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Given that the graph of $y = f(x)$ is an upside-down parabola with vertex $(2,3)$ and $x$-intercept at $(4,0)$. Find the value of $k$ for which the graph of $y=f(x)+k$ intersects the $x$-axis at exactly one point. | nvidia |
Let $\omega$ be a complex number such that $\omega^3=1$ and $\omega\neq 1$. If $x$ and $y$ are real numbers such that
$$x(\omega+1)+y\omega=1,$$
determine the value of $x+y$. | nvidia |
Find all three-digit numbers with different digits such that the number is divisible by the sum of its digits. | nvidia |
Find the number of positive integers less than 30 that are perfect numbers, where a perfect number is a positive integer equal to the sum of its proper factors, excluding the number itself. | nvidia |
Let $g$ be a function such that $g(x^2+x+1)=3x-1$ for all real numbers $x$. Find $g(5)$. | nvidia |
Let $f(n)$ denote the number of distinct prime factors of a positive integer $n$, where $f(1)=0$. For example, $f(2)=1$ and $f(6)=2$. Define $T(n)$ by $T(n)=f(1)+ f(2) + \cdots + f(n).$ Determine the smallest positive integer $n$ such that $T(n)$ is a perfect square. | nvidia |
Let $n$ be a positive integer. Consider the function $f(n)$ defined as the sum of all divisors of $n$. For which positive integers $n$ does the function $f(f(n))$ always divide $n^2+n$? | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-2bx+b^2-1=0$$
are such that
$x_1^2+x_2^2=64$.
Determine $b$. | nvidia |
Consider two monic quadratic polynomials $f(x)$ and $g(x)$ and a real number $r$. The roots of $f(x)$ are $r+1$ and $r+2$, while the roots of $g(x)$ are $r+3$ and $r+4$. Additionally, we have the relationship $f(x^2)-g(x^2)=r$ for all real numbers $x$. Find the value of $r$. | nvidia |
A hyperbola has its center at the point $(3,2)$ and one focus at the point $(2,3)$. The vertex of the hyperbola is at the point $(3,3)$. Find the equation of the asymptotes of the hyperbola. | nvidia |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 50\}$ have a perfect square factor other than one?
Find the number of positive integers less than or equal to 100 that can be expressed as the sum of two perfect cubes. | nvidia |
Let $f(x)=|2x+3|-|x+4|$.
Find the sum of all the values of $x$ where the graph of $y=f(x)$ intersects the line $y=x-6$. | nvidia |
Find the largest integer value of $k$ such that the quadratic expression $kx^2+4x-7$ is negative for all real values of $x$ in the interval $(2, 5)$. | nvidia |
Determine all real numbers $a$ such that $a=e^{a-1}$. | nvidia |
What is the largest integer that must divide the sum of the squares of any 5 consecutive integers? | nvidia |
Find the smallest positive integer $b$ such that $x^3 - b^2$ is not a perfect square for any integer $x$. | nvidia |
How many different positive integers can be represented as the sum of two distinct members of the set $\{1, 2, 3, \ldots, 10, 11, 12\}$? | nvidia |
The ellipse $\frac{x^2}{4}+\frac{y^2}{5}=1$ has two vertices with different y-coordinates. Find the coordinates of the vertex with the smaller y-coordinate. | nvidia |
Let $a_1,$ $a_2,$ $\dots,$ $a_{12}$ be positive real numbers such that $a_1 \cdot a_2 \cdot \dots \cdot a_{12} = 1.$ Find the minimum value of
\[a_1 + a_2 + \dots + a_{12}.\] | nvidia |
Find the sum of the rational roots of $g(x)=x^3-9x^2+16x-4$. | nvidia |
Let $\alpha$ and $\beta$ be two positive real numbers such that $\alpha + \beta = 2$.
Determine the least possible value of $\alpha$ given that $\sqrt{\alpha} - \sqrt{\beta} < \frac{1}{2}$. | nvidia |
Find the number of integer values of $n$ between 1 and 99 inclusive for which the decimal representation of $\sqrt{n}$ is a terminating decimal. | nvidia |
What is the sum of all of the solutions of the equation $\frac{4x}{20}=\frac{5}{x}$? | nvidia |
Problem:
The sum of the squares of the real and imaginary parts of the complex number $z=a+bi$ is equal to 145. If the argument of $z$ is equal to $\frac{\pi}{4}$, find the value of $a$. | nvidia |
The points $A$ and $B$ are located in a coordinate system at the points $(6,0)$ and $(0,8)$ respectively. Find the area of the rectangle with the sides parallel to the axes of the coordinate system and with the vertices at the points $A$ and $B$. | nvidia |
Find the number of ordered pairs of positive integers $(a,b)$ with $a<b$ for which the geometric mean of $a$ and $b$ is equal to $2^{15}$. | nvidia |
Find an integer m that satisfies 0 ≤ m < 23 and
$$
73192 \equiv m \pmod{23}
$$ | nvidia |
What is the largest value of n such that there exists an n-digit positive integer that has no 1's or 2's as digits? | nvidia |
Determine the number of positive integers $n$ such that $n,$ $n+1,$ and $n+2$ form a geometric sequence in some order. | nvidia |
Determine the sum of the cubes of the first 2007 positive numbers whose units digit is either 1 or 7. | nvidia |
Find the largest value of $a$, if $a$ and $b$ are positive integers such that the equation
$$\frac{7}{13}=\frac{b}{5a-b}$$
holds true. | nvidia |
Find all the positive integral solutions of the equation $x^3 + y^3 = z^3$ such that $x^2 + y^2 - z^2 = 0$. | nvidia |
Determine the smallest positive integer $n$ such that $3^{n+1}$ leaves a remainder of 1 when divided by 41. | nvidia |
Find the sum of all the even numbers in the interval [1, 23] that are not divisible by 3. | nvidia |
Find the smallest positive integer $n$ such that the sum of the digits of $F_n$, the $n$th Fibonacci number, is divisible by 7. | nvidia |
Let $a$ and $b$ be nonnegative real numbers such that
\[\cos (ax + b) = \cos 23x\]for all integers $x.$ Find the smallest possible value of $a.$ | nvidia |
The integers $M$ and $N$ are chosen such that
\[\frac{M}{x-3}+\frac{N}{x^2+2x}=\frac{x^2+5x-2}{x^3-x^2-6x}\]for all real values of $x$ except $3$, $0$, and $-2$. Find $N/M$. | nvidia |
Two circles with centers $O_1$ and $O_2$ intersect at points $C$ and $D$. The measure of $\angle O_1CO_2$ is $70^\circ$, and $\angle O_2DO_1$ is $40^\circ$. Find the measure of $\angle CAB$, where $A$ and $B$ are points on the circles such that $AB$ is tangent to both circles at $A$ and $B$ and intersects line $O_1O_2$... | nvidia |
Suppose the set $X=\{1,2,\dots,100\}$ is randomly partitioned into two subsets $A$ and $B$, such that for every $k\in\{1,2,\dots,50\}$ there is exactly one number in $A$ that is congruent to $k$ modulo $50$.
Find the expected value of the number of elements in $A$ that are divisible by 25. | nvidia |
Find the smallest three-digit prime number such that reversing the digits of the number produces a palindrome. | nvidia |
An equilateral triangle with side length 6 is inscribed in a circle and the circle is circumscribed by a square. Find the area of the region outside the triangle and inside the square. | nvidia |
A rectangular table has 6 rows and an unknown number of columns. The table is filled with two types of candies, white and black, such that each cell contains exactly one candy. The candies are arranged in such a way that the number of pairs of adjacent cells (horizontally or vertically) with the same color is equal to ... | nvidia |
Let a circle centered at the point $B$ intersect a
circle centered at the point $F$. Determine the radius of the circle centered at $B$ if the radius of the circle centered at $F$ is equal to $2$, and the distance between the centers of the circles is equal to $\sqrt{6}$,
and the tangent drawn to the circle centered at... | nvidia |
What is the smallest prime factor of the sum of the cubes of the first $n$ positive integers, where $n$ is a positive integer greater than 1? | nvidia |
Let $\diamond (x)$ be the sum of the digits of a positive integer $x$. $\mathcal{T}$ is the set of positive integers such that for all elements $n$ in $\mathcal{T}$, we have that $\diamond (n)=9$ and $0\le n< 10^{5}$. If $m$ is the number of elements in $\mathcal{T}$, compute $\diamond(m)$. | nvidia |
Let $n$ be a positive integer such that $n$, $n + 1$, $n + 2$, and $n + 3$ are all composite. What is the smallest possible value of $n$? | nvidia |
The lengths of the sides of a triangle with positive area are $\log_{10}12$, $\log_{10}75$, and $\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$. | nvidia |
In the equation $ax^2+bx+c=0$, the sum of the roots is equal to the product of the roots, and the value of $a$ is 3. If the equation has real and distinct roots, what is the value of $b+c$? | nvidia |
The sum of the squares of two consecutive even numbers is 100. Find the difference between the numbers. | nvidia |
In a triangle, the sum of the lengths of the three sides is 36 cm, and each side has an integer length. How many non-congruent triangles meet these criteria, assuming that the triangle inequality holds for all three sides (i.e., the sum of the lengths of any two sides is greater than the length of the third side)? | nvidia |
Find the smallest positive integer that can be written in the form $1000a + 7000b$, where $a$ and $b$ are integers such that $a$ is odd and $b$ is even. | nvidia |
A square is inscribed in a regular hexagon, with the vertices of the square located at the midpoints of the sides of the hexagon. Find the matrix that represents a rotation of the square by 120 degrees counterclockwise about its center. | nvidia |
Find the sum of all even integers between 21 and 101. | nvidia |
What is the smallest positive integer $n$ such that $n!$ is divisible by $2^n$? | nvidia |
A polynomial with integer coefficients is of the form
\[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 12 = 0.\]Find the number of different possible rational roots of this polynomial, given that the sum of the roots is equal to -1. | nvidia |
The local library has two types of books: fiction and non-fiction. The number of fiction books on a shelf is at most three times the number of non-fiction books. Additionally, the total number of books on the shelf is between 30 and 50, inclusive. If the library adds 15 more fiction books to the shelf, the number of fi... | nvidia |
Find all real values of $x$ that satisfy $\frac{x^2+1}{x+1} \ge \frac{3x-1}{x-1}$. (Give your answer in interval notation.) | nvidia |
Problem:
Find the sum of all complex numbers $z$ such that $|z+1-i|=|z-1+i|$. | nvidia |
Find the value of $c$ for which the equation $x^2 + 2x + c = 0$ has two solutions that differ by exactly $4$ units. | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
What is the sum of all integer values of $N$ such that $\sqrt{N}$ is strictly between $3\sqrt{2}$ and $4\sqrt{3}$? | nvidia |
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | nvidia |
Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 0.$ Find the set of all possible values of $ab + ac + bc.$ | nvidia |
Determine the largest possible degree of the polynomial $p(x)$ such that the graph of the function $p(x)/x^4$ has no vertical asymptotes. | nvidia |
Let $x$ and $y$ be positive real numbers such that
\[\frac{1}{x + 3} + \frac{1}{y + 3} = \frac{1}{4}.\]
Find the minimum value of $x + 3y$. | nvidia |
Find the sum of all integers between 300 and 500 that are multiples of 5. | nvidia |
Given that $f(x) = \frac{a}{x^2 - 1}$, where $a$ is a constant, find the product of all possible values of $a$ such that the point $(2, f(2))$ lies on the graph of $f^{-1}$. | nvidia |
The hyperbola $x^2-y^2=1$ and the ellipse $4x^2+y^2=4$ intersect at two points $A$ and $B.$ Find the area of the triangle formed by points $A$, $B$ and the origin $O$. | nvidia |
Find all values of $x$ such that $\arctan x > \arctan \frac{1}{x}$. | nvidia |
Find all real values of $x$ such that $-5 < x^3 + 2x^2 - 3x + 1 < 5$. | nvidia |
The function $f(x)$ satisfies the equation
\[f^2(x) + g(x) = -4g(-x)\]for all real numbers $x.$
Let $M$ be the maximum value of $f(x)g(x)$ and let $m$ be the minimum value of $f(x)g(x).$
Find $M - 3m.$ | nvidia |
A box contains three different colored balls: red, blue, and green. The balls are drawn one by one, and after each draw, the ball is returned to the box. The probability that the red ball is drawn is $\frac{1}{7}$, the probability that the blue ball is drawn is $\frac{2}{7}$, and the probability that the green ball is ... | nvidia |
A line passes through the points $(-2,7)$ and $(1,-1)$. Find the coordinates of the point on this line that is equidistant from the points $(3,0)$ and $(0,5)$. | nvidia |
Let $f(x)$ be a function defined by $f(x) = \frac{x+1}{x-1}$ for all real numbers $x\neq 1$. Find the value of $f(f(f(2)))$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 1.$ Find the minimum value of $x + y + z.$ | nvidia |
Let $g(n)$ be the smallest positive integer $m$ such that $\frac{1}{m}$ has exactly $n$ zeros before the decimal point. How many positive integer factors does $g(2005)$ have? | nvidia |
The sum of two consecutive odd integers is 24, and the larger integer is twice the smaller. What are the integers? | 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 |
The function $g(x)$ is defined by $g(x)=\sin(x)-2x$. Find the value of $x$ such that $g(x)=0$. | 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 |
If $x$ and $y$ are real numbers such that $x+y=3$ and $x^3+y^3=27$, find the value of $x^2+y^2$. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x+y) + f(xz) - f(x) f(y+z) \le 2\]for all real numbers $x,$ $y,$ and $z.$ | nvidia |
Find the value of $a$ such that $(a^2 + 1)^4 = 4^4 \cdot 5^4$. | nvidia |
A pyramid with a square base is given. The length of the base is $a$, and the altitude is $h$. A cylinder whose base diameter is equal to $2a$ is cut in half and is placed on the base of the pyramid so that the two bases are parallel. Find the ratio of the volume of the cylinder to the volume of the pyramid if the top ... | nvidia |
Find the smallest positive integer that is a multiple of 7 and has exactly 5 positive divisors. | nvidia |
A committee of five members is formed from a group of ten students, with each student having an equal chance of being selected. Let $p$ be the probability that all five members are from the same school, assuming that there are two schools with five students each. Let $q$ be the probability that three members are from o... | nvidia |
Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed.
| nvidia |
Five teams are participating in a tournament. Each team plays exactly two games against each of the other teams. Find the largest possible number of teams that win exactly two games. | nvidia |
Find all real solutions to $x^4+(2-x)^4=34$. Enter all the solutions, separated by commas. | nvidia |
Find the product of all real numbers that are tripled when added to twice their reciprocals. | nvidia |
Determine the smallest positive integer $k$ such that for any integer $n,$ $0 \le n \le 180,$ $\cos (n^\circ + k^\circ) = \cos n^\circ.$ | nvidia |
The domain of the function $f(x) = \arccos(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2025}$, where $m$ and $n$ are positive integers and $m>1$. Find the the smallest possible value of $m+n.$ | nvidia |
Find the number of all non-congruent squares modulo 12 among the first 500 positive integers. | nvidia |
What is the sum of all real numbers that are tripled when subtracted from their reciprocals? | nvidia |
Find the sum of all positive divisors of 18 that are also divisors of 24. | nvidia |
Find the values of $a$ and $b$ such that $\frac{x-a}{x+b} = \frac{x^2-54x+623}{x^2+23x-270}$, and then compute $a+b$. | nvidia |
A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes? | nvidia |
Let $A$ be the set of all 3-digit numbers that are divisible by 5 but not by 7. Let $B$ be the set of all 3-digit numbers that are divisible by 7 but not by 5. Which set has more elements? | nvidia |
The product of the first $n$ positive integers is $n!$. The product of the first $n$ positive even integers is $2^n \cdot n!$. The product of the first $n$ positive odd integers is $\frac{(2n)!}{2^n \cdot n!}$. Find the product of the first $n$ positive integers that are multiples of 3. | nvidia |
Given a positive integer $n$, find the least possible value of $x+y$ such that $\frac{x}{n} + \frac{y}{2n} = \frac{1}{3}$ and $x$, $y$ are positive integers. | nvidia |
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