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Find the number of positive divisors of 216 that are also perfect squares. | nvidia |
Find the sum of all integers from 1 to 100 that are not divisible by either 5 or 8. | nvidia |
Let $a$, $b$, and $c$ be real numbers such that $\sqrt{a}+\sqrt{b}+\sqrt{c} = 5$. Determine the minimum value of $a+b+c$ such that all square roots involved are defined. | nvidia |
Find the number of permutations of the letters in the word "PERMUTATION" such that at least one of the letters "P", "E", or "R" is in its original position. | nvidia |
A right pyramid has a square base with edge length 20. Four spheres are tangent to each of the edges of the pyramid's base, and all four spheres are internally tangent to a sphere centered at the apex of the pyramid with radius 70. Find the total surface area of the pyramid that lies outside all of the five spheres. | nvidia |
Let's $a,ar,ar^2,ar^3, \dots$ be a geometric sequence with common ratio $r$ such that $|r|<1$. If the sum of all the terms of this sequence equals 3 and the sum of all the terms occupying odd places is equal to 2, find $a$. | nvidia |
The vertices of a triangle are $(-2,3)$, $(5,-1)$ and $(k,7)$. Determine the value of $k$ such that the perimeter of the triangle is equal to 20 units. | nvidia |
A box contains 10 marbles, each marked with a different positive integer. If two marbles are randomly selected without replacement, what is the probability that the sum of the numbers on the marbles is divisible by 3? | nvidia |
Let $a_n$ be a sequence of positive integers such that $a_1=1$ and $a_n=2a_{n-1}+a_{n-2}$ for $n\geq 2$.
Determine the largest positive integer $k$ such that the sum $\ds\sum_{n=1}^k\dfrac{1}{a_n}$ doesn't exceed unity. | nvidia |
Let $f(x)$ be a continuous function on the interval $[0,2\pi]$. Determine all real numbers $a$ such that the inequality
$$f(x+a) \geq f(x)+f(a)$$
holds for every $x$ in $[0,2\pi]$. | nvidia |
The product of two positive integers is 72 and their sum is 18. What is the value of the positive difference of the cubes of the integers? | nvidia |
Let $f(x)$ be a polynomial with rational coefficients and roots $\{r_1,r_2,\dots,r_n\}$ such that for each root $r_i$, there exists an integer $m_i$ satisfying $r_i+m_i=\sqrt{m_i^2+1}$. Find the smallest possible value of $n$ if $f(x)$ is divisible by $x^2-3x-4$ and has no rational roots. | nvidia |
Find the number of positive integers $n$ less than or equal to 100 that are relatively prime to 12. | nvidia |
Determine a positive integer such that the sum of all its divisors is equal to twice its prime factorization. | nvidia |
The set of all integers $a$ that satisfy the following three conditions contains exactly three elements: $a$ is divisible by $4$, $a$ is a divisor of $24$, and $a$ is positive. What is the sum of all such values of $a$? | nvidia |
Find the smallest positive integer $n$ such that the sum of the digits of $n^2$ is equal to the sum of the digits of $n$. | nvidia |
Determine the number of integers n such that the expression $2n^2 + 3n + 1$ takes a value between 50 and 100. | nvidia |
Let $f(x) = \log_{a} x$ where $a$ is a positive integer greater than 1. Find the value of $x$ such that $f(f(x)) = f^{-1}(f^{-1}(x))$. | nvidia |
In the figure, $ABCD$ is a rectangle and $E$ and $F$ are points on $BC$ and $CD$ such that $\triangle ABE \cong \triangle BCF$ and $\angle BAE$ is a right angle. If $AB=2$, $BC=3$, and $AE=4$, find the area of the quadrilateral $AFCE$. | nvidia |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | nvidia |
Determine the value of $x$ such that $\sinh(\log_{5}(x+1)) = \frac{3}{2}$. | nvidia |
How many four-digit numbers have the property that the first two digits are the same and the last two digits are also the same, but the two pairs of identical digits are not the same? | nvidia |
Let $g$ be a function taking the positive integers to the positive integers, such that
(i) $g$ is a periodic function with period $5$ (i.e. $g(n + 5) = g(n)$ for all positive integers $n$)
(ii) $g(m + n) = g(m) + g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m + n$ is divisible by $5,$ then... | nvidia |
Find the number of ways to arrange 8 red and 6 blue balls in a row such that no two blue balls are adjacent and no red ball is at the start or end of the row. | nvidia |
Let $S$ be the sum of the first $100$ positive even integers. Find the largest integer $k$ such that $S$ is divisible by $4^k$. | nvidia |
In a certain number system, the sum of the squares of all single-digit numbers is equal to the product of the two largest single-digit numbers. Determine the base of this number system. | nvidia |
Let $f(x)$ be a function such that
\[f(x+y) = f(x) + yf(1)\]for all real numbers $x$ and $y.$ If $f(2) = 12,$ find $f(10).$ | nvidia |
Problem:
Let $a$, $b$, and $c$ be positive real numbers such that $\sqrt{a}+\sqrt{b}+\sqrt{c}=6$. Find the minimum value of $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}.$ | nvidia |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | nvidia |
The vertices of a cube are labeled with 8 different integers such that the sum of the numbers at the endpoints of each edge is the same. Find the smallest possible sum of the labels. | nvidia |
Find the number of subsets of the set $\{1, 2, 3, \ldots, 12\}$ that contain no three consecutive integers. | nvidia |
Given a polynomial $P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$ with integer coefficients, find the smallest positive integer $n$ such that $P(2^n)$ is divisible by 11 for all possible values of $a, b, c, d,$ and $e$. | nvidia |
Find the three-digit number that is divisible by 5 prime numbers, with the additional condition that the sum of its digits is equal to the product of the 5 prime numbers. What is the sum of the five prime numbers? | nvidia |
Find the number of functions $g(x)$, defined on real numbers, such that $g(\frac{x+y}{2}) = g(x)g(y)$ for all real numbers $x$ and $y$. | nvidia |
The rational function
$$y=\frac{3x^4-7x^3+17x^2-12x+5}{x^2-2x+3}$$
is such that the line $y=1$ is tangent to its graph.
Find the sum of the roots of the numerator that belong to the
first and fourth quadrants. | nvidia |
Consider a function $g : \mathbb{Z}^+ \to \mathbb{Z}^+$ such that for every positive integer $n,$ we have that $g(n)$ divides $n$ and for every pair of positive integers $m$ and $n,$
$$\text{lcm}(g(m), g(n)) \cdot \text{gcd}(m, n) = m + n.$$
Let $N$ be the number of possible values of $g(12),$ and let $S$ be the sum ... | nvidia |
Let $\{b_k\}$ be a sequence of real numbers such that $b_1=1$ and $b_{m+n}=b_m \cdot b_n + \frac{1}{m+n},$ for all positive integers $m$ and $n.$ Find $\lim_{k\to\infty} b_k.$ | nvidia |
The graph of the parabola described by the equation $y=-3x^2-30x-81$ intersects the graph of the circle described by the equation $(x+5)^2+(y+6)^2=25$ at two points, one of which is $(p,q)$. What is $q$? | nvidia |
The product of two positive integers is 18. The positive difference of these two integers is 3. What is the sum of the two integers? | nvidia |
Find the largest possible value of $n$ such that $12n$, $15n$, and $21n$ are all perfect squares. | nvidia |
Let $f(n)$ be the remainder left when $n$ is divided by 17, $g(n)$ be the remainder left when $n$ is divided by 19, and let $h(n)$ be the maximum value between $f(n)$ and $g(n)$. Find all positive integers $m$ such that for every integer $n$, $$h(23m + n) = h(n).$$ | nvidia |
Determine the value of $x$ that satisfies the equation $$\left|\frac{x^2-1}{x+1}\right|=\frac{2}{x}$$ when $x \neq -1$. | nvidia |
Find $t$ such that $(t,5)$ lies on the line through $(0,3)$ and $(-8,0)$. | nvidia |
Determine the degree of the polynomial $(1+x)^n-1$ for a positive integer $n$. | nvidia |
Given that an integer can be expressed as the product of its prime factors, determine the number of distinct factors of the integer 60. Consider both positive and negative factors in your count. | nvidia |
Find the number of combinations of letters that satisfy the property of being five-letter words that start with "T" and end with the letter "S". | nvidia |
Determine all real numbers $a$ such that the expression $\frac{a+i}{a-i} + \frac{a+2i}{a-2i} + \frac{a+3i}{a-3i}$ is a real number. | nvidia |
In a certain triangle, the lengths of the three sides are consecutive integers. The area of the triangle is 10 square units. Find the lengths of the sides. | nvidia |
Let ABCD be a rectangle with integer side lengths and area equal to 100. Two points M and N are chosen inside the rectangle so that the triangles AMB, BNC, CMD, and DNA are similar and have the same area. Find the minimum perimeter of the rectangle ABCD. | nvidia |
The squares of four consecutive positive integers have the same sum as the squares of the first n positive integers, where n > 4.
Find n. | nvidia |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | nvidia |
Find the number of integers $k$ for which the equation $x^3 - 2x^2 + kx - 1 = 0$ has at least one root that is a perfect square. | nvidia |
Find all real numbers x such that the inequality \[\frac{(x^2+2x-8)(x^2-3x+2)}{x^2+2} \le 0\] holds true. Express the solutions in interval notation. | nvidia |
Find the smallest positive integer that is divisible by 5 and whose hexadecimal representation does not contain the digit 1. | nvidia |
Consider a polynomial
$$P(x) = x^4 + ax^3 + bx^2 + cx + d$$
of degree 4 with distinct roots. Suppose that the product of the roots that lie in the first quadrant is 1. Which of the following is true about the coefficients of $P(x)$?
$\textbf{(A)}\ a > 0 \qquad \textbf{(B)}\ b > 0 \qquad \textbf{(C)}\ c > 0 \qquad \t... | nvidia |
Let $\mathbf{A}$ be a $3 \times 3$ matrix with entries in the set $\{-1, 0, 1\}$ such that $\mathbf{A}^2 = \mathbf{A}$. Find the maximum possible value of the sum of all the entries of $\mathbf{A}$. | nvidia |
Let $g : \mathbb{N} \to \mathbb{N}$ be a function such that
\[g(x) g(y) - g(\text{lcm}(x,y)) = \gcd(x,y)\]for all positive integers $x$ and $y.$
Let $m$ be the number of possible values of $g(6),$ and let $t$ be the sum of all possible values of $g(6).$ Find $m \times t.$ | nvidia |
Find the number of positive integer solutions to the equation $2^a \cdot 3^b \cdot 5^c = 3600$, where $a, b,$ and $c$ are natural numbers. | nvidia |
Let $a_1$, $a_2$,..., $a_n$ be the terms of an arithmetic sequence. The sum of the odd-numbered terms of this sequence is 2010, while the sum of the even-numbered terms is 3020. Determine the sum of all $n$ terms of the sequence. | nvidia |
How many 4-digit numbers have the property that the sum of the digits is equal to the product of the digits? | nvidia |
Let $p(x)=x^3+ax^2+bx+c$ be a polynomial with real coefficients and $a^2=3b$. Determine the value of $a$ given that the equation $p(x)=0$ has three real roots. | nvidia |
Let $f$ be a non-constant polynomial such that
\[f(x - 1) + f(x) + f(x + 1) = \frac{[f(x)]^2}{2013x}\]for all nonzero real numbers $x.$ Find the sum of all possible values of $f(1).$ | nvidia |
A circle is inscribed in an isosceles right triangle $ABC,$ where $AB=BC.$ The diameter of the circle is equal to the length of side $AC.$ Find the ratio of the area of triangle $ABC$ to the area of the circle. | nvidia |
Given the geometric sequence:
$\log_{\frac{1}{3}}{27}, \log_{\frac{1}{3}}{9}, \log_{\frac{1}{3}}{3}, ...$
where the $n$th term is $\log_{\frac{1}{3}}{\left(\frac{1}{3}\right)^n}$, find the value of $n$ such that the sum of the first $n$ terms is equal to $-30$. | nvidia |
Determine the quadratic polynomial with real coefficients that has a root of $1 + 2i$ and where the constant term is $13$. | nvidia |
Tetrahedron $ABCD$ has $AD=BC=28$, $AC=BD=44$, and $AB=CD=52$. For any point $X$ in space, suppose $f(X)=AX+BX+CX+DX$. The least possible value of $f(X)$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
| nvidia |
Determine the value of $a+b$ where the value of $\log_{5}{327}$ is between the consecutive integers $a$ and $b$. | nvidia |
In triangle $ABC$, $AC=3\sqrt{2}$ and $BC=6$. Points $A$ and $B$ are on the coordinate axes, and the angle $BAC$ is a right angle. Find the sum of the coordinates of point $C$. | nvidia |
$n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}$. Find $n$. | nvidia |
The sum of two positive integers is 50 and their difference is 12. What is the value of the positive difference of the squares of the integers? | nvidia |
If $a$ and $b$ are positive integers such that the decimal representation of $\frac{a}{b}$ repeats every 6 digits, what is the maximum possible value of $b$? | nvidia |
A rectangular prism is sliced into two rectangular prisms, one of them being a cube. The total surface area of the original rectangular prism is $900$ square centimeters. The ratio between the surface areas of the non-overlapping parts of the two prisms and the ratio between their volumes are both equal to $k$. Given t... | nvidia |
Two dice are rolled. What is the probability that the sum of the numbers on the dice is 9 and that one of the dice shows an even number? | nvidia |
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ w... | nvidia |
Count the number of positive integers less than 500 that are divisors of 1000. | nvidia |
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$. | nvidia |
Determine the largest integer value of $c$ for which the expression $\frac{5x^2+3x+2}{x^2+cx+4}$ has a domain of all real numbers. | nvidia |
Given a function $f(x) = x^2 - kx + 1$, find the average value of $x$ for which $f(x) = 0$, given that the sum of the roots of the equation is equal to the product of the roots. | nvidia |
Determine the smallest integer $n$ such that the inequality $3^n > 4n$ holds. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) + y) = g(x + y) + xg(y) - xy - x + 2\]for all real numbers $x$ and $y.$
Determine the number of possible values of $g(0),$ and find the product of all possible values of $g(0).$ | nvidia |
Let $n$ be a positive integer such that the polynomial $x^2+nx+1$ is divisible by the polynomial $x^2+x+1$. Find the smallest possible value of $n$. | nvidia |
The graph of the function
\[y=\frac{x^2+1}{x^2-1}\]
consists of two separate pieces, as it is undefined at x=1 and x=-1.
Find all points on the graph that are an equal distance from (1,2) and (-1,2). | nvidia |
The sum of the squares of two consecutive even integers is 340. Find their difference. | nvidia |
Let $f(n)$ be the number of distinct prime factors of $n.$ For example, $f(1)=0$ and $f(6)=2.$ Define $T(n)$ by $T(n)=f(1)+f(2)+\cdots+f(n).$ Let $c$ denote the number of positive integers $n \leq 2005$ with $T(n)$ congruent to $1$ modulo $3,$ and let $d$ denote the number of positive integers $n \leq 2005$ with $T(n)$... | nvidia |
Given a triangle with side lengths $a$, $b$, and $c$ that form an arithmetic sequence, find the maximum possible area of the triangle if the sum of the side lengths is 60. | nvidia |
Determine the number of ways to color the vertices of a cube with 4 different colors, such that each face of the cube contains all 4 colors. | nvidia |
Let a sequence $(a_n)$ of natural numbers satisfy the following condition: there exists a natural number $m$ so that $S(n)-S(m)$ is divisible by $n-m$, where $S(n)=a_1+...+a_n$. For example, the sequence of natural numbers $2,4,6,...$ satisfies this condition because $m$ can be taken to be 1. Does the sequence $(a_n)$ ... | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 4.$ Find the minimum value of
\[\frac{x^2}{y} + \frac{y^2}{x}.\] | nvidia |
The product of three different positive integers is equal to $7^3$. What is the sum of the three integers? | nvidia |
Find the length of the shortest path that starts at the origin, goes through the points $(1,2)$ and $(3,4)$, and ends at the point $(5,6)$, where the path consists of line segments that can only be horizontal or vertical. | nvidia |
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 |
The center of a circle is at the point $C(1,-2)$ in the complex plane. The point $A(3,2)$ lies on the circle. Find the equation of the circle. | nvidia |
Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ | 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 |
Problem:
Let $g(x)$ be a function defined by $g(x) = x^2 + 2x$. A sequence $(g_n)$ of functions is defined by $g_1 = g$ and
\[g_n = g \circ g_{n-1}\] for all $n \ge 2$. Let $T$ be the set of all real numbers $x$ such that
\[g_n(x) = -x\] for some positive integer $n$. Determine the number of elements in $T$. | nvidia |
The probability that a randomly selected positive integer $n$ satisfies
$\sin\left(\ds\f{1}{n}\right)=\ds\f{1}{\lfloor n\sin(1)\rfloor}$ is
$\ds\f{a}{b}$
where $\ds\f{a}{b}$ is a fraction in lowest terms.
Find $\lfloor 1000b\rfloor$.
[asy] size(100);
import graph; real f(real x) { return (1/x); } real g(real x) { retur... | nvidia |
Three standard dice are rolled, and the numbers $a,b,c$ are obtained. What is the probability that $a+b+c=9$ and at least two of the numbers are equal? | nvidia |
The expression $\ds\sqrt[3]{3^3+2^3+1^3}+\sqrt[3]{3^2\cdot2+3^2\cdot1+2^2\cdot1}+\sqrt[3]{3\cdot2^2+3\cdot1^2+2\cdot1^2}$ can be simplified to a rational number $m/n$ where $m$ and $n$ are relatively prime. Find $m+n$. | nvidia |
Given the quadratic equation $ax^2+bx+c=0$, determine the relationship between the coefficients $a$, $b$, and $c$ such that the ratio of the sum of the squares of the roots to the product of the roots is a constant value $k$. | nvidia |
Find all solutions $x$ (real and otherwise) to the equation
\[x^4+64=0.\]Enter all the solutions, separated by commas. | nvidia |
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