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Given that the graphs of the functions
$$y=x^2+5x+7 \quad \text{and} \quad y=(x+2)^2+(x+1)^2$$
intersect at a single point, determine the $x$-coordinate of this point. | nvidia |
Find the number of distinct arrangements of the letters in the word "STARRYS" such that the two R's are not adjacent. | nvidia |
Determine the number of pairs of distinct positive integers $(m, n),$ $1 \le m < n \le 1000,$ such that $m^2 + n^2$ can be expressed as the sum of squares of two other distinct positive integers. | nvidia |
How many five-digit positive integers have at least two digits that are the same? | nvidia |
Find the sum of all possible positive integer values of $c$ such that the quadratic equation $x^2 - 4x + c = 0$ has rational roots. | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 1.$ Find the minimum value of
\[(x + 2y)(y + 2z)(xz + 1).\] | nvidia |
Two dice are rolled together, and the sum of the numbers shown on the dice is noted. What is the probability that the sum obtained is a prime number? | nvidia |
The function $f(n)$ is defined as the product of all positive divisors of $n$ that are less than or equal to $\sqrt{n}$. For example, $f(12) = 1 \cdot 2 \cdot 3 = 6$. Find the smallest positive integer $m$ such that $f(m)$ is a perfect square. | nvidia |
There are values $C$ and $D$ such that
\[\frac{Dx+15}{x^2+8x+12}=\frac{2}{x+2}+\frac{C}{x+6}.\]
Find $C+D$. | nvidia |
Find the number of positive integers, including 1, that are divisors of both 48 and 64. | nvidia |
How many positive integers $n$ satisfy the inequality $\cos 2n < \cos n < \sin 3n$? | nvidia |
What is the greatest positive integer that must divide the sum of the first twelve terms of any arithmetic sequence whose terms are positive integers and the common difference is even? | nvidia |
The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
The real function $f$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a + b = 2^n,$ the equation
\[f(a) + f(b) = n^2\]holds. What is $f(2002)$? | nvidia |
Find the positive integer value of $n$ for which the equation
$$x^2 + y^2 + z^2 = x^3 + y^3 + z^3$$
has exactly $n$ solutions $(x,y,z)$, where $x, y, z$ are integers. | nvidia |
Determine the number of distinct factors of the number $n$, where $n$ is the product of the first 5 prime numbers, that are not multiples of 3. | nvidia |
Find the number of perfect squares that can be expressed as the sum of two consecutive prime numbers less than 20. | nvidia |
There are two pairs $(x,y)$ of real numbers that satisfy the equation $x-y = 2xy = 5$. Given that the solutions $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$? | nvidia |
What is the greatest number of distinct triangles that can be formed using 5 points in a 3-dimensional space such that no four points are coplanar? | nvidia |
Let $f(n)$ denote the smallest integer such that $n!$ is a factor of $f(n)!$. Find all positive integers $n$ such that $f(f(n)) = n$. | nvidia |
The function $f(x)$ is defined by
$$f(x)=\prod\limits_{i=1}^{5}\frac{x^4-2^ix+5}{x^4+2^ix+5}.$$Let $k$ be the number of different real solutions $r$ of the equation $f(x)=1$ in the range $0 \leq r \leq 6$. Find $k$. | nvidia |
Find all real numbers $b$ such that the roots of the polynomial
$$x^3 - 4x^2 + 15x + b$$
form an arithmetic progression and are not all real. | nvidia |
Let $m$ be a positive integer. Find the number of integers $n$ such that $n!+m$ is a perfect square. | nvidia |
Consider the function
$$f(x)=\sum_{n=3}^\infty \frac{1}{n^2-x}$$
defined for all real numbers $x$ except for $x=n^2$ with $n=3,4,5,\dots$.
Determine $f(x)$ for a value of $x$ where the series has a closed form. | nvidia |
Let $p(x) = x^4 - 360x^2 + 400$. The four complex roots of $p(x)$ are plotted in the complex plane. Let $R$ be the sum of the distances from the origin to each root. Find $\frac{1}{R}$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
rea... | nvidia |
Find the smallest integer $m$, greater than $2$, such that $m^{-1}\pmod{200}$ is defined. | nvidia |
Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 25$, $b_1 = 40$, $b_n=0$, and $$ b_{k+1} = b_{k-1} - \frac{5}{b_k} $$for $k = 1,
2, \dots, n-1$. Find $n$. | nvidia |
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the equation
\[f(x + f(y)) = f(x) + f(y) + f(f(x) - y)\]for all real numbers $x$ and $y.$ Let $m$ be the number of possible values of $f(0)$ and let $t$ be the maximum possible value of $f(0).$ Find $m + t.$ | nvidia |
Find the sum of all solutions to the equation $(x-6)^2=25$. | nvidia |
The integer $n$ is the largest positive multiple of $15$ such that every digit of $n$ is either $8$ or $0$. Compute $\frac{n}{15}$.
| nvidia |
Find all positive integers whose digits are non-decreasing, and for which the sum of the square of their digits is equal to the square of the integer itself.
Note: A non-decreasing sequence is a sequence where each element is greater than or equal to the previous one. For example, 112345 is a non-decreasing integer. | nvidia |
Find the largest value of $x$ that satisfies the inequality $|x-3| + |x-5| < 4$. | nvidia |
Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$
\text A. \ \ 21 \qquad \text B. \ \ 60 \qquad \text C. \ \ 119 \qquad \text D. \ \ 180 \qquad \text E. \ \ 231
$$ | nvidia |
Let $b_1,$ $b_2,$ $\dots$ be a sequence of positive real numbers such that
\[b_n = \sqrt{b_{n-1} + \sqrt{n}}\]for all $n > 1.$ Find the largest possible value of $b_1$ such that the sequence is bounded. | nvidia |
Determine the value of the parameter $a$ such that the function $f(x)=\dfrac{ax^2-7x+2}{(x^2-4)(x^2-9)}$ has the greatest number of extrema. | nvidia |
Let $f(x)$ be a function defined as \[f(x) = \left\{
\begin{array}{cl}
x+1 & x<2, \\
x^2 & 2\le x<5, \\
2x-3 & x\ge 5.
\end{array}
\right.\] Find all real values of $x$ such that $f(f(f(f(x))))=x$. | nvidia |
A portion of the graph of a cubic function $f(x)$ is shown below.
Let $g(x)=-f(x)$ and $h(x)=f(2-x)$. If $a$ is the number of points where the graphs of $y=f(x)$ and $y=g(x)$ intersect, and $b$ is the number of points where the graphs of $y=f(x)$ and $y=h(x)$ intersect, then what is $5a+2b$?
[asy]
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real tic... | nvidia |
Determine all possible values of $a$ for which there exists a triangle with side lengths $1, a, b$, where $a < b$, and for which the following conditions hold: the circumradius of the triangle formed by the two shorter sides and the median to the longest side is equal to the circumradius of the triangle itself, and the... | nvidia |
A triangle has sides $a$, $b$, and $c$, where $a=15$, $b=14$, and $c=13$. The inradius of the triangle is 2.5. Find the area of the triangle. | 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 |
Determine the largest possible value of $a$ for which the expression $\frac{2a}{a^2+1}$ is an integer. | nvidia |
Find the number of positive integers less than or equal to 30 that are relatively prime to 30.
Answer:
There are 8 positive integers that are less than or equal to 30 and are relatively prime to 30. These are the integers that are not divisible by 2, 3, or 5. | nvidia |
Find the smallest positive integer $k$ such that the sum of the reciprocals of the first $k$ positive integers is greater than 2. | nvidia |
The function $g(x)$ satisfies
\[x^2g(y) = y^2g(x)\]for all positive real numbers $x$ and $y.$ If $g(8) = 12,$ find $g(2).$ | nvidia |
Let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d$ be two distinct polynomials with real coefficients such that the product $f(x)g(x)$ can be factored as the product of five distinct linear factors with integer coefficients. Find the smallest possible value of $|a|+|b|+|c|+|d|$. | nvidia |
The sum of the cubes of two positive integers is 504. The sum of the two integers is 14. What is the product of the two integers? | nvidia |
Find the largest positive integer that cannot be represented in the form $3p + 5q$, where $p$ and $q$ are non-negative integers. | nvidia |
Let $m$ and $n$ be positive integers such that $n > 2m$. Determine the remainder when
$$\sum_{k=0}^{\lfloor \frac{n}{m} \rfloor} \binom{n}{km}$$
is divided by $n$. | nvidia |
Given a sphere with center (0,0,0) and radius $\sqrt{65}$ and a plane defined by the equation $2x+3y-z=k$, find $k$ so that the plane intersects the sphere exactly at one point. | nvidia |
In the rhombus $ABCD$, the diagonals are perpendicular bisectors of each other and intersect at point $O$. The measure of angle $DAB$ is twice that of angle $DAO$. If the length of segment $AO$ is 6, find the area of the rhombus. | nvidia |
What is the smallest positive integer $n$ such that $n$ is divisible by $2^2$, $3^2$, and $5^2$? | nvidia |
Suppose $x$ and $y$ are integers such that $xy+5x+4y=-5$. Find the greatest possible value of $y$. | nvidia |
Find the smallest positive integer with exactly 5 distinct positive divisors such that the sum of its divisors is less than 3 times the integer itself. | nvidia |
What is the largest positive integer $n$ for which the polynomial $x^n + x + 1$ has a factor of the form $x^k + x + 1$, where $k$ is a positive integer less than $n$? | nvidia |
Find all real values of $x$ that satisfy $\frac{x^2-2x+1}{x^2+2x+2} < \frac{3}{4}$ (Give your answer in interval notation.) | nvidia |
What is the smallest number that can be obtained by rearranging the digits of 2004 such that the sum of the digits is 6? | nvidia |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find $|z|.$ | nvidia |
Find the number of integers between 62 and 215 that are not divisible by 3 or 5. | nvidia |
Determine the minimum of
\[2(x+5)+\frac{1}{\sqrt{x^2+4x+5}}\]
for all real numbers $x$. | nvidia |
Find the smallest positive integer $n$ such that the product of its digits in base $8$ is equal to $n$ itself. | nvidia |
Find the equation of the line passing through the points (2,3) and (4,5), given that the slope of the line is twice the slope of the line passing through the points (1,2) and (3,4). | nvidia |
Given that $a$ and $b$ are positive integers and that $a+b=24$, what is the value of $ab$ if $2ab + 10a = 3b + 222$? | nvidia |
Find the smallest positive integer $x$ such that $x+1$ is divisible by 3, $x+2$ is divisible by 5, and $x+3$ is divisible by 7. | nvidia |
Let $x$ and $y$ be positive real numbers such that $\frac{1}{x+2} + \frac{1}{y+2} = \frac{1}{3}.$ Find the minimum value of $xy(x+y).$ | nvidia |
Find the product of the $y$-values that satisfy the equation $7=\frac{y^3-3y^2-18y}{y-3}$. | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $abc = 1000.$ Find the largest possible value of $b.$ | nvidia |
Let \(a\) and \(b\) be two positive integers such that their product is 156 and their arithmetic mean is equal to the geometric mean of the two numbers whose sum is equal to \(a+b\), and whose product is equal to the difference of the squares of \(a\) and \(b\). Find the least possible value of the sum \(a+b\). | nvidia |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $n-1$, $n$, and $n+2$ cents, the largest amount of postage that cannot be formed is exactly three times the number of different stamp denominations available. | nvidia |
Find the largest number of consecutive, positive odd integers whose sum is 99. | nvidia |
Determine the number of digits in the base-9 representation of the number 1458. | nvidia |
Consider the equation
$$\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} = \frac{1}{d^2}$$
where $a$, $b$, $c$, and $d$ are positive integers. Find all possible values of $(a, b, c, d)$ such that $a < b < c$ and $d$ is minimal. | nvidia |
Find all real values of $b$ such that the polynomial
\[x^4 + 2x^3 + bx^2 + 3x + 1 = 0\]
has at least one root that is a rational number of the form $\frac{m}{n}$, where $m$ and $n$ are integers with $n \neq 0$. Enter all possible values of $b$ as a comma-separated list. | nvidia |
Let $x$ be a positive integer such that $2x^2+x$ is a perfect square. How many integer values of $x$ satisfy the condition $2x^2+x<1000$? | nvidia |
Determine the complex number $z$ satisfying the equation $2z-3i\bar{z}=-7+3i$. Note that $\bar{z}$ denotes the conjugate of $z$. | nvidia |
Find the smallest positive integer $m$ such that
\[\tan^m x + \cot^m x \ge \frac{2}{m}\]for all real numbers $x$ where $\tan x$ and $\cot x$ are defined. | nvidia |
A line containing the point (2,3) is such that the sum of its x and y intercepts is equal to 10. Determine the slope of the line. | nvidia |
Given that $a$ and $b$ are positive integers such that $\gcd(a,b) = 18$, find the smallest possible value of $\gcd(a^2,b^3)$. | nvidia |
A set of $n$ cities is connected by a network of roads, with each city connected to exactly three other cities. The network is such that it is possible to travel from any city to any other city by a sequence of roads. Let $M$ be the number of ways to color the roads with two colors, red and blue, such that no two roads... | nvidia |
Four 4-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal? | nvidia |
Find the smallest positive integer greater than 20 such that it is neither a prime number nor a composite number with any prime factor greater than 3. | nvidia |
A complex number $z$ is such that $|z|=1$ and $|z-3|=5$. Determine the real part of $z$. | nvidia |
Consider the system of equations:
\begin{align*}
a^x+b^y & =k, \\
a^{x+y}+b^{x+y}&=m,
\end{align*}
where $a,b,k,m$ are known constants and $x,y$ are unknown variables.
Prove that there exists a unique solution for the system if and only if $a\neq b$.
Can you find the condition on the constants for the soluti... | nvidia |
Find the largest integer $m$ such that $3003^m$ divides $3003!$. | nvidia |
What is the smallest positive integer that can be written in the form $1503m + 18181n$, where $m$ and $n$ are integers? | nvidia |
Find the number of positive integers less than or equal to 100 that have at least one digit repeated. | nvidia |
Find the number of positive real solutions to the equation $x^5+ax^4+bx^3+cx^2+dx+e=0$, where $a, b, c, d, e$ are positive integers such that $a^2+b^2+c^2+d^2+e^2=2024$. | nvidia |
Let $x$ be a positive real number such that $x + \frac{1}{x} = 98.$ Find $\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^3$ | nvidia |
Find the largest integer $M$ such that the numbers $M$, $M+1$, and $M+2$ have different prime factors, and their product is a perfect square. | nvidia |
Let $n$ be the greatest integer such that $\left(8^n\right)\left(10^{16}\right)$ divides the product
$\left(2^{10}\right)\left(3^{12}\right)\left(5^{15}\right)$. Find $n$.
The final answer is 4. | nvidia |
Let $z$ be a complex number such that
$|z-1|=|z+1|=|z-i|$. Determine $|z|^2$. | nvidia |
Find the value of $x$ such that $(2010-x)^3 = x^3$. | nvidia |
For how many four-digit positive integers is the sum of the digits equal to $8$? | nvidia |
Find the product of the sequence: $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \cdot \frac{1}{243} \cdot \frac{729}{1}$. | nvidia |
Find the smallest possible area of a scalene triangle with integer side lengths, where the area is also an integer. | nvidia |
Find the maximum value of
\[\frac{|a+b|+|b+c|+|c+d|+|d+a|}{|a|+|b|+|c|+|d|}\]
where $a,b,c,d$ are non-zero real numbers and $a+b+c+d=0$. | nvidia |
Given a quadratic equation $ax^2+bx+c=0$ with roots $x_1$ and $x_2$, the sum of the squares of its roots can be found using the formula $x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2$. This can be expressed in terms of the coefficients $a$, $b$, and $c$ as $x_1^2+x_2^2=\left(\frac{-b}{a}\right)^2-2\left(\frac{c}{a}\right)$.
Problem... | nvidia |
Find the sum of all real solutions to the equation $x^2+9=6x+\frac{9}{x}$. | nvidia |
Find the largest value of $c$ such that the graph of $f(x) = x^3 - 2x^2 + cx + 1$ has exactly one real root. | nvidia |
Find the sum of all positive integer solutions less than or equal to 15 to the congruence 11(2x+1)≡ 22(mod 5). | nvidia |
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