x^2 + y^2) = x^2 g\left(\frac{1}{x}\right) + y^2 g\left(\frac{1}{y}\right)\]for all non-zero real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $m \ti... | nvidia |
Determine the smallest positive integer $n$ such that the sum of the squares of the first $n$ positive multiples of 13 exceeds 10000. | nvidia |
In a city, there are two parks located at points A(2,3) and B(6,8). A new park is to be built such that the sum of the distances from the new park to A and B is minimized. Where should the new park be built? | nvidia |
If $a$, $b$, and $c$ are three positive integers such that the average value of any two sides of the triangle they define when used as side lengths form a geometric sequence, what is the smallest possible sum $a+b+c$? | nvidia |
Let $a$, $b$, $c$, $d$ be real numbers such that $(a+bi)(c+di)=7+i$. Find the smallest possible value of $a^2+b^2$. | nvidia |
The difference of squares of two consecutive natural numbers is 165. Find the sum of these numbers. | nvidia |
Eight more than twice the number of parakeets minus the difference between the number of parakeets and the number of canaries is equal to 120. If the number of parakeets is 25, how many canaries are there? | nvidia |
A hyperbola and a circle share a center, and the transverse axis of the hyperbola is the line containing the diameter of the circle. The hyperbola and circle intersect at four points. Given that the equation of the circle is $(x-2)^2 + y^2 = 9,$ find the area of the quadrilateral formed by those four points. | nvidia |
Find the number of integers $n$ that satisfy
\[10 < n^3 < 99.\] | nvidia |
Let
\[\begin{array}{lcl}
a_{11}&=&1\\
a_{21}&=&2\\
a_{22}&=&3\\
a_{31}&=&4\\
a_{32}&=&5\\
a_{33}&=&6\\
&\vdots&\\
a_{nn}&=&\frac{n(n+1)}{2}.
\end{array}\]Let $f(n)$ be the value of the sum
\[a_{11}-a_{21}-a_{22}+a_{31}+a_{32}+a_{33}-a_{41}-a_{42}-a_{43}-a_{44}+\dots+a_{nn},\]where the signs change after each perfect sq... | nvidia |
Let $f(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} + \cdots + \frac{1}{x-n}$. Find the largest value of $n$ such that the equation $f(x) = x$ has at least one real solution. | nvidia |
Let $S$ denote the set of all numbers of the form $a_1+a_2+\cdots+a_{12}$ where $a_1, a_2, \ldots, a_{12}$ are distinct integers such that $1 \le a_i \le 18$ for all integers $i$ satisfying $1 \le i \le 12.$ Find the greatest common divisor of all members of $S$. | nvidia |
Find the smallest positive integer $m$ such that $2^m \equiv 1 \pmod{17}$. | nvidia |
Determine the sum of all integer divisors of the number $2^{3} \cdot 3^{4} \cdot 5^{2}$. | nvidia |
Let $f(x) = \frac{1}{x} + \frac{1}{x-1} + \frac{1}{x-2} + \cdots + \frac{1}{x-n}$. Find the largest value of $n$ such that the equation $f(x) = x$ has at least one real solution. | nvidia |
Given a 3x3 matrix $\mathbf{B}$ with integer entries, where each row and column sums to 0, find the maximum possible value of the sum of all the entries in $\mathbf{B}^2$. | nvidia |
Determine the product of the roots of the equation $x^3 - 4x^2 + 7x - 12 = 0$, given that the sum of the squares of the roots is equal to 30. | nvidia |
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