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Let $n$ be a positive integer. Determine the largest value of $n$ such that $1!+2!+\cdots+n!$ is a perfect square. | nvidia |
Consider the inequality
\[\frac{x^2+6x-1}{x^2+x+1}>2.\]
Find the values of $x$ that satisfy this inequality. | nvidia |
Find all real numbers $y$ such that \[2 \le \frac{y}{3y-2} < 5.\](Give your answer in interval notation.) | nvidia |
Find the largest three-digit number that is not divisible by any of the first three prime numbers. | nvidia |
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? | nvidia |
Let $g : \mathbb{N} \to \mathbb{N}$ be a function such that $g(1) = 1$ and
\[g(\text{lcm}(x,y)) = g(x) + g(y)\]
for all positive integers $x$ and $y$ that are relatively prime.
Let $m$ be the number of possible values of $g(12),$ and let $t$ be the maximum value of $g(12).$ Find $m + t.$ | nvidia |
Two pairs $(x,y)$ of real numbers satisfy the equation $x+y=3xy=5$. 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 |
Let $A$ be a set of 10 distinct integers, and let $B$ be a set of 5 distinct integers. How many ways are there to choose 5 integers from $A$ and 2 integers from $B$ such that the 5 integers from $A$ are in increasing order, the 2 integers from $B$ are in increasing order, and the 5 integers from $A$ are all less than t... | nvidia |
Find all real numbers $b$ such that the roots of the polynomial
$$x^3 - 9x^2 + 36x + b$$
form a geometric progression and are not all real. | nvidia |
Let $ABCDEF$ be a convex hexagon such that $\angle FBA = 120^\circ,$ $\overline{AB}\|\overline{DE},$ $\overline{BC}\|\overline{EF,}$ $\overline{CD}\|\overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{-8,-4,0,4,8,12\}.$ The area of the hexagon can be written in the form $m\sqrt {n}... | nvidia |
Find the maximum number of distinct integer values of $c$ such that the quadratic expression $x^2 - 25x + c$ can be factored into the product of two linear binomials with integer coefficients, where the sum of the constants in the binomials is positive. | nvidia |
Let $A$, $B$, and $C$ be the three points on a circle with center $O$, such that $\angle AOB = \angle BOC = 60^\circ$. The radius of the circle is 3 cm. Let $D$ be the midpoint of arc $BC$, and let $E$ be the midpoint of arc $AC$. A square is inscribed in quadrilateral $ABDE$, such that one of its vertices is at point ... | nvidia |
Given a square grid of size $n \times n$, where $n$ is a positive integer greater than 2, find a closed-form expression for the number of ways to color the vertices of the grid using two colors, say black and white, such that no two adjacent vertices have the same color and exactly one vertex remains uncolored. Assume ... | nvidia |
Find all values of $n$ for which there exist unique positive integers $a,$ $b,$ $c$ such that
\[ \left(\sqrt{3} + \frac{1}{\sqrt{3}}\right)^n = a\left(\sqrt{3} + \frac{1}{\sqrt{3}}\right)^{n-1} + b\left(\sqrt{3} + \frac{1}{\sqrt{3}}\right)^{n-2} + c\left(\sqrt{3} + \frac{1}{\sqrt{3}}\right)^{n-3}. \] | nvidia |
Find the largest positive integer $n$ such that $3n^2 + 13n - 31$ is divisible by 7 for all positive integers $n$. | nvidia |
Consider the function $f(z) = z^2 + az + b$, where $a$ and $b$ are complex numbers. Determine all possible values of $a$ and $b$ such that the equation $f(f(z)) = z$ has exactly three distinct complex roots. | nvidia |
[asy] draw(circle((0,8),4),black+linewidth(.75)); draw(circle((0,4),2),black+linewidth(.75)); draw((-2,4)--(0,0)--(2,4),dot); draw((-4,8)--(0,0)--(4,8),dot); MP("C",(-2,4),W);MP("C'",(2,4),E); MP("D",(-4,8),W);MP("D'",(4,8),E); MP("Q",(0,0),S); [/asy]
Two circles are externally tangent. Lines $\overline{QCD}$ and $\ove... | nvidia |
Find the smallest constant $k$ so that
\[x^2 + 2y^2 + 3 \ge k(x - 2y)\]for all real numbers $x$ and $y.$ | nvidia |
Find the product of all $x$ such that the expression $\frac{x^2+2x+1}{x^2+2x-3}$ is undefined. | nvidia |
Find the number of positive multiples of 11 that are less than 2000 and end with the digit 5. | nvidia |
Find the minimum value of
\[\frac{x^2 + 7}{\sqrt{x^2 + 3}}\]over all real numbers $x.$ | nvidia |
Find the smallest positive real number $x$ such that
\[\frac{x - \lfloor x \rfloor}{x} = \frac{1}{\lfloor x \rfloor}.\] | nvidia |
The set of points on a coordinate plane is given that are all the same distance away from the two points (2,3) and (8,7). This set of points is not just a straight line on the coordinate plane.
How many integer x-coordinates do the points on the coordinate plane have that are all equidistant to points (2,3) and (8,7)? | nvidia |
Given the function $f(x)=2x^3-5$, find the value of $x$ such that $f^{-1}(x)=2$. | nvidia |
Find the smallest positive integer $n$ such that $n^2+1$ is divisible by exactly three distinct prime factors. | nvidia |
Given a quadratic equation $ax^2+bx+c=0$, the sum of the squares of its solutions can be found using the formula:
$$x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2=\left(\frac{-b}{a}\right)^2-2\frac{c}{a}.$$
Using this formula, the sum of the squares of the solutions of the given equation $x^2-13x+4=0$ is:
$$x_1^2+x_2^2=(13)^2-2(4)... | nvidia |
Determine the largest possible degree of the polynomial $p(x)$ such that the graph of the rational function $\frac{p(x)}{x^4-3x^2+2}$ has a slant asymptote. | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a + b + c = 24.$ Find the largest possible value of $b.$ | nvidia |
Find the product of the values of $y$ that satisfy the equation $y^2 - 1992y + 1993 = 0$, given that one of the solutions is an integer. | nvidia |
Find the smallest positive integer whose sum of factors is equal to twice the number itself. | nvidia |
Let $a$ and $b$ be positive integers. Determine the smallest value of $ab$ such that the equation $x^2+ax+2b=0$ has rational roots and $a+b$ is a perfect square. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 4.$ Find the minimum value of
\[x + \frac{xy}{2} + \frac{xyz}{4}.\] | nvidia |
Let the function $f$ be defined by $f(x)=|x-1|-2x$ and let $g$ be a linear function that satisfies $g(0)=1$ and $g(1)=f(1)$. Find the product of all $x$ such that $g(x)=f(x)$. | nvidia |
Given a regular hexagon, find the length of its side if its vertices are located on the lattice points of a coordinate plane with integer coordinates and all the coordinates of the vertices are between 0 and 10. | nvidia |
Find the number of positive integers less than or equal to 100 that have a prime factorization with exactly two distinct prime factors. | nvidia |
The sides of rectangle $PQRS$ have lengths $12$ and $15$. A right isosceles triangle is drawn so that no point of the triangle lies outside $PQRS$. Find the maximum possible area of such a triangle. | nvidia |
Find the sum of all the positive integer divisors of 15. | nvidia |
The function $g$ is defined on the set of positive real numbers and satisfies \[g(x)= \begin{cases} 2x & \mbox{if }x\le 1 \\ g(g(x^{-1})) & \mbox{if }x>1. \end{cases}\]Find $g(2)$. | nvidia |
Find the area of the region enclosed by the parabola $y^2=x$ and the line $x+y=4$. | nvidia |
Find all triples $(a,b,c)$ of integers such that $a<b<c$, $a,b,c>1$ and $c^3-a^3=b^3$. | nvidia |
Let $f(x)=ax^2-bx+c$ be a quadratic function that intersects its own inverse, graphed on the same axes, at exactly two distinct points. Determine the largest possible value of $b$. | nvidia |
In the equation
$$x^2-kx+1=0$$
determine $k$ such that the sum of the roots is equal to 5.
Note: The new problem requires a different approach as it asks for the sum of the roots instead of the difference, and the equation is a quadratic equation instead of a rational equation. | nvidia |
The product of the first five positive odd integers is 945. What is the value of the product of the first five positive even integers divided by the product of the first three positive even integers? | nvidia |
Let $a, b,$ and $c$ be positive real numbers such that $abc = 64.$ Find the minimum value of $a^2 + b^2 + c^2.$ | nvidia |
Three consecutive integers are such that the sum of the integer and the square of the next integer is 46 less than twice the square of the third integer. What is the value of the smallest integer? | nvidia |
A square and a regular hexagon are inscribed in the same circle. The side length of the square is $12$ cm. Find the area of the hexagon. | nvidia |
Find a positive integer $m$ such that $m!$ is divisible by $2^{10} \cdot 5^3$. | nvidia |
Let $x$, $y$ and $z$ be three positive real numbers such that $xyz=8!$. Compute the smallest possible value of
$$\frac{1}{x}+\frac{2}{y}+\frac{3}{z}.$$ | nvidia |
Find the largest three-digit number that is divisible by each of the integers from 1 to 6. | nvidia |
What is the smallest positive integer $n$ such that $3n \equiv 1356 \pmod{22}?$ | nvidia |
The area of the triangle $ABC$ is 32 square centimeters. The area of the triangle $ABD$ is 24 square centimeters. The area of the triangle $ACD$ is 25 square centimeters. What is the area of the triangle $BCD$? | nvidia |
The numbers $a,$ $ar,$ and $ar^2$ form a geometric sequence, in that order. Find all values of $r$ such that $\sin a,$ $\sin ar,$ and $\sin ar^2$ form an arithmetic sequence, in that order. | nvidia |
Given that $x+\frac{1}{x}=-5$, find the value of $\sin^2{x}+\cos^2{x}$. | nvidia |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a + b + c = 21.$ Find the largest possible value of $b.$ | nvidia |
Find the largest integer $n$ for which $12^n$ evenly divides $20!$. | nvidia |
How many different convex pentagons are there with integer sides and perimeter equal to 35, such that at least one of the internal angles is a right angle? | nvidia |
Suppose $x$ and $y$ are positive integers such that $\gcd(x,y)$ is divisible by exactly $3$ distinct primes and $\mathop{\text{lcm}}[x,y]$ is divisible by exactly $9$ distinct primes.
If $x$ has fewer distinct prime factors than $y$, then $x$ has at most how many distinct prime factors? | nvidia |
A regular polyhedron has a total of 540 degrees in the sum of its interior angles. If each face of the polyhedron is a regular polygon, and the number of sides of each face is equal to the number of faces of the polyhedron, determine the number of faces of the polyhedron. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 1.$ Find the maximum value of
\[\frac{abc}{a^2 + b^2 + c^2}.\] | nvidia |
Let $x$ and $y$ be real numbers such that $\frac{x}{y} + \frac{y}{x} = 4$. Find all possible values of $\frac{x^3}{y^3} + \frac{y^3}{x^3}.$ | nvidia |
A positive integer is called "prime-like" if it has exactly three distinct prime divisors. Determine the largest "prime-like" number less than 1000. | nvidia |
Find the number of positive integers greater than 10 but less than 100 that are relatively prime to 21. | nvidia |
What is the largest positive integer $n$ for which $n!$ can be expressed as the product of two or more consecutive positive integers? | nvidia |
Find the minimum value of
\[\frac{x^2 + 7}{\sqrt{x^2 + 3}}\]over all real numbers $x.$ | nvidia |
Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $9$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
| nvidia |
Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number. | nvidia |
Find the sum of all positive integral values of $n$ such that $n^3-2n^2+5n+1=m^2$ for some perfect square $m^2$. Note that there is at least one such $n$. | nvidia |
Problem:
Consider a polygon in the complex plane formed by the points 0, $z$, $2z$, $3z$, $\ldots$, $(n-1)z$, where $n$ is a positive integer greater than or equal to 3. The area of the polygon is known to be $\frac{n(n-1)}{2} |z|^2 \sin \theta$, where $\theta$ is the angle between the positive real axis and the line s... | nvidia |
Let $g(x)$ be a function such that $g(0) = 0$ and
\[g(xy) = g \left( \frac{x^2 + y^2}{2} \right) + (x + y)^2\]for all real numbers $x$ and $y.$ Find $g(x).$ | nvidia |
If $a$ and $b$ are complex numbers such that $a\neq 0$, $|a|=1$, and $|b|=\sqrt {5}$, compute $|a^2b^3a^{-1}|$. | nvidia |
Let $a$, $b$, $c$, $d$ be positive integers such that $a>b>c>d$. Compute the value of $(a^3-b^3)-(c^3-d^3)$, given that $a^2-b^2=c^2-d^2$. | nvidia |
Let $T$ be the set of all real values of $x$ with $0 < x < 1$ such that $x$, $\sqrt{1-x^2}$, and $\frac{x}{\sqrt{1-x^2}}$ form the side lengths (in some order) of an isosceles triangle. Compute the sum of $\frac{x^2}{1-x^2}$ over all $x$ in $T$. | nvidia |
There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is the value of $b?$ | nvidia |
The product of two positive integers is 48. The positive difference of these two integers is 4. What is the sum of the two integers? | nvidia |
Consider the function
$$F(x)=\begin{cases}
\sqrt{x}+2, & \text{if } x\geq 0\\
-1, & \text{if } x<0.
\end{cases}$$
Determine all values of $x$ for which $F(F(F(x)))$ is a real number. | nvidia |
A regular pentagon with side length 1 is inscribed in a circle. Points $A_0, A_1, A_2,...$ lie on the circle, such that for every positive integer $n$, the angle $A_{n-1}O A_n$ is a right angle, where $O$ is the center of the circle. Furthermore, for every positive integer $n$, the quadrilateral $A_{n-1}BA_n A_{n+1}$ i... | nvidia |
In a sequence of five consecutive positive integers, the sum of the squares of the integers is equal to the sum of the cubes of the integers. What is the middle integer? | nvidia |
What is the sum of all positive integers $\nu$ for which $\mathop{\text{lcm}}[\nu,15]=90$? | nvidia |
Consider the expansion of $(x^2 + 2x + 1)^n$. Prove that the coefficient of the $x^{2n}$ term is always equal to 1, regardless of the value of $n$. | nvidia |
Let $x$ and $y$ be positive integers such that $xy = 24.$ Find the minimum value of $x + 3y.$ | nvidia |
Determine all possible values of $y$ for which the equation $(y^3-3y+1)^{\sqrt{2-y}}=1$ holds true. | nvidia |
Find the smallest positive integer that has four different prime factors, with the additional condition that the sum of these prime factors is a perfect square. | nvidia |
Determine the sum of all single-digit replacements for $w$ such that the number ${32{,}w16}$ is divisible by 4. | nvidia |
What is the smallest positive integer $m$ such that $m^2$ is equal to the product of the squares of two distinct prime numbers and can be expressed as the difference of two consecutive cubes? | nvidia |
For real numbers $x$ and $y$ such that $x+y>4$, find the minimum value of
\[\frac{x^2+y^2}{x+y-4}.\] | nvidia |
What is the product of all positive integer divisors of 108 that are also perfect squares? | nvidia |
Find the largest real number $c$ such that \[x_1^2 + x_2^2 + \dots + x_{101}^2 \geq cM^2\]whenever $x_1,x_2,\ldots,x_{101}$ are real numbers such that $x_1+x_2+\cdots+x_{101}=0$ and $M$ is the median of $x_1,x_2,\ldots,x_{101}.$ | nvidia |
The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers? | nvidia |
The function f(x) satisfies f(x+y) = f(x) + f(y) for all real numbers x and y. Find all possible values of f(0). | nvidia |
Determine the number of distinct, natural-number factors of $n$ that are also multiples of $k$, given that $n=2^3 \cdot 3^2 \cdot 5$ and $k=2^2 \cdot 3$. | nvidia |
Let $f(x)$ be a polynomial such that for all real values of $x$ except $-2$ and $3$ the following equality holds:
\[\frac{f(x)}{(x+2)(x-3)} = \frac{1}{x-3} + \frac{x^2-2x-5}{x^2-x-6}.\]Find $f(-1)$. | nvidia |
Let $a$, $b$, and $c$ be three positive integers such that $a$ is the arithmetic mean of $b$ and $c$. If the sum of all possible values of $a$ is 15, and $a^2$ is equal to the sum of the squares of $b$ and $c$, find the largest possible value of $a^2$. | nvidia |
How many four-digit numbers with distinct digits are there such that the sum of the digits is divisible by 3? | nvidia |
Find the product of two real numbers $\sqrt{2^5\cdot 3^7}$ and $\sqrt{3^5\cdot 2^7}$, and then simplify the result. | nvidia |
A triangle has three sides of the following side lengths: $9$, $15$, and $y^3$. What are all of the positive integer values of $y$ such that the triangle exists? Separate your answers using commas and express them in increasing order. | nvidia |
Given that the sum of the roots of $x^2 - 4x + 3 = 0$ is 4, find the value of $k$ such that the sum of the roots of $x^2 - kx + 3 = 0$ is twice the sum of the roots of $x^2 - 4x + 3 = 0$. | nvidia |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | nvidia |
Let p be a prime such that for some 1<m<n
p^m + p^n + p^(m + n) are all perfect squares. Show that there is a common positive integer that is a factor of all three squares greater than 1 and find that common factor. | nvidia |
Find the sum of all positive integers that divide both numbers $$432, 576.$$ | 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 |
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