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Find the positive number $x$ such that the geometric mean of $1+x, 2+x, 3+x, \dots, 98+x, 99+x$ is $10x$. | nvidia |
Given that $\mathbf{P}$ is a $2 \times 2$ matrix such that $\mathbf{P}^3 = \mathbf{I}$ and the sum of the squares of its elements is 7, find the largest possible value of the determinant of $\mathbf{P}$. | nvidia |
How many 5-digit numbers can be formed using the digits 0-9, with the condition that the digits in the units and tens places are even, and the digits in the hundreds and thousands places are prime numbers? | nvidia |
The smallest integer greater than 1 that has exactly 12 divisors can be written in the form $p^a \cdot q^b$, where $p$ and $q$ are prime numbers and $a$ and $b$ are non-negative integers. Find $p + q$. | nvidia |
Let $p(x) = ax^3+bx^2+cx+d$ be a cubic polynomial with integer coefficients. Find the number of positive integers $n \le 1000$ for which there exists a real number $x$ such that $n = p(x) + p(-x)$. | 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 |
Determine the maximum value of
\[x^3 + y^3 + z^3\]over all real numbers $x,$ $y,$ and $z$ such that
\[x^2 + y^2 + z^2 = 1.\] | nvidia |
Let p be a prime number such that p>3. For how many positive integer divisors of 2p^2 does 1/n yield a terminating decimal that has a nonzero digit at the ten-thousands place? | nvidia |
The rooms in a hotel are numbered consecutively from 1 to 1000. Three rooms are selected at random, and it is found that the sum of their numbers is 2044. What is the number of the third room if the number of the first room is one more than the number of the third and the number of the second room is 18 more than the... | nvidia |
Find the value of $x$ such that $\sqrt[5]{16^x}=2\sqrt[3]{8^x}$ | nvidia |
The graph of the equation $x^2+y^2-2x+4y=0$ intersects the graph of the line $y=mx$ at two distinct points $A$ and $B$. Determine $m$ such that the line $y=mx$ is tangent to the circle defined by the equation $x^2+y^2-2x+4y=0$. | nvidia |
The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ | nvidia |
Let b be a real number randomly selected from the interval $[-17,17]$. Then, m and n are two relatively prime positive integers such that m/n is the probability that the equation $x^4+25b^2=(4b^2-10b)x^2$ has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$.
| nvidia |
Determine the values of $x$ for which the expression $\frac{3x^2-5x+2}{x^2+ax+b}$ is undefined, where $a$ and $b$ are constants to be determined, given that the sum of the roots of the denominator is $-2$ and the product of the roots is $-3$. | nvidia |
Let $G$ be the set of all positive integers. The function $g:G \to G$ satisfies the following two properties:
(i) First,
$g(m) + g(n) = g(\mathrm{lcm}(m,n))$
for all $m,n \in G$ such that $\gcd(m,n)=1$.
(ii) Second,
$g(mn)=g(m) + g(n) - 1$
for all $m,n \in G$.
Let $p$ be the number of possible values of $g(2)$, ... | nvidia |
If $0.8= \frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers.
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 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 4.$ Find the maximum value of
\[\frac{x^2}{2} + \frac{y^2}{3} + \frac{z^2}{4}.\] | nvidia |
What is the smallest positive integer $n$ such that $531n \equiv 1067n \pmod{24}?$ | nvidia |
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$? | nvidia |
In the complex plane, the graph of the equation $|z-2|+|z-6|=10$ forms an ellipse. Find the area of the rectangle whose vertices are the foci of this ellipse and the endpoints of the major axis. | nvidia |
Find the number of distinct subsets of the set $\{1, 2, 3, \ldots, 25\}$ whose sum is a perfect square. | nvidia |
Find the smallest positive integer $n$ such that when $n$ is divided by $5$, the remainder is $2$, and when $n$ is divided by $11$, the remainder is $8$. | nvidia |
Find the sum of the squares of the roots of the polynomial $x^5 + 3x^4 - 7x^3 + 2x^2 + 4x - 6 = 0$, given that the sum of the roots of the polynomial is equal to -3. | nvidia |
Find the smallest positive integer such that the sum of its proper divisors (excluding the number itself) is equal to the number multiplied by 3/2. | nvidia |
The integers $a$, $b$, and $c$ are the roots of the polynomial
$$x^3-10x^2+23x-14=0$$
and satisfy $0<a<b<c<15$. Determine all possible ordered triples $(a,b,c)$. | nvidia |
Find the number of positive integers less than or equal to 100 that are either perfect squares or multiples of 5. | nvidia |
Choose the smallest of the following products, and express it as a decimal number:
$$\sqrt{2}\cdot\sqrt{3}, \ \ \sqrt{2}\cdot\sqrt{5}, \ \ \sqrt{2}\cdot\sqrt{7}, \ \ \sqrt{2}\cdot\sqrt{11}, \ \ \sqrt{2}\cdot\sqrt{13}$$ | nvidia |
$\triangle ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$\textbf{(A)}\ s^2\le8r^2\qquad \textbf{(B)}\ s^2=8r^2 \qquad \textbf{(C)}\ s^2 \ge 8r^2 \qquad\\ ... | nvidia |
The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number? | nvidia |
The volume of a cube with edge length $x$ inches is twice the volume of a cube with edge length $y$ inches. Find the edge length of the larger cube in the form $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ has no perfect cube factors. Compute $c+d$. | nvidia |
Determine all real numbers $a$, $b$, $c$ such that the minimum value of the expression
$$ax^2+by^2+cxy+a^2x+b^2y+c^2z$$ over all real numbers $x$, $y$, $z$ satisfying $x+y+z=0$ is equal to $0$. | nvidia |
Find the number of distinct positive integers $n$ such that $n^2+2n+1$ can be expressed as a sum of two perfect squares in exactly three different ways. | nvidia |
Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$, find $z^{2000}+\frac 1{z^{2000}}$. | nvidia |
Let $a$, $b$, $c$ be real numbers such that
$a+b+c=0$ and $a^2+b^2+c^2=2$. Determine $a^4+b^4+c^4$ | nvidia |
Let $x$ and $y$ be positive integers such that $xy = 1337$.
Find the minimum value of $x + y$. | nvidia |
Let $x$ be a real number such that $x^{1992} + x^{1991} + \ldots + x + 1 = 1993^{1992}$. Find the largest integer that does not exceed $x$. | nvidia |
In the expression $$5^{\frac{1}{2n}}+5^{\frac{3}{2n}}$$ determine n such that the given expression be equal to $$2\sqrt {15}$$. | nvidia |
The sum of the digits of the number $1111^n$ is $4$ for some positive integer $n$. Find the smallest such $n$. | nvidia |
There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$. | nvidia |
Let $p(x)=x^4+7x^3-11x^2-18x+16$ be a polynomial function such that $p\left(\cos\theta\right)=\cos(2\theta)$ for all $\theta$. Find the least positive integer $k$ such that $\cos(2\theta) = 2\cos(k\theta)$. | nvidia |
Determine the sum of all prime numbers $p$ for which there exists no integer solution in $x$ to the congruence $3(6x+1)\equiv 4\pmod p$.
New Problem:
Find the smallest positive integer $n$ such that there exists a polynomial $f(x)$ with integer coefficients, satisfying the condition that $f(x) \equiv 0 \pmod{x^2-1}$ ... | nvidia |
Find the smallest positive integer that can be expressed as the sum of three distinct perfect squares in at least two different ways. | nvidia |
Let $a$ and $b$ be two 3-digit numbers such that $a+b=999$.
Determine the largest possible value of the product $a\cdot b$. | nvidia |
What is the sum of all real numbers that are tripled when multiplied by their reciprocals? | nvidia |
Three different positive integers are chosen. The sum of the reciprocals of any two of the integers is always greater than the reciprocal of the third. How many such triples of positive integers are there whose sum is 14? | nvidia |
Count the number of positive integers less than $10^6$ that can be expressed as the difference of two perfect squares, where one square is a multiple of 24. | nvidia |
In a triangle $ABC$, the point $D$ is on the side $BC$ so that $BD:DC=1:2$. Given that $\overrightarrow{AB} = \mathbf{u}$, $\overrightarrow{AC} = \mathbf{v}$, and the area of the triangle $ABC$ is $9$, find $\|\mathbf{u} \times \mathbf{v}\|$. | nvidia |
The complex numbers $x+\sqrt{y}$ and $x-\sqrt{y}$ are the roots of the quadratic equation $ax^2+bx+c=0$. Find $a+b+c$ if the sum of the squares of the roots is equal to 12. | nvidia |
A sequence of real numbers $b_1, b_2, b_3, \ldots$ is chosen so that $b_n = 2b_{n - 1} - 3b_{n - 2}$ for each $n \ge 3.$ The sum of the squares of the first $100$ terms of this sequence is $1337$, and the sum of the cubes of the first $100$ terms is $23456$. What is the smallest integer $k$ such that the sequence is pe... | nvidia |
In the equation
$$2x^2-kx+2=0$$
determine k such that the product of the roots equals the negative sum of the roots modulo 13. | nvidia |
Find the largest number of nonnegative real numbers $x_1,x_2,\dots,x_n$ that satisfy both $\sum_{i=1}^{n} x_i^2=100$ and $\prod_{i=1}^{n} x_i^2=1$. | nvidia |
The arithmetic mean of four numbers is 15. Two of the numbers are 10 and 18 and the other two are equal. What is the product of the two equal numbers? | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a^2 + b^2 + c^2 = 4.$ Find the largest possible value of $a + b + c.$ | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$ Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\] | nvidia |
How many positive four-digit integers less than 3000 have at least two digits that are the same? | nvidia |
Find the value of $b$ that satisfies the equation $234_{b}+567_{b}=93C_{b}$, where $C_{b}=12_{10}$. | nvidia |
Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution. | nvidia |
Let $g(x) = f(x) + xf(1/x)$ for all real numbers $x.$ Find the sum of all real numbers $x$ such that $g(x) = 2$ and $g(x^2) = 1.$ | nvidia |
Determine the smallest possible degree of the polynomial $p(x)$ such that the graph of the function $p(x)/e^x$ has no horizontal asymptote. | nvidia |
Let $x$ be a positive integer. Find the number of different prime factors of $x$ such that $x$ and $11x$ have the same number of divisors. | nvidia |
The function g(x) satisfies
\[g(x \cdot y) = g(x) \cdot g(y)\]for all positive real numbers x and y, and $g(2) = 3.$ Find $g(4)$. | nvidia |
Find the smallest positive integer that has exactly eight positive integer divisors, including 1 and itself, such that none of the divisors are consecutive integers. | nvidia |
Let $S$ be the sum of all real numbers $x$ such that $x^3 - 6x^2 + 21x - 26 = (x-1)^3$. Compute the value of $(2S-3)^3$. | nvidia |
Find the function $f(x),$ with real coefficients, such that $f(1) = 3$ and
\[f(x) + f(y) = f(xy) + f\left(\frac{x}{y}\right)\]for all real numbers $x$ and $y.$ | nvidia |
Find the largest real number $a$ such that
\[\frac{1}{\sqrt{a}} \le \frac{\sin^n x}{\sqrt{\cos^2 x + 1}} + \frac{\cos^n x}{\sqrt{\sin^2 x + 1}}\]for all real numbers $x$ and positive integers $n$. | nvidia |
Find the value of $x$ such that the point $(x,-3)$ lies on the circle centered at $(2,-1)$ and passing through the point $(5,4)$. | nvidia |
Let $p(x)$ be a polynomial with integer coefficients such that $p(1)=p(3)=p(5)=p(7)=p(9)=13$ and for every integer $n$, if $p(n)$ is a perfect square, then $p(n+10)$ is also a perfect square.
Find the smallest possible value of $p(11)$. | nvidia |
Let $a_n$ be the number of ways to color the vertices of a regular $n$-gon using two colors, such that no two adjacent vertices have the same color. Determine the value of $a_7$ modulo $343$. | nvidia |
Let $a,b,c$ be positive real numbers. When you simplify $\sqrt[3]{\frac{a^6b^4}{c^{11}}}$, what is the sum of the exponents of the variables that are outside the radical? | nvidia |
Find the sum of the reciprocals of the roots of $x^2-13x+4=0$. | nvidia |
What is the maximum number of different letters that can appear in a 10-character string such that no two adjacent characters are the same? | nvidia |
A circle of radius $2$ is drawn in the Cartesian plane with its center at the origin. A point $(x,y)$ is randomly selected on the circle such that $x$ and $y$ are both integers. What is the probability that the point lies in the first quadrant? | nvidia |
Find the number of pairs $(m,n)$ of positive integers satisfying $m+n=2015$ and $m\mid n$. | nvidia |
For all prime numbers $p$, what is the smallest integer that always divides into the sum of $p$ and the square of $p$? | nvidia |
If the integer part of the cube root of a positive integer $n$ is $m$, then the sum of the digits of $n$ can be expressed as $am^3+bm^2+cm+d$, where $a$, $b$, $c$, and $d$ are integers. Find the sum of the values of $a$, $b$, $c$, and $d$. | nvidia |
Triangle $PQR$ has an inradius of $7$ and a circumradius of $21$. If $2\sin{Q} = \sin{P} + \sin{R}$, then the area of triangle $PQR$ can be expressed as $\frac{p\sqrt{q}}{r}$, where $p, q,$ and $r$ are positive integers such that $p$ and $r$ are relatively prime and $q$ is not divisible by the square of any prime. Comp... | nvidia |
What is the positive integer $N$ for which $$ 22^2 \times 55^2 = 10^2 \times N^2 \ ? $$ | nvidia |
How many positive integers $n<2010$ can be written in exactly two different ways as a sum of two squares, that is, how many can be written as $a^2+b^2$ and $c^2+d^2$ with $a, b, c,$ and $d$ all positive integers and $a\neq c$? | nvidia |
In the figure, a square is inscribed in a circle and a triangle is circumscribed around the circle, where the diameter of the circle is $10$. The height of the triangle is also $10$. If the area of the triangle is twice the area of the square, what is the length of the side of the square?
[asy]
draw((0,0)--(5,0)--(5,5... | nvidia |
Problem:
What is the smallest positive integer $n$ such that $y=\cos (x+n)$ is symmetric about the line $x=-\frac{n}{2}$? | nvidia |
Let $a$ and $b$ be positive integers such that $a^2+2b^2$ is divisible by 5. What is the smallest positive integer $n$ such that $a^4+4b^4+4ab+n$ is divisible by 5? | nvidia |
How many 4-digit positive integers exist that satisfy the following conditions: (A) Each of the first two digits must be 1, 4, or 5, and (B) the last two digits cannot be the same digit, and (C) each of the last two digits must be 5, 7, or 8? | nvidia |
If $z$ is a complex number such that $|z^2-1|=|z-1|$, find $|z+1|$. | nvidia |
Find the largest integer $n$ such that the product of the first $n$ terms of the sequence
$\sqrt{1+\frac{1}{k}}$ is less than $2$.
I hope this problem is to your liking. Please let me know if you need anything else. | nvidia |
Find the value of $x$ in the expression
$$\frac{240}{x+15\cdot\frac{3}{5}}$$
such that the result is a whole number. | nvidia |
Find the maximum value of the sum $S_n=\frac{1}{a_n}+\frac{1}{a_{n+1}}$, where $a_n=2^n+1$ and $n$ ranges through the positive integers. | nvidia |
Find the length of the segment on the number line whose endpoints satisfy $|x-\sqrt[3]{27}|=2$. | nvidia |
Let $x$ be a positive real number such that $x + \frac{1}{x} = 98.$ Find\[\sqrt{x} + \frac{1}{\sqrt{x}}.\] | nvidia |
A rectangular prism box is filled with unit spheres arranged such that each sphere touches its four nearest neighbors and the sides of the box. Find the volume of the box that contains exactly $n$ spheres for a given positive integer $n$. | nvidia |
The operation $\oplus$ is defined as $a \oplus b = a \sqrt{b-a}$, where $a$ and $b$ are positive real numbers. Find the value of $x$ such that $x \oplus (x+1) = 3 \oplus 4$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x + y\sqrt{g(z)}) = \sqrt{g(x)} + yg(z)\]
for all real numbers $x,$ $y,$ and $z.$
Let $k$ be the number of possible values of $g(4),$ and let $t$ be the product of all possible values of $g(4).$ Find $k + t.$ | nvidia |
Given a positive integer $m$, let $f(m)$ be the number of factors of $m$ that are multiples of $m-1$. Find the smallest value of $m$ such that $f(m) = 100$. | nvidia |
The three sides of a right triangle have lengths $a$, $b$, and $c$, where $a<b<c$ and $a$, $b$, and $c$ are positive integers. Given that the sum of the areas of the squares formed on each side of the triangle is $145$ square units, find the length of the hypotenuse. | nvidia |
How many ways can the letters of the word "committee" be arranged such that the two T's are never together and the word does not start or end with the letter "E"? | nvidia |
Find the sum of all integers $k$ such that the quadratic equation $x^2+kx+k^2+1=0$ has exactly one integer solution. | nvidia |
Find the greatest value of $m$ such that the number $10^{100}-m$ is divisible by 56. | nvidia |
Determine the value of $x$ such that $\cos (2x + 30^\circ) = \sin x$. | nvidia |
A rectangular prism is cut by a plane in such a way that the cross-section is a triangle with vertices on the vertices of the prism. The dimensions of the prism are 5, 7, and 11. How many unit cubes of the prism have interiors lying entirely on one side of the plane? | nvidia |
What is the largest number of non-overlapping unit squares that can be placed inside a $6 \times 6$ grid, such that no square shares a side with any other square? | nvidia |
Given that $a$ and $b$ are nonzero real numbers such that $a^2 + \frac{1}{b} = 7$ and $b^2 + \frac{1}{a} = \frac{11}{2}$, find all possible values for $a$. | nvidia |
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