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Let $S$ be the set of all real numbers $a$ such that $a^2-3\sin(\sqrt{a})\geq 0$. Find $\max\limits_{a \in S} |a|$ | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x+y+z=1$. What is the minimum value of \[\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}?\] | nvidia |
Let $g(x)$ be a monic cubic polynomial with integer coefficients such that $g(x)$ is a divisor of the polynomial $(x+1)^3 + 1$. Determine the smallest possible value of $g(0)$. | nvidia |
Find the area of the triangle with vertices $(0,7,10),$ $(-1,6,6),$ and $(-4,9,6).$ | nvidia |
In $\triangle ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.
$\textbf{(A)}\ 4\qquad \textbf{(B)}\ \sqrt{17} \qquad \textbf{(C)}\ 4.25\qquad \textbf{(D)}\ 2\sqrt{5} \qquad \textbf{(E)}\ 4.5$
| nvidia |
Find the smallest positive integer $m$ such that the product $2^{2005} \cdot 3^{2005} \cdot 4^{2005} \cdots 2005^{2005}$ is congruent to 1 modulo $m$. | nvidia |
There are $36$ different complex numbers $w$ such that $w^{36}=1$. For how many of these is $w^9$ a real number? | nvidia |
Solve the inequality $|x^2-2x+3|<x-1$ for all real numbers $x$. | nvidia |
The diagonal of a cube with twice the surface area of a cube with edge length 8 inches can be expressed in the form $m\sqrt[2]{n}$ where $m$ and $n$ are positive integers and $n$ contains no perfect square factors. Compute $m+n$. | nvidia |
The squares of two consecutive positive integers differ by 43. Determine the smaller integer. | nvidia |
The number of permutations of the digits of the number $1234$ is 24, which can be factorized as $2 \cdot 2 \cdot 2 \cdot 3$ such that 2, 2, 2, and 3 are distinct prime factors. How many other four-digit numbers have exactly four distinct prime factors, $p_1$, $p_2$, $p_3$, and $p_4$, such that $p_1 \cdot p_2 \cdot p_3 ... | nvidia |
Find the value of $a$ such that $\log_{a^2}2x=3$ and $x=9$. | nvidia |
What is the sum of all real numbers that are tripled when added to twice their reciprocals? | nvidia |
Let $f(n)$ be the sum of the proper positive integer divisors of $n$. For how many values of $n$ does $f(n)$ divide $n$, given that $2 \le n \le 100$? | nvidia |
Find the sum of all the even divisors of 420 that are not divisible by 7. | nvidia |
What is the units digit of the sum of the squares of all the even positive integers between 20 and 220? | nvidia |
Find the number of positive integers less than 1000 that are divisible by both 7 and 13.
Note: This problem requires a different approach than the original problem, as it involves finding the least common multiple of 7 and 13, and then counting the number of multiples of that LCM that are less than 1000. | nvidia |
If $m$ and $n$ are positive integers such that $\gcd(m,n) = 12$, then what is the smallest possible value of $\gcd(10m,15n)$? | nvidia |
Find the number of distinct ordered-pair solutions $(m, n)$ of the equation $m^2 + n^2 = 100$, where $m$ and $n$ are non-negative integers. | nvidia |
Let $s_n$ denote the sum of the first $n$ positive integers. How many of the first 100 values of $n$ are such that $s_n$ is a perfect square? | nvidia |
Find all real values of $x$ that satisfy $\frac{x^3-2x^2-5x+6}{x^2-4x+3} \le 0.$ (Give your answer in interval notation.) | nvidia |
Given positive integers $m$ and $n$ such that $\frac{1}{m} - \frac{1}{2n} = \frac{1}{5}$, what is the largest possible value of $m+n$? | nvidia |
Let $f(x)$ be a function such that $f(x+y)=\frac{f(x)f(y)}{f(x)+f(y)}$ for all real numbers $x$ and $y$. If $f(8)=15$, determine $f(4)$. | nvidia |
On the Cartesian plane, let $P_1 = (a_1,b_1), P_2 = (a_2,b_2), \dots , P_n = (a_n,b_n)$ be points that satisfy the conditions:
(1) $0<a_1 < a_2 < \dots < a_n$
(2) For $1\leq k \leq n,$ $b_k > 0$
(3) For $2 \leq k \leq n,$ $a_k - a_{k-1} > b_k+b_{k-1}$
(4) $\sum_{k=1}^{n}b_k = 2003$
For $1\leq k < n,$ let $P_kQ_k$ ... | nvidia |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | nvidia |
Compute the number of ways to arrange 10 students in a row such that the first 5 students are all boys and the last 5 students are all girls. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x + y = 10.$ Find the minimum value of $\frac{1}{x} + \frac{1}{y}.$ | nvidia |
Find the smallest four-digit integer that is divisible by the sum of its distinct, non-zero digits. | nvidia |
Determine the number of integers $m$ such that \[2+\left\lfloor\dfrac{50m}{51}\right\rfloor=\left\lceil\dfrac{49m}{50}\right\rceil.\] | nvidia |
Determine all real numbers x for which $$\left|x-3\right|-\left|x+2\right|=1$$ | nvidia |
Let $x$ be a real number such that $\lfloor x \rfloor + x = 27.85$. Find $x$. | nvidia |
Compute the number of ordered triples of integers $(x,y,z)$ with $1\le x<y<z\le 100$ such that $x^2+y^2+z^2$ is a perfect square. | nvidia |
Find the smallest possible degree of a nonzero polynomial with rational coefficients that has all of the numbers \[1+\sqrt{2}, \; 2+\sqrt{3}, \;3+\sqrt{4},\; \dots, \;1000+\sqrt{1001}\]as roots and also has the additional property that the sum of its coefficients is a positive integer. | nvidia |
Point $P$ lies on the circle $x^2+y^2=16$ and is 5 units from the point $(0,0)$. Find the sum of all possible x-coordinates that satisfy the given conditions. | nvidia |
How many positive integers $n$ less than $500$ can be expressed as the product of $i$ consecutive prime numbers for exactly $3$ values of $i\ge 2$? | nvidia |
A sequence of positive integers $a_1, a_2, a_3,...$ is defined such that $a_n$ is the smallest positive integer that is not the sum of any two distinct previous terms. Find the smallest $n$ such that $a_n > 100$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) + y) = g(x^2 + y) - 2g(x) y\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the product of all possible values of $g(2).$ Find $m + t.$ | nvidia |
Determine all real numbers x such that the equation
(x^2 + 3x + 2)^((x^2-4)/(x+2))= 1
holds true. | nvidia |
Let $T$ be the set of all ordered quadruples of real numbers $(b_1,b_2,b_3,b_4)$ such that $b_1,b_2,b_3,b_4>0$ and $b_1+b_2+b_3+b_4=1$. Each ordered quadruple in $T$ generates a sequence according to the rule $b_n=b_{n-1}\cdot \min(b_{n-2},b_{n-3},b_{n-4})$ for all $n\ge 5$. Find the smallest possible value of $n$ such... | nvidia |
Given that $\frac{1}{a} - \frac{1}{a+2} < \frac{1}{20}$, what is the least possible positive integer value of $a$ such that $a$ is a multiple of $3$? | nvidia |
Find the sum of all positive integer solutions to the equation $x^2+y^2=5^{12}$, where $x$ and $y$ are both divisible by 5. | nvidia |
Find the sum of all values of $z$ that satisfy the equation $\frac{4}{2z^2+5z-2}+\frac{5}{z^2+6z+9}=1$. | nvidia |
The first three positive even composite numbers whose sum is a prime number are a, b, and c, where a < b < c. What is the sum of the units digits of a, b, and c? | 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 |
The number $34!$ ends with $1,\!024$ zeroes. Determine the number of zeroes at the end of the number $55!$. | nvidia |
Given that $\frac{10!}{7!4!}=210$, find the smallest positive integer $m$ for which $\frac{n!}{(n-m)!m!}$ is not divisible by $n$, for some $n$. | nvidia |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? | nvidia |
Let $a$ and $b$ be two positive integers such that $\ds \frac{a+b}{a-b}$ is an integer and $ab=360$. Find the maximum value of $a+b$. | nvidia |
Compute the product of two numbers $a$ and $b$ that differ by 8, knowing that their sum is 100. | nvidia |
A box contains 25 balls, each labeled with a different number from 1 to 25. If two balls are drawn at random without replacement, what is the probability that the sum of the numbers on the two balls is less than 30? | nvidia |
A set of $n$ points in the plane is defined such that each point is connected to the other points by a line segment, and each line segment has a unique length. The points are arranged such that the line segments form a cycle, and the length of each line segment is equal to the sum of the lengths of the two line segmen... | nvidia |
Find the smallest possible four-digit palindrome number $abba$ such that $a$ and $b$ are digits and $a$ is even, and $abba$ is divisible by 9. | nvidia |
Let $Q$ be the product of the first $200$ positive even integers. Find the largest integer $m$ such that $Q$ is divisible by $2^m$. | nvidia |
What is the smallest positive integer $n$ such that $17n \equiv 1234 \pmod{7}$ and $n$ is a multiple of $5$? | nvidia |
The function $f$ defined by $f(x)= \frac{ax+b}{cx+d}$, where $a$,$b$,$c$ and $d$ are nonzero real numbers, has the properties $f(19)=19$, $f(97)=97$ and $f(f(x))=x$ for all values except $\frac{-d}{c}$. Find the unique number that is not in the range of $f$. | nvidia |
Find the absolute value of the difference between the sum of the squares of the first 10 odd numbers and the sum of the squares of the first 10 even numbers. | nvidia |
A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively ... | nvidia |
The polynomial $Q(x),$ with real coefficients, satisfies
\[Q(\cos^2 x) \ge Q(\sin^2 x)\]for all real numbers $x.$ Find the sum of the coefficients of $Q(x).$ | nvidia |
Let $p$ and $q$ be the roots of the quadratic equation $x^{2}-4x+m=0$. Determine $m$ such that $(p-2)^{2}+(q-2)^{2}=24$. | nvidia |
A portion of the graph of $y = g(x)$ is shown in blue below, where $g(x)$ is a quadratic function. The distance between grid lines is $1$ unit.
What is the product of all distinct numbers $x$ such that $g(g(g(x)))=-2$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=... | nvidia |
Let $f(x)$ be a function defined on positive integers that satisfies $f(1)=5$ and
\[\log_{25}(f(n+1))-\log_{25}(f(n))=\frac {n}{n^2+3n+3}\]
for every positive integer $n$. Determine all values of $n$ such that $f(n)$ is an integer. | nvidia |
Find the largest constant $C$ so that
\[|x| + |y| + 1 \ge C|x + y|\]for all real numbers $x$ and $y.$ | nvidia |
Find all real numbers $a$ such that for every real polynomial $p(x)$ of degree $n \ge 2$, the following equation holds:
$$p(x - a) + p(x + a) = p(x) + p(x + 2a)$$
for all real numbers $x$. | nvidia |
Let $M$ be the product of the digits of the number $3210$. What is the smallest integer $n$ such that $Mn$ is a perfect square? | nvidia |
Let $B$ be the set of all numbers which can be represented as the product of three consecutive positive integers. What is the smallest number that is a common factor of all numbers in $B$, but is not a common factor of all positive integers? | nvidia |
What is the smallest number of spheres that can be packed into a cube of side length 12 cm such that each sphere is tangent to at least 4 other spheres and the spheres are all the same size? | nvidia |
Find all real numbers $m$ for which there exists a nonzero, 2-dimensional vector $\mathbf{u}$ such that
\[\begin{pmatrix} 3 & 6 \\ 1 & 2 \end{pmatrix} \mathbf{u} = m \mathbf{u}.\]Enter all the solutions, separated by commas. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x + f(y)) = x - y\]for all real numbers $x$ and $y.$ | nvidia |
The points $A=(1,0)$, $B=(2,0)$ and $C=(p,q)$ are three points on the parabola $y=x^2-3x+2$, where $1\le p\le2$. Find the largest possible area of the triangle $ABC$.
In this problem, we need to find the largest possible area of the triangle $ABC$, where the points $A$ and $B$ are fixed and the point $C$ is variable a... | nvidia |
The coefficients of a polynomial $P(x)$ are all 1, and its roots are all integers. If the number of terms of $P(x)$ is $15$, how many distinct roots can $P(x)$ have? | nvidia |
Let $n$ be a positive integer. Find the largest integer that always divides the expression
\[n^3(n+1)^2 + 2n^2(n+1) + n + 1\] | nvidia |
Find the smallest positive integer $n$ such that $n^3 + 2n^2 + 3n + 1$ is divisible by 7. | nvidia |
The product of two positive integers plus their sum is 95. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Determine the value of $m$ in the quadratic equation $2x^2-5x+m=0$, if it is known that the product of the roots is half the product of the coefficients of the quadratic term and the linear term. | nvidia |
Consider the inequality $x^2 + cx + 3 \leq 0$. How many integer values of $c$ are possible such that there are exactly two integer solutions for $x$? | nvidia |
Let $C$ be a circle with center $O$ and radius $r$. Points $A$ and $B$ are on the circle such that the length of minor arc $AB$ is $\theta$, where $0<\theta<\pi$. Let $C$ be a point on the circle such that $\angle ABC=\angle BCA=\theta/2$. The line $AC$ intersects the circle again at point $D$. If the length of minor a... | nvidia |
The sum of two positive integers is 80 and their difference is 20. What is the value of the positive difference of the cubes of the integers? | nvidia |
A system of two linear equations in three variables has infinitely many solutions. Determine the value of $a$ such that the system
$$\begin{cases}
x-ay+3z=2\\
2x+(a-1)y-3z=3
\end{cases}$$
has infinitely many solutions. | nvidia |
Let $S_n$ be the sum of the first $n$ terms of the geometric series
$$3,\frac{3}{2},\frac{3}{4},\ldots$$Find the smallest value of $n$ for which $S_n$ is an integer. | nvidia |
Find all the values of $x$ such that the number $54x27$ is divisible by $9$, then determine the product of these values. | nvidia |
If the number $1-\sqrt[3]{53}$ can be expressed in the form $\sqrt[3]{a}-\sqrt[3]{b}$, where $a$ and $b$ are positive integers, find $a+b$. | nvidia |
Find the sum of the first 50 positive integers whose square roots are integers. | nvidia |
Determine the largest two-digit positive integer that is not divisible by any prime number less than 10. | nvidia |
What is the smallest possible $x$-coordinate of the graph of the equation \[\frac{(x-1)^2}{25} + \frac{(y-2)^2}{16} = 1,\] if the only points on the graph that are considered are those that also satisfy the inequality \[x^2 + y^2 \le 100?\] | nvidia |
If $x$ is a positive real number such that $x\cdot\{x\}=2$, where $\{x\}$ denotes the fractional part of $x$, find the value of $x$ as a decimal. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(xy) + f(xz) - f(x) f(yz) = 1\]for all real numbers $x,$ $y,$ and $z.$ | nvidia |
An ellipse has its foci at $(2,1)$ and $(2,7)$ and its major axis is parallel to the y-axis. What is the sum of the x-coordinates of the two points on the ellipse that are also on the line x=5? | nvidia |
A sequence $a_n$ of positive integers is defined by the initial values $a_0 = 1, a_1 = 2$ and the recursive formula $a_{n+1} = a_n + a_{n-1}$ for $n \geq 1$. Prove that the number of positive integers less than or equal to 2022 that are not divisible by any of the terms of the sequence is even. | nvidia |
There are 2023 positive numbers with both their product and the product of their reciprocals equal to 2024. Let $y$ be one of these numbers. Find the minimum value of $y \cdot \frac{1}{y}.$ | nvidia |
In a trapezoid, the lengths of the two parallel sides are $12$ and $30$, and the distance between them is $10$. Find the length of the segment that connects the midpoints of the two non-parallel sides. | nvidia |
Find the value(s) of $x$ such that $8xy-12y+2x-3=0$ is true for all values of $y$. | nvidia |
Let $T$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{2}}{2} \le |x| + |y| \le \frac{\sqrt{3}}{2}.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in T$ such that $z^n = -1.$ | nvidia |
Let $\theta$ be an acute angle such that $\sin^n \theta + \cos^n \theta = 1,$ where $n$ is a positive integer. Determine the possible values of $n.$ | nvidia |
Find the area of the triangle with vertices $(3,2,4),$ $(1,1,1),$ and $(2,-1,5)$. | nvidia |
Compute the ordered pair of positive integers $(m,n)$ such that
\begin{align*}
m^n+3&=n^m,\\
4m^n&=n^m+11.
\end{align*} | nvidia |
Let $a,b,c,d,e,f,g$ be real numbers such that \begin{align*}
abc&=def\\
bde&=fgc\\
cdf&=ega.
\end{align*}Find the smallest possible value of
$$(ae)^2+(bf)^2+(cg)^2.$$ | nvidia |
Find the smallest positive integer that has exactly 10 positive integer factors, with the additional constraint that its prime factorization contains only two distinct prime numbers. | nvidia |
Consider the sequence $b_n = \sum\limits_{k=1}^n \cos{k}$, where $k$ represents radian measure. Find the smallest positive integer $n$ such that $b_n > 1$ and $b_{n+1} < 1$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $ab = 1$. Find the minimum value of
\[\frac{1}{a} + \frac{1}{b} + \frac{a}{b} + \frac{b}{a}.\] | nvidia |
Let $n$ be the smallest positive integer such that there exist positive relatively prime integers $a$ and $b$ such that $ab+a+b=n$. Prove that $n$ is prime and determine the number of distinct pairs of integers $(a,b)$ that satisfy the equation. | nvidia |
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