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Let $a_1,$ $a_2,$ $\dots,$ $a_{n}$ be a sequence of $1$'s and $-1$'s such that the sum of the terms in every contiguous subsequence is nonnegative. Find the maximum value of $n$. | nvidia |
Find a positive integer that is divisible by 24 and whose square root is a number between 10 and 10.2. | nvidia |
What is the smallest positive integer that must be a factor of the sum of any three consecutive positive integers, given that the sum is greater than 10? | nvidia |
What is the sum of all positive integers less than 100 that are squares of perfect squares? | nvidia |
Find the product of all integer divisors of $105$ that also divide $14$. (Recall that the divisors of an integer may be positive or negative.) | nvidia |
Find all positive integer values of $a$ such that $\frac{a^2+1}{a^2}$ is an integer. | nvidia |
Let $p(x) = x^2 + bx + c$ be a quadratic polynomial with real coefficients such that the graphs of $y=p(x)$ and $y=p^{-1}(x)$ are symmetric about the line $y=x$. Determine the value of $b$. | nvidia |
Let $a$, $b$, $c$ be positive numbers such that
$a^2+b^2+c^2=14$ and
$(a+b+c)^2=89$.
Find $ab+bc+ca$. | nvidia |
Find the greatest integer n such that n^2 + 3n + 2 is divisible by n+1 for every positive integer n. | nvidia |
Simplify the expression $\sqrt{12x+5}\cdot\sqrt{3x-2}$, assuming $x$ is a real number such that the expression is defined. | nvidia |
Given a sequence of functions, $\sigma_n(x) = x + \frac{1}{n}$, where $n$ is a positive integer, find $x$ such that the composition of $\sigma_1(x)$, $\sigma_2(x)$, and $\sigma_3(x)$, in that order, equals 5. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=18$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=6$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
The sum of two positive integers $a$ and $b$ is 1001. What is the largest possible value of $\gcd(a,b)$? | nvidia |
Determine the maximum value of the expression $\frac{x^2 + y^2}{xy - 1}$ for real numbers $x$ and $y$ satisfying the inequality $xy > 1$. | nvidia |
Find the smallest positive integer n such that 12n is congruent to 1 (mod 17). | nvidia |
Let \(a\) and \(b\) be real numbers such that the equation
\[x^2-4x+2=0\]
has two real roots \(x_1\) and \(x_2\), and the equation
\[ax^2+bx+c=0\]
has the roots \(\ds \frac{1}{x_1}\) and \(\ds \frac{1}{x_2}\).
Find \(a+b+c\). | nvidia |
There are three positive integers that are divisors of each number in the list $$24, 48, -8, 80, 60.$$Find the sum of these three positive integers. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
$$f(x) + f(f(y)) = x + y$$for all real numbers $x$ and $y.$ Find the number of such functions $f.$ | nvidia |
Find $a$ such that the polynomial $x^2 - 4x + a$ can be written as the sum of the squares of two linear polynomials. | nvidia |
Find the sum of all integers from 100 through 999, inclusive, that are divisible by 7. | nvidia |
Find $m$ such that $3^4 \cdot 2^3 \cdot m = 10!$. | nvidia |
The perimeter of a triangle is 21 cm. The sum of the squares of the lengths of its sides is 135 cm². Find the length of the longest side of the triangle. | nvidia |
Let $x$ and $y$ be real numbers such that $\frac{\tan x}{\tan y} = 2$ and $\frac{\sin x}{\sin y} = 3$. Find the value of $\frac{\cos^2 x}{\cos^2 y}$. | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 3. If $\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. | nvidia |
Let $h(n)$ be the sum of the proper positive integer divisors of $n$. For how many values of $n$ is $h(n)$ less than $n$, given that $2 \le n \le 75$? | nvidia |
A sequence $\{a_n\}$ is defined such that the sum of the first $n$ terms is $n^2$ for all positive integers $n$. Determine the value of $\frac{a_{n+1}}{a_n}$ for any positive integer $n$. | nvidia |
Let $m$ and $n$ be positive integers such that $m+n=9$ and the least common multiple of $m$ and $n$ is $3m$. What is the product of the two numbers? | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 1000^\circ.$ | nvidia |
Let $A$ be a $3 \times 3$ matrix with integer entries such that $A^3 = I$, where $I$ is the $3 \times 3$ identity matrix. Find the smallest possible value of the sum of the absolute values of the entries of $A$. | nvidia |
The value of
\[\frac{2x}{3} + \frac{24}{x}\]is smallest for which positive integer $x$? | nvidia |
Find all positive values of $a$ such that the equation $x^4-4x^2+a=0$ has at least one real solution for $x$. Express your answer in interval notation. | nvidia |
Find all positive integers $n$ such that the equation
$$x + \lfloor x \rfloor = n$$
has exactly three solutions, and the largest solution is an integer. | nvidia |
What is the sum of the values of $x$ that satisfy the equation $x^2-5x+5=9$? | nvidia |
Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.
| nvidia |
Find $x^3+y^3$ if $x$ and $y$ are positive integers such that \[\begin{aligned} xy+2x+2y&=45 \\ x^2y+2xy^2 &= 420.\end{aligned}\] | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 10.$ Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c}.\] | nvidia |
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.
| nvidia |
What is the sum of the cubes of the three smallest prime numbers that are greater than 10? | nvidia |
Find the smallest positive integer $m$ such that $7^m \equiv m^7 \pmod 4$. | nvidia |
Find the number of three-digit numbers that are multiples of 7 but have digits that add up to a multiple of 11. | nvidia |
How many positive integers less than $1000$ can be written as the sum of two positive perfect fourth powers? | nvidia |
A positive integer is randomly chosen from among the numbers formed by summing all the elements in each row of Pascal's Triangle from rows 0 through n. What is the probability that the value of the integer chosen is a power of 2, if n = 20? | nvidia |
In $\triangle DEF$ the ratio $DE:EF$ is $2:5$. The bisector of the interior angle at $F$ intersects $ED$ extended at $Q$ ($D$ is between $E$ and $Q$). The ratio $QE:ED$ is:
$\textbf{(A)}\ 1:2 \qquad \textbf{(B)}\ 2:3 \qquad \textbf{(C)}\ 3:2 \qquad \textbf{(D)}\ 5:2 \qquad \textbf{(E)}\ 5:1$ | nvidia |
There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=3$, $e=-a-c$, and the sum of the numbers is $2i$, find $d+f$. | nvidia |
Find the smallest positive integer $n$ such that $\prod_{k=1}^{n} \cos (2^k)$ is a rational number. | nvidia |
Let $A$ be the set of nonnegative integers less than 32 that are relatively prime to 32. An element $a \in A$ is called "invertible" if there exists an element $b \in A$ such that $ab \equiv 1 \pmod{32}$. Find the number of elements in $A$ that are not invertible. | nvidia |
Determine $a$, if any, such that the equation $$x^2+ax+\frac{1}{4}=0$$ has roots whose product is equal to the sum of their squares. | nvidia |
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the recurrence relation
$$a_{n+1}=\sqrt{2a_n+1}$$for all positive integers $n.$ Additionally, it is known that the sequence $b_n=a_n^2+2a_n+1$ is an arithmetic progression. Enter the ordered pair $(a_1,a_2).$ | nvidia |
Let $f(x)$ be a rational function such that the inequality $f(x) > 0$ is satisfied if and only if $x \in (-\infty, -2) \cup [3, 4]$. Find the sum of the roots of the numerator of $f(x)$. | nvidia |
A polynomial with integer coefficients is such that the product of its first and last coefficients is equal to the constant term. Determine the value of the second coefficient if the degree of the polynomial is 5. | nvidia |
Determine the value of $k$ such that the equation
$\ds\cos^3x-\sin^3x=k\cos 2x$
holds true for all $x$ in the interval $(0^\circ, 90^\circ)$. | nvidia |
Find the value of $b$ such that the quadratic expression $x^2 + bx + 25$ can be factored into the product of two binomials with integer coefficients. | nvidia |
Find the sum of all integer divisors of $360$ that leave a remainder of 1 when divided by 3. | nvidia |
Let $P_n$ be the product of all positive integers less than or equal to $n$ that are either a prime number or have exactly one prime factor. What is the smallest positive integer $n$ such that $P_n$ is a multiple of 1000? | nvidia |
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$
| nvidia |
Find the largest integer value of $m$ such that $m^2-8m+16$ is negative. | nvidia |
Find the number of three-digit numbers that have exactly two identical digits, where the identical digits are adjacent. | nvidia |
Find the number of positive divisors of 2002. | nvidia |
Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix} \renewcommand{\arraystretch}{1}.\] | nvidia |
Consider the quadratic equation:
$$x^2 - mx + 20 = 0$$
Find the sum of all positive integer values of $m$ for which this equation has only integer solutions. | nvidia |
Find the number of values of $x$ that satisfy the inequality $\frac{1}{x} > \sin x$ for $-2\pi < x < 2\pi$. | nvidia |
Find the smallest positive integer $n$ such that there exist exactly three distinct positive integers $a, b, c$ satisfying the equation $a^3 + b^3 + c^3 = n^3$. | nvidia |
Let $x,y,z$ be positive real numbers such that $x+y+z=12$ and $xy+yz+xz=36$. Let $k=\max\{xy,yz,xz\}$. Find the smallest possible value of $k$. | nvidia |
What is the smallest positive integer $n$ such that $3n$ is a perfect square and $2n$ is a perfect cube? | nvidia |
Let $x$, $y$, and $z$ be positive integers such that $x$ and $y$ have exactly the same digits, but in different order, and $z=|x-y|$.
For what numbers $n$ of digits is the smallest possible value of $z$ unique? | nvidia |
For any positive integer $n$, let $f(n)$ denote the sum of the squares of its digits (e.g., $f(18) = 1^2 + 8^2 = 65$).
Determine the maximum value of $f(a-b)$, where $a$ and $b$ are positive integers such that $ab=333$. | nvidia |
Let $A$ be a set of positive integers, such that for any two distinct elements $a$ and $b$ of $A$ we have $2\leq\lvert a-b\rvert\leq 3$.
Find the smallest possible number of elements of $A$ that guarantees that $A$ will contain two distinct elements $a$ and $b$ with $\lvert a-b\rvert=2$. | nvidia |
Problem:
Find the number k such that the graph of the function
$$y = 3x^2 + kx + 2$$
passes through the point (2, 5) and the vertex of the parabola is on the x-axis. | nvidia |
Find the sum of all positive integers less than 100 that are cubes of perfect cubes. | nvidia |
Determine the largest value of $n$ for which $13^n$ has exactly three digits. | nvidia |
Find the number of positive integers $n$ such that $n$ and $S(n)$ have the same remainder when divided by 5. | nvidia |
Find all possible values of $z$ such that $\sin z=\cos(z+10)$, where $z$ is a complex number. | nvidia |
Let $\p_k(x)=x^3-x+k$. Find the constant $k$ such that for every value of $c$, $p_k(c)=0$ implies that $2p_k(1)+p_k(-2)=0$. | nvidia |
Find the sum of the positive divisors of 18 that are not multiples of 3. | nvidia |
Given that the sum of two fractions is $\frac{3}{4}$, and one of the fractions has a numerator that is twice as large as its denominator, find the value of the other fraction.
Express your answer in the form of $\frac{a}{b}$. | nvidia |
Find $s$ such that $(s,-2)$ lies on the line through $(3,4)$ and $(-1,-5)$. | 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 |
Let $f(n)$ denote the number of ordered pairs of positive integers $(a,b)$ such that $\text{lcm}(a,b)=n$. Determine all prime numbers $p$ for which $f(p^{2019})$ is a perfect square. | nvidia |
Find the number that is the harmonic mean of the two numbers 12 and 15. | nvidia |
Let $a$ and $b$ be any two prime numbers, with $b$ greater than $a$. The largest integer which divides all possible numbers of the form $a^3+b^3$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 11$ | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) \ge 1\]for all real numbers $x,$ $y,$ and $z.$ | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $a+b+c=1$. Find the maximum value of $\sqrt{a+b} + \sqrt{b+c} + \sqrt{c+a}$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a+b+c=1$. Find the maximum value of $\frac{1}{a^2+b^2+c^2}$. | nvidia |
For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers?
| nvidia |
The vertices of a triangle are located at $(0,0), (4,0),$ and $(0,y)$ with $y>0$. A rectangle is inscribed inside the triangle such that two of its corners rest on the hypotenuse of the triangle and one of its sides rests on the base of the triangle. Find the maximum possible area of this rectangle. | nvidia |
Let a and b be real numbers such that
\[(a-1)^2=3(b-2)^2,\quad 2a+b=4,\]
and
\[\frac{1}{\sqrt{a^2+1}}+\frac{1}{\sqrt{b^2+4}}=1.\]
Determine a+b. | nvidia |
Let $f(x)$ be a polynomial function such that for every integer $x$,
$f(x)$ and $f(x+1)$ are both integers. Determine the maximum possible degree of the polynomial function $f(x)$. | nvidia |
Find the area of the shaded region formed by removing an isosceles right triangle with leg length 8 from the interior of a 16 x 16 square.
[asy]draw((0,0)--(32,0)--(32,32)--(0,32)--cycle);
label("16'",(16,32),N);
label("16'",(32,16),E);
filldraw((32,0)--(32,16)--(16,0)--cycle,gray(.6));
[/asy]
Note: The area can be ... | nvidia |
A function $g$ satisfies the relation $g(x+y)=g(x)+g(y)$ for all real numbers $x$ and $y$. If $g(3) = 1$ and $g(4) = 2$, find the value of $g(7)$. | nvidia |
How many three-digit numbers have the property that the sum of their digits is a prime number and the product of their digits is a perfect square? | nvidia |
Let $g$ be a function defined on the integers as follows:
\[g(n) = \left\{
\begin{array}{cl}
2n & \text{if $n$ is odd}, \\
n - 3 & \text{if $n$ is even}.
\end{array}
\right.\]Given that the range of $g(g(g(n)))$ is $\{-3,-6\}$, find the smallest positive integer $k$ that is in the domain of $g$. | nvidia |
How many integers satisfy the following system of inequalities?
\[
\begin{array}{cccc}
(1) & -2x & \geq & x+5 \\
(2) & -x & \leq & 8 \\
(3) & -3x & \geq & x+11
\end{array}
\] | nvidia |
Let $y = f(x)$ be a periodic function whose graph is shown below, where the period is $T=8$. Note that the graph is symmetric about $x=0$, $y=0$, and the line $y=x$. The equation $f(f(f(x)))=1$ has 16 solutions in $[-8,8]$. How many of them satisfy $f(x)>1$? [asy]
size(150);
real ticklen=3;
real tickspace=2;
real... | nvidia |
Let $a^2=\frac{16}{44}$ and $b^2=\frac{(2+\sqrt{5})^2}{11}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | nvidia |
Consider a 4-by-4 grid of unit squares, where each square can be colored either blue or red. The color of each square is determined randomly, with equal probability of being either color. Find the expected number of 2-by-2 squares within the grid that have all four squares of the same color. | nvidia |
What is the largest possible integer k such that, for any positive integer n, the sum of the cubes of the first n positive integers is divisible by k? | nvidia |
Find the product of the possible values of $y$ that satisfy the equation $\frac{y-1}{y+1} \cdot \frac{y+2}{y-2} = 3$. | nvidia |
Let $S$ be the set of all positive integers whose digits are non-decreasing from left to right (for example, 1188, 2222, 3479). For each positive integer $n$, let $c(n)$ be the number of elements of $S$ that are less than or equal to $n$. Evaluate
\[\lim_{n\to\infty}\frac{c(n)}{n}.\] | nvidia |
Given the equation of a circle $(x-2)^2 + (y-3)^2 = 4$, find the sum of the $x$-coordinates of the points where the circle intersects the $x$-axis and the sum of the $y$-coordinates of the points where the circle intersects the $y$-axis. | nvidia |
A three-digit number has the property that the sum of its digits is a perfect square, and when its digits are reversed, the resulting number is a multiple of 7. What is this three-digit number? | nvidia |
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