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The perimeter of the largest regular hexagon that can be inscribed in a circle of radius 5 is what? | nvidia |
How many positive integers less than $1000$ can be expressed as the sum of three distinct positive perfect squares? | nvidia |
Find a positive integer $n$ such that $n^3 + 2n^2 + 3n + 1$ can be expressed as the sum of four distinct perfect cubes. | nvidia |
Given 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 $a$. | nvidia |
In a certain type of pyramid, each row of blocks has one more block than the row above it. Each block in the pyramid is labeled with a positive integer that is the sum of the labels on the two blocks below it, except for the bottom row where all the blocks have a label of 1. On which row of the pyramid will there be a ... | nvidia |
Find all real solutions to the equation:
$$\sqrt{x} + \sqrt{x + 1} = \frac{7}{\sqrt{2}}$$
where $x$ is a positive integer. | nvidia |
Let $ T $ be the set of all edges and diagonals of a regular hexagon. A pair of elements of $ T $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | nvidia |
Find the sum of all positive integers less than 500 that are cubes of perfect cubes, excluding those that are also perfect squares. | nvidia |
Given that $\gcd(75, 360) = 15$, find the smallest positive integer $n$ such that $\gcd(360n, 75n+120) = 15$. | nvidia |
The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?
| nvidia |
A number is written with the digits 0, 1, 2, 4, 5 and 7. Each digit can be used only once.
Is it possible to form three two-digit numbers such that the square of the biggest exceeds the sum of the squares of the other two numbers by 50?
If yes, find the numbers, if no, justify the answer. | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $\text{lcm}(a,b,c)=\text{lcm}(a,b)\cdot\text{lcm}(b,c)$. Find the smallest possible value of $\text{gcd}(a,b,c)$. | nvidia |
Let $x$ be a positive real number such that $\sqrt{x} + \frac{1}{\sqrt{x}} = 10.$ Find $x^2 + \frac{1}{x^2}$. | nvidia |
Find the smallest positive real number $c,$ such that for all nonnegative real numbers $x$ and $y,$
\[\sqrt{xy} + c |x - y| \ge \frac{x + y}{2}.\] | nvidia |
Find the smallest positive integer $m$ such that $5m$ is a perfect cube and $7m$ is a perfect square. | nvidia |
Given that $a,$ $b,$ $c,$ and $d$ are real numbers such that
$\mathbf{A}=\begin{pmatrix}a&b\\c&d\end{pmatrix},$
$\mathbf{B}=\begin{pmatrix}-3&1\\1&-3\end{pmatrix},$
and $\mathbf{AB}=\mathbf{BA},$ determine the largest possible value of
$\mathbf{det(A)}.$ | nvidia |
Find the smallest possible value of the product of five consecutive composite numbers that is divisible by 4. | nvidia |
Let $x$ be a real number such that
$2\cdot e^x+e^{-x}=5$.
Determine $x$. | nvidia |
Find the largest value of $m$ such that $3x^2+mx+70$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
How many three-digit numbers have the property that the sum of their digits is equal to the number of divisors they have? | nvidia |
There are $30$ different complex numbers $z$ such that $z^{30}=1$. For how many of these is $z^{15}$ a real number? | nvidia |
Find the sum of all $x$ that satisfy the equation $\frac{x^2 + x}{x(x^2-1)} = \frac{1}{x+1} - \frac{2}{x}.$ | nvidia |
Given a sequence of positive real numbers $(a_n)$ such that $\cos a_n + \cos (a_n + 1) + \cos (a_n + 2) = 0$ for all $n$, find the sum of all possible values of $\sin 3a_n + \sin 3(a_n + 1) + \sin 3(a_n + 2)$. | nvidia |
Given that the product of two consecutive integers is 30 more than their sum, find the smaller integer. | nvidia |
A positive integer $n$ has only two prime factors, namely $p$ and $q$, and the product of its positive divisors is $864$. What is the number of distinct positive divisors of $n$? | nvidia |
Find the sum of the lengths of the two longest altitudes in a triangle with sides $9$, $20$, and $22$. | nvidia |
Find the smallest positive integer $x$ such that $5^x+7^x$ is divisible by 64. | nvidia |
Consider the functions $g$ and $h$ defined on the set of positive integers, satisfying the conditions:
$$g(mn)=mg(n), \ g(1)=10, \ g(x)=h(x)$$for every positive integer $m, n, x$. Find $h(3)$. | nvidia |
Find the minimum value of
\[\sin^4 x + \frac{3}{2} \cos^4 x,\]as $x$ varies over all real numbers. | nvidia |
Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$. | nvidia |
The roots of the equation $x^3 + ax^2 + bx + c = 0$ are in geometric progression. Determine the value of $a$ given that the product of the roots is equal to $27$ and the sum of the roots is equal to $-6$. | nvidia |
Let $n$ be a positive integer such that
$$(n-1)!(n+1)^2=n!(n^2-n+1).$$
Determine $n$. | nvidia |
Determine $b$ such that the product of the numbers $32_{b}$ and $4B_{b}$ is equal to $1542_{b}$, where $B_{b}=11_{10}$. | nvidia |
Given that 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 exterior angles of a concave polygon with $n+2$ sides? | nvidia |
Let's determine all possible values of the expression
\[\sqrt {\frac {x+y}{x+y-1}},\]
provided that
\[\sqrt {x-1}+\sqrt {y-1}=1.\] | nvidia |
Let $A$ be the set of integers that can be expressed as a sum of two squares of rational numbers. What is the number of square-free positive integers that divide 24 and are in the set $A$? | nvidia |
When three positive integers are divided by $47$, the remainders are $25$, $20$, and $3$, respectively.
When the sum of the three integers is divided by $47$, what is the remainder? | nvidia |
Determine the sum of the coefficients of the polynomial
$$(x^2-4x+2)(x^2+5x+1).$$ | nvidia |
Let $T$ be the triangle formed by the centers of three spheres that are externally tangent to each other. Prove that $\frac{3V}{S^2} = \frac{\pi}{12} \sqrt{\frac{r_1r_2r_3}{(r_1+r_2)(r_2+r_3)(r_3+r_1)}}$, where $r_1,r_2,r_3$ are the radii of the three spheres, $S$ is the area of triangle $T$, and $V$ is the volume of t... | nvidia |
Find the sum of the squares of the roots of $x^2-7x+1=0$. | nvidia |
What is the smallest positive number that is a perfect square and $10$ more than a number that can be expressed as the sum of two distinct prime numbers? | nvidia |
A certain right triangle has the property that when its two legs are used as the diameters of two semicircles, the area of the larger semicircle is three times the area of the smaller semicircle. If the length of the shorter leg of the triangle is 3 cm, what is the length of the hypotenuse? | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a + b + c = 27.$ Find the largest possible value of $b.$ | nvidia |
The equation $x^2-kx-12=0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? | nvidia |
Let the differentiable function $f(x)$ satisfy $f'(x)=2x+1$ for $x$ on the real number line. Determine the value of $x$ so that the tangent line to the graph of $y=f^{-1}(x)$ at point $(x,f^{-1}(x))$ has slope $\frac{1}{5}$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 2.$ Find the maximum value of
\[\frac{x}{\sqrt{1+x^2}} + \frac{y}{\sqrt{1+y^2}}.\] | nvidia |
Find the sum of all real solutions to the equation \[\frac{x-2}{x^2+4x+1} = \frac{x-5}{x^2-10x}.\] | nvidia |
The positive integer $n$ has 24 positive factors. If $4n^2$ has 45 positive factors, what is the number of positive factors of $6n^2$? | nvidia |
A circle with center at $(2, 3)$ and radius $4$ intersects the line $y = 2x - 1$ at two points. Find the coordinates of the point of intersection that is farther away from the origin. | nvidia |
Find the smallest positive integer $n$ such that $3n$ is a multiple of 7 and $5n$ is a multiple of 11. | nvidia |
Find all integer values of $x$ such that $x^2+2x+2$ is greater than 10 and less than 20. | nvidia |
Let set $\mathcal{B}$ be a 60-element subset of $\{1,2,3,\ldots,80\},$ and let $T$ be the sum of the elements of $\mathcal{B}.$ Find the number of possible values of $T$. | nvidia |
Find the sum of all the positive integer divisors of 72 that are also perfect squares. | nvidia |
An integer $m$ is said to be a "divisor" of another integer $n$ if there exists an integer $k$ such that $n=km$. Given an integer $n$, let $d(n)$ denote the number of positive divisors of $n$. Prove or disprove: $d(n^2)=2d(n)+1$ for every positive integer $n$. | nvidia |
The number 432 can be represented as a product of three numbers $p$, $q$, and $r$, where $p$, $q$, and $r$ are integers greater than 1. Determine $p$ such that $q$ and $r$ are primes and the product of $q$ and $r$ is minimized. | nvidia |
Let $P(z)$ be a polynomial of degree $n$ with complex coefficients such that $P(z)=P(-z)$ for all complex numbers $z$. Suppose that $P(z)$ has $n$ distinct roots. What is the maximum number of roots of $P(z)$ that can be real? | nvidia |
Someone observed that $7! = 7 \cdot 8 \cdot 9 \cdot 10 \cdot 11 \cdot 12$. Find the largest positive integer $m$ for which $m!$ can be expressed as the product of $m - 4$ consecutive positive integers. | nvidia |
How many positive integers $p$ less than 200 have a corresponding integer $q$ divisible by 4 such that the roots of $x^2-px+q=0$ are consecutive even positive integers? | nvidia |
Let $g(x)$ be a function defined on the set of integers, such that
$$g(g(x))=x.$$Given that $g(99)=12$, find $g(12)$. | nvidia |
Determine the number of positive integers less than or equal to 30 that are not divisible by 2 or 3. | nvidia |
Suppose that $a$, $b$, and $c$ are positive integers such that $a^2$, $b^2$, and $c^2$ are the three terms (in some order) of a geometric progression. If $a+b+c=2000$, how many possible values of $a$ are there? | nvidia |
A cube has a volume that is numerically equal to the sum of the lengths of all its edges. Determine the length of one edge of the cube. | nvidia |
Let $a$ and $b$ be positive real numbers such that
\[a^2 + b^2 = 8ab.\]Find the minimum value of $a^3 + b^3$. | nvidia |
Find the sum of all positive integral values of $m$ such that the equation $m^2+km+\ell=0$ has integral solutions for some prime number $\ell$, where $k$ is a fixed positive integer.
Note: In this problem, we are given a quadratic equation with a fixed coefficient $k$ and a prime number $\ell$. We need to find the sum... | nvidia |
In a four-dimensional space, a hypersphere is tangent to the hyperplane $x_1 + x_2 + x_3 + x_4 = 0$ at the point $(a, b, c, d)$, where $a, b, c,$ and $d$ are positive integers. The equation of the hypersphere can be written in the form $(x_1-a)^2 + (x_2-b)^2 + (x_3-c)^2 + (x_4-d)^2 = r^2$, where $r$ is a positive real ... | nvidia |
What is the sum of all positive integer divisors of 91? | nvidia |
What is the largest positive integer $n$ such that $\sqrt{n}$ leaves a remainder of $1$ when divided by each of the prime numbers $3,5,7$ and $11$? | nvidia |
Two integers, $m$ and $n$, are randomly selected from the set of positive integers less than 100, with replacement. If the probability that the sum of the two integers is even is $\frac{3}{5}$, what is the probability that the sum of the two integers is a multiple of 3? | nvidia |
In the figure, three circles are mutually tangent and are circumscribed by a fourth circle. The two smaller circles have radii of 2 and 3 units, respectively, and the larger circle has a radius of 5 units. Find the radius of the fourth circle such that the area of the shaded region is equal to the sum of the areas of t... | nvidia |
In a 2 by 2 grid, the diagonals are drawn. Let a point $P$ be on one of the diagonals such that the circles inscribed in the two triangles formed by $P$ and the adjacent sides have the same radius. If the sides of the grid have length 5, find the length of the diagonal that doesn't contain point $P$. | nvidia |
Consider the quadratic equation $x^2 + bx + c = 0$, where $b$ and $c$ are integers. What is the sum of all possible integer values of $b$ such that the roots of the equation are rational numbers (not necessarily integers), and the sum of the squares of the roots is an integer? | nvidia |
What is the sum of all the positive integers between 1 and 100 that are not divisible by 2, 3, or 5? | nvidia |
Let \(x, y,\) and \(z\) be three positive real numbers such that \(x^2+y^2+z^2=3.\) Find the maximum value of the expression
\[\frac{x^2y^2+y^2z^2+z^2x^2}{xyz}.\] | nvidia |
Determine the number of distinct $x$-intercepts of the graph of the equation $y = (x-2)(x^2+3x+2)$. | nvidia |
Let $n$ be the number of positive integers less than 25 that are either a multiple of 3 or a multiple of 5, and let $m$ be the number of positive integers less than 25 that are both a multiple of 3 and a multiple of 5. Compute $(n - m)^2$. | nvidia |
What is the sum of the $x$-values that satisfy the equation $5=\frac{x^3-2x^2-8x}{x+2}$? | nvidia |
Find the largest real number $\lambda$ such that
\[\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \ge \lambda \left(\frac{a}{c} + \frac{c}{a}\right)\]for all positive real numbers $a,$ $b,$ $c,$ $d.$ | nvidia |
Consider two real numbers $x$ and $y$ such that $-3 < x - y < 7$. If $\frac{x+y}{y}$ is an integer, determine its value. | nvidia |
A set of four points in the plane has the property that the area of the triangle formed by any three of the points is always a positive integer. If three of the points are $(0,0), (1,0), (0,1)$, what is the smallest possible distance between the fourth point and the origin? | nvidia |
There are four positive integers that are divisors of each number in the list $$36, 72, -12, 114, 96.$$Find the sum of these four positive integers. | nvidia |
Three-fifths of all the birds in a park are sparrows and the rest are pigeons. If there are 240 birds in the park, what's the total number of legs of pigeons that are in the park? | nvidia |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | nvidia |
Determine the smallest whole number greater than 1 that has exactly 5 positive factors. | nvidia |
Find all values of $a$ such that the inequality
$$\frac{1}{x^2 - 4x + 5} < a$$
holds for all real numbers $x$. | nvidia |
Determine the sum of all possible values of $x$ such that the number ${x2{,}74x}$ is a multiple of 11. | nvidia |
Determine the number of vertices of a polyhedron, knowing that each face is a triangle and that every edge is shared by exactly two faces, if the total number of edges is 24. | nvidia |
The area of a square inscribed in a circle of radius $r$ is equal to $\frac{2r^2}{\pi}$.
Determine the ratio of the area of the square inscribed in a circle of radius $2r$ to the area of the square inscribed in a circle of radius $r$. | nvidia |
Find the smallest positive integer $m$ such that
$$m^3 \equiv 12 \pmod{15}.$$ | nvidia |
The sum of the digits of a three-digit number is 16, and the sum of the squares of its digits is 124. Find the number. | nvidia |
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| nvidia |
Find the minimum value of $a^2+b^2+c^2$ given that $a,b,c$ are positive real numbers satisfying $ab+ac+bc=1$. | nvidia |
A circle centered at (6,0) and a radius of $\sqrt{5}$ is inscribed inside a parabola $y^2=4x$. Find the largest possible area of a rectangle inscribed in the circle with one vertex at the point of tangency of the parabola and the circle. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(f(x) + y) = x + f(y)\]for all real numbers $x$ and $y.$ | nvidia |
Find the number of solutions to the equation $x^2 + y^2 = 100$ where $x$ and $y$ are integers and $x > 0$. | nvidia |
Find the smallest positive integer that is divisible by each of the first eight positive integers, and also has a remainder of 1 when divided by 9. | 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 |
If $\sqrt{3x^2+4}=\sqrt{28}$, find the sum of all the possible values of $x.$ | nvidia |
Compute the largest integer $k$ such that $2004^k$ divides $2004!$.
| nvidia |
A positive integer $m$ has the property that the product of its digits is equal to the square of their sum.
Determine the minimum value of $m$ such that the number of its divisors is a perfect cube. | nvidia |
Find the number of positive integers less than 1000 that have a remainder of 5 when divided by 11. | nvidia |
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