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There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
Consider a function $g$ that satisfies the equation $g(x) + g(-x) = |x|$ for all real numbers $x$ in its domain. What is the largest set of real numbers that can be in the domain of $g$? | nvidia |
Find the largest integer $n$ such that the equation \[\lceil x \rceil^n - \lfloor x \rfloor^n = 2023\] has a solution $x$ in the interval $(0, 1)$. | nvidia |
Find the largest three-digit number that is a multiple of 12 and has exactly three digits that are prime numbers. | nvidia |
Let $f(x)=x^2+\frac{1}{x^2}+ax+b$, where $a$ and $b$ are real numbers, be a function defined for all $x\neq 0$. Given that $f(1)$, $f(-1)$, $f(i)$, $f(-i)$ are all integers, what is the smallest possible value of $|a|+|b|$? | nvidia |
Consider the function
$$f(x) = \frac{1}{\frac{1}{x^2} + \frac{1}{x} + \frac{1}{x+1}}$$
and determine the product of all the real numbers $x$ that are not in the domain of $f(x)$. | nvidia |
Consider a sequence of five positive integers $a$, $b$, $c$, $d$, $e$, such that $a+b+c+d+e=2010$, where the sum of any four consecutive terms in this sequence is always divisible by 3. What is the minimum possible number of distinct values among the elements $a$, $b$, $c$, $d$, $e$? | nvidia |
Determine the largest possible real value of $x^3-6x^2+6x+2$. | nvidia |
A certain phone company produces phone numbers consisting of seven digits. Each digit can be one of the numbers from 0 to 9. However, the phone company has a restriction that no two digits in a phone number can be the same. Find the number of possible phone numbers the company can produce. | nvidia |
Let $a, b, c$ be real numbers such that
\begin{align*}
a+b+c&=3,\\
a^2+b^2+c^2&=5,\\
a^3+b^3+c^3&=13.
\end{align*}
Determine $ab+bc+ca$. | nvidia |
Find the smallest positive integer k such that 12! + k \* 11! is divisible by 13. | nvidia |
Solve
$$\frac{2x+4}{x^2+4x-5}=\frac{2-x}{x-1}$$
for $x$.
Consider the following problem:
Let $a$ and $b$ be two positive real numbers such that their arithmetic mean is 5 and their geometric mean is 3. Find the sum of the roots of the quadratic equation
$$(a+1)x^2 + (b-1)x + 1 = 0$$ | nvidia |
There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? | nvidia |
Find the number of three-digit positive integers that have at least one digit repeated.
Note: This problem requires a different approach than the original one, as it involves finding the number of integers with at least one repeated digit, rather than counting the number of integers with only two specific digits (2's ... | nvidia |
Let $a,$ $b,$ and $c$ be three positive real numbers whose sum is $2$ and $a^2+b^2+c^2=3.$ If no one of these numbers is more than twice any other, then find the maximum value of the sum $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. | 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 |
How many different rectangles have integer side lengths and an area of 120 square units, such that the length of the diagonal is an integer as well? | nvidia |
Let $S$ be the set of all points in the plane that are a fixed distance $d$ from a given point $P$. A triangle $ABC$ with $AB=40$, $BC=50$, and $CA=30$ has its circumcenter $O$ lying in the set $S$. If $P$ is equidistant from the three vertices of the triangle, determine $d$. | nvidia |
What is the sum of all of the two-digit primes that are greater than 12 but less than 99 and are still prime when their two digits are interchanged? | nvidia |
Find the greatest value of $x$ such that $|x^2-3x+1| \le 2$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $2x + 3y = 4.$ Find the minimum value of
\[\frac{2}{x} + \frac{3}{y}.\] | nvidia |
Let $n$ be a positive integer such that $n^3 + 1$ is divisible by 10. What is the largest possible value of $n$? | nvidia |
A cube is randomly inscribed in a sphere such that the vertices of the cube touch the surface of the sphere. What is the probability that no three vertices of the cube form an obtuse angle at the center of the sphere? | nvidia |
Let $a$ and $b$ be real numbers. If the polynomial $x^2 + ax + b$ has exactly one real root and $a = 2b - 1$, find the value of the sum of all possible values of $b$. | nvidia |
Determine the base of the number system in which the equation $32_{b}+15_{b}=47_{b}$ holds, if the sum of the digits in each term is equal to the number representing the term. | nvidia |
Determine the sum of all positive integer values of $a$ such that the quadratic equation $6x^2 - 7x + a = 0$ has two distinct real roots. | nvidia |
Let $x$, $y$, and $z$ be positive numbers such that $xyz = 1$. Determine the minimum value of
$x+y+z+\ds\f{1}{x}+\ds\f{1}{y}+\ds\f{1}{z}.$ | nvidia |
Let $P$ be a point inside a unit square such that $P$ is equidistant from the sides of the square. Let $Q$ and $R$ be points on adjacent sides of the square such that triangle $PQR$ is an obtuse triangle with $QR$ as its longest side. Find the minimum area of triangle $PQR$. | nvidia |
Compute the value of x such that
$$\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots$$.
Determine the sum of the infinite series
$$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$$
and use this result to co... | nvidia |
Assume that $y_1,y_2,\ldots,y_6$ are real numbers such that
\[\begin{aligned} y_1+9y_2+25y_3+49y_4+81y_5+121y_6 &= 3 \\
9y_1+25y_2+49y_3+81y_4+121y_5+169y_6 &= 20 \\
25y_1+49y_2+81y_3+121y_4+169y_5+225y_6 &= 220. \end{aligned}\]
Find the value of $49y_1+81y_2+121y_3+169y_4+225y_5+289y_6$. | nvidia |
A positive real number $y$ is such that \[
\sqrt[4]{1-y^4} + \sqrt[4]{1+y^4} = 2.
\]Find $y^8$. | nvidia |
Find the sum of all real values of $m$ for which $\frac{3}{5x} = \frac{m - 2x}{9}$ has exactly one real solution. | nvidia |
A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the square to the area of the circle? | nvidia |
In a triangle $ABC$, the angle bisectors of angles $A$ and $B$ intersect at point $I$. The side lengths $AB$, $AC$, and $BC$ are positive integers such that $AB < AC < BC$. If $AB=12$, what is the smallest possible value of $BC$ such that no circle can be inscribed in triangle $ABC$? | nvidia |
Find the number of positive integers $n$ such that $2n^3 + 13n^2 - 17n - 8$ is a perfect cube. | nvidia |
Let $a$ and $b$ be two positive numbers such that $\frac{1}{a} + \frac{1}{b} = 2$ and $\frac{1}{a^2} + \frac{1}{b^2} = 6$. Determine the value of $\frac{a+b}{ab}$.
$$
a=\frac{y}{2x}
$$ | nvidia |
The smallest positive integer that has exactly eight distinct positive factors is 24.
Here is another problem:
What is the largest positive integer that has exactly four distinct positive factors? | nvidia |
Find the maximum value of the expression
\[\frac{(x+y)(y+z)}{(x+y)^2+(y+z)^2}+\frac{(y+z)(z+t)}{(y+z)^2+(z+t)^2}+\frac{(z+t)(t+x)}{(z+t)^2+(t+x)^2}+\frac{(t+x)(x+y)}{(t+x)^2+(x+y)^2},\]
over all positive values of $x,$ $y,$ $z,$ $t.$ | nvidia |
How many four-digit positive integers are there such that the product of the digits is equal to 12? | nvidia |
Let $x$ be a real number such that $\left\lfloor x\right\rfloor + x = 7.2$. Determine the value of $\left\lceil 2x\right\rceil$. | nvidia |
Find all positive values of $d$ such that the inequality $x^2-4x+d<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Compute the value of $x$ such that
$\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots$. | nvidia |
For any integer $n>2$, the number of prime numbers greater than $n!+n$ and less than $n!+2n$ is:
$\text{(A) } 0\quad\qquad \text{(B) } 1\quad\\ \text{(C) } \frac{n}{2} \text{ for n even, } \frac{n+1}{2} \text{ for n odd}\quad\\ \text{(D) } n-2\quad \text{(E) } n-1$ | nvidia |
Find all functions $f(x)$ such that $f(x) + f(-x) = x^2$ for all real numbers $x$. | nvidia |
The polynomial equation $x^3 - ax^2 + bx - 100 = 0$ has three distinct integer roots. How many possible values of $a$ are there, if it is known that $a$ is an integer such that $0 < a \leq 100$? | nvidia |
Let $f(x)=16x+3$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f((2x)^{-1})$. | nvidia |
Find the number of three-digit numbers of the form $\_\_2$ that are divisible by 4. | nvidia |
The function $g(n)$ is defined on the positive integers such that $g(g(n)) = 3n$ and $g(3n + 2) = 3n$ for all positive integers $n.$ Find the smallest value of $n$ such that $g(n) = 99.$ | nvidia |
What is the least positive integer that is divisible by three distinct primes? | nvidia |
Find the least positive integer $n$ such that $$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$$ | nvidia |
The sum of two consecutive even integers is 50, and the larger integer is twice the smaller. Find their product. | nvidia |
Find the sum of all integer values of $n$ such that $n^2+1$ is a perfect square and $\frac{n^2+1}{2n}$ is an integer. | nvidia |
What is the sum of the first 10 positive integers that are not divisible by 3? | nvidia |
A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The prob... | nvidia |
The roots of the polynomial
\[x^5+x^4+x^3+x^2+x+1\]
are all different. Three of them are complex conjugate pairs. Find the sum of all the real roots of this polynomial. | nvidia |
Given that the function $f(x)$ satisfies the equation
$f(x+y)+f(y+z)+f(z+x)=3f(x)+3f(y)+3f(z)$,
for all real numbers $x$, $y$, and $z$, find the value of $f(2007)$ if $f(1) = 2023$. | nvidia |
Find the sum of all positive integer values of $c$ such that the graph of the function $y = \frac{2x^2 + 3x + c}{x^2 + 2x + 1}$ has two x-intercepts. | nvidia |
Find the number of positive integers less than or equal to $360$ that have a remainder of either $2$ or $3$ when divided by $7$. | nvidia |
Find the smallest positive integer that has exactly five different prime factors, one of which is squared, and is a multiple of 100. | nvidia |
A set of positive integers contains exactly one number that is a perfect square and exactly two numbers that are perfect cubes. If the sum of all the numbers in the set is 317, what is the largest possible value of the perfect square number? | nvidia |
In the city of Circeville, there are $n$ neighborhoods, each containing $k$ houses, where $n$ and $k$ are positive integers. Each house is painted with one of the three colors: blue, red, and green. The houses in each neighborhood are painted in such a way that there is at least one house of each color, and no two adja... | nvidia |
Find the sum of all constants $c$ such that the quadratic $x^2 + 3x + c$ can be expressed as the square of a binomial with integer coefficients. | nvidia |
A positive integer $k$ can be represented as the product of three distinct factors $a$, $b$, and $a^2+b^2$, where $a$ and $b$ are each one-digit numbers. What is the largest possible value of $k$ that is a three-digit number? | nvidia |
Find $w$, such that $5^65^w=25$. | nvidia |
Find all real numbers $x$ such that
\[\frac{2^x + 3^x}{2^{x+1} + 3^{x+1}} = \frac{5}{7}.\]Enter all the solutions, separated by commas. | nvidia |
Find the coefficient of $x^2$ in the expansion of $(x+1)^n + (x-1)^n$, where $n$ is a positive integer. | nvidia |
Let $x$ and $y$ be positive numbers such that $x^{\sqrt y}=y^{\sqrt x}$. Find the maximum value of $x+y$. | nvidia |
Find the sum of all the positive divisors of 91 that are not multiples of 7. | nvidia |
Determine the sum of the coefficients of the polynomial formed by the product of the denominators of all possible fractions that can be expressed as a finite decimal with a length of 5 digits after the decimal point, where the numerator is a prime number less than 100 and the denominator is a power of 2. | nvidia |
Let $c$ and $d$ be positive real numbers such that each of the equations $x^2 + cx + 3d = 0$ and $x^2 + 3dx + c = 0$ has real roots. Find the smallest possible value of $c + d.$ | nvidia |
Let $g(x)$ be a periodic function with period $T$. Find the smallest positive number $b$ such that the function $g(kx)$ has a period $4T$, given that $k$ and $b$ are integers. | nvidia |
Let $a,$ $b,$ $c,$ and $d$ be distinct real numbers such that
$$(x-a)(x-b)+(x-c)(x-d)=0$$
has exactly one real root. Enter all possible values of $a+b+c+d$, separated by commas. | nvidia |
In a triangle $ABC$, $\angle B$ is a right angle, $AB=1$ and $BC=2$. A square is drawn on each side of the triangle so that each square shares a side with the triangle. The squares are then rotated about their shared vertices so that each square overlaps the triangle and one other square. Find the area of the region in... | 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 |
Let $a$ and $b$ be the roots of the equation $3x^2-10x+3=0$. Find the maximum value of $a^2+b^2$ such that $a^3+b^3<45$. | nvidia |
Find the smallest positive integer $n$ such that $n!$ can be expressed as the sum of $n$ consecutive positive integers. | nvidia |
Let's assume that three intersecting chords are drawn in a circle. What is the least possible number of equal parts the circle can be divided into, if each chord divides the circle into an integer number of parts? | nvidia |
The sides of a cube are covered with red paint, and then it is cut into smaller cubes, each with a side length equal to one-tenth of the original cube's side length. Determine the ratio of the total surface area of the smaller cubes that have red paint on at least one of their sides to the total surface area of the sma... | nvidia |
Find all real values of x such that:
$$(2\sin(x)+3\cos(x))^2+(2\sin^2(x)+3\cos^2(x))=1$$ | nvidia |
Find the value of $k$ for which the function $f(x)=\frac{k}{x^2}+x^2$ attains its minimum value at $x=a$, where $a$ is a fixed positive real number. | nvidia |
What is the maximum number of distinct chords that can be drawn in a regular polygon with 24 sides, such that no two chords intersect inside the polygon? | nvidia |
Find the sum of all positive integral values of $m$ such that $m^2 - 18m + 64 = q$ for some prime number $q$. Note that there is at least one such $m$. | nvidia |
The letters of the English alphabet are distributed randomly among 26 distinguishable boxes. Find the probability that there is at least one letter that is placed in the box with the corresponding letter of the alphabet. | nvidia |
A heptagon is inscribed in a circle. Six of the sides have length 56 and the seventh, denoted by BC̄, has length 28. Find the sum of the lengths of the four diagonals that can be drawn from B. | nvidia |
There are $n$ identical boxes, and $k$ identical marbles are placed randomly into the boxes. The probability that each box contains exactly $m$ marbles is $\frac{1}{k+1}$. Determine $n$ in terms of $k$ and $m$. | nvidia |
Find the number of different polynomials $p(x)$ with integer coefficients such that $p(0) = 0$, $p(1) = 1$, and $p(x^2) - p(x)$ is a constant for all real numbers $x$. | nvidia |
Let the real numbers $x$ and $y$ satisfy $\frac{x}{y}+\frac{y}{x}=3$. Determine the value of $\frac{x^3}{y^3}+\frac{y^3}{x^3}+6$. | nvidia |
Find the largest value of $m$ such that $2x^2 + mx + 120$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Find the largest positive integer $n$ such that $\sqrt[4]{2016^n}$ is an integer. | nvidia |
Let the integers $a$ and $b$ be such that
\[f(a^x) + xf(b^{-x}) = x\]for all real numbers $x.$ Determine $f(a).$ | nvidia |
The number 270000 can be expressed as the product of two integers $a$ and $b$, where $a$ is the largest possible integer such that $a^3$ divides 270000. Find the smallest integer $c$ such that $abc$ is a perfect fifth power. | nvidia |
The sum of the four numbers is 720. If the first number is 120, and the value of the second number is three times the value of the fourth number, and the third number is 150, find the value of the fourth number. | nvidia |
Define the function $g(x)=\frac{c}{3x+2}$. If $g(-1)=g^{-1}(c-2)$, find the sum of all possible values of $c$. | nvidia |
Given that $$a = \log_{2006} 8030$$ and $$b = \log_{2006} 6157$$
and knowing that $${2006}^{1/11}=16$$ determine integer $n$ from the equality $$a^n=b.$$ | nvidia |
A function g(n) is defined for all positive integers n, and satisfies the following conditions:
g(1) = 1,
g(m + n) = g(m) + g(n) for all positive integers m and n.
Find all possible values of g(5). | nvidia |
How many positive integers $x$ satisfy $2^x < 500 < 3^x$? | nvidia |
Find the smallest positive integer $n$ such that $4n$ leaves a remainder of $1$ when divided by $21$.
Express your answer as an integer from $0$ to $20$, inclusive. | nvidia |
Find the area of the ellipse defined by the equation $\frac{(x-2)^2}{9}+\frac{(y+3)^2}{16}=1$ | nvidia |
Find the smallest positive integer that is divisible by 7 and whose digits add up to 9. | nvidia |
Find the $2 \times 2$ matrix $\mathbf{A}$ such that
\[\mathbf{A} \mathbf{x} = \mathbf{x} \mathbf{A}\]for all $2 \times 2$ matrices $\mathbf{x}.$ | nvidia |
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