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The function $f(x)$ satisfies
\[f(xy) = \frac{f(x)}{y}\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ | nvidia |
A number is divisible by $9$ if the sum of its digits is divisible by $9.$ For example, the number $19\,836$ is divisible by $9$ but $19\,825$ is not.
If $D\,767\,E89$ is divisible by $9,$ where $D$ and $E$ each represent a single digit, what is the sum of all possible values of the sum $D+E?$ | nvidia |
Find the smallest multiple of $11$ that is greater than $250$ and has the sum of its digits equal to $11$. | nvidia |
The graph of $y=f(x)$ is shown below, with $1$ unit between grid lines. Assume $f(x)$ is defined only on the domain shown.
What is the sum of all integers $c$ for which the equation $f(x)=c$ has exactly $6$ solutions?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm... | nvidia |
The average of ten numbers is 18. If five of the numbers are removed, the mean of the remaining five numbers becomes 10. What is the sum of the five numbers that were removed? | nvidia |
In a certain convex polygon, the sum of the interior angles that are multiples of 30 degrees is equal to twice the sum of the interior angles that are not multiples of 30 degrees. How many sides does the polygon have? | nvidia |
An arithmetic sequence has the property that the sum of its first $n$ terms is $2n^2$ for any positive integer $n$. Find the 2008th term of this sequence. | nvidia |
A circle with center $(2,5)$ and radius $6$ intersects the line $y=-2x$. Determine the sum of all possible x-coordinates of the points of intersection. | nvidia |
Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$ | nvidia |
Let $g(x)=2x^2+5$. Find the sum of all $x$ that satisfy the equation $g^{-1}(x)=g(\frac{x+1}{2})$. | nvidia |
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. For how many values of $n$ is $n+S(n)+S(S(n))=3008$? | nvidia |
Let $A$ be a set of 20 distinct points in the plane, no three of which are collinear. Consider the set of all triangles whose vertices are in $A$. How many triangles in this set have a specific property that there is at least one point among the remaining 17 points that is equidistant to all three vertices of the trian... | nvidia |
If $x$ is a positive integer such that $(2x+1)^2-(2x+1)-1962=0$, find $x$. | nvidia |
Find the area of the largest rectangle that can be inscribed in the ellipse \[3x^2 + 2y^2 = 6.\] | nvidia |
Count the number of positive integers less than 250 that are multiples of 7 but not multiples of 21. | nvidia |
A function $g$ has the property that $g(2x+3)=x^2-2x-3$ for all real numbers $x$. What is $g(7)$? | nvidia |
Let $a$ and $b$ be positive integers such that their sum is 450 and their least common multiple is 480. Find the smallest possible value of $|a-b|$. | nvidia |
Find the number of positive integers less than 360 that are not divisible by any of the numbers 2, 3, or 5. | nvidia |
What is the smallest positive integer $n$ such that $\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\cdots\left(\frac{n-1}{n}\right)\left(\frac{n}{n+1}\right)$ is less than $\frac{1}{100}$? | nvidia |
Let $n$ be a positive integer. A positive integer $m$ is said to be $n$-flat if $m$ is divisible by $n$ and the decimal representation of $m$ ends with $n$ zeros. How many positive integers less than 1000 are 12-flat but not 24-flat? | nvidia |
Find the length of the longest sequence of consecutive digits in the decimal expansion of $e$ such that the sum of the digits is equal to 7. | nvidia |
Find the value of $a$ such that the graphs of the equations $y=2x^2-ax+1$ and $y=8x^2+ax+1$ intersect at exactly one point. | nvidia |
Let $P(x)$ be a polynomial of degree 6 such that
\[P(x) = P(2016-x).\]
Find the maximum number of real roots that $P(x) - 1$ can have. | nvidia |
Find the value of $y$ that satisfies $\sqrt[3]{y\sqrt{y^2}}=4$. | nvidia |
Suppose $x$, $y$, and $z$ are positive real numbers such that $x>y>z$. Which of the following must be true?
$xyz > x+y+z$
$\sqrt{x}>\sqrt{y}+\sqrt{z}$
$\frac{x}{y}>\frac{y}{z}$
$\frac{x+y}{2}>\sqrt{yz}$
$x^2+y^2+z^2>2xy+2yz+2zx$
Enter your answer as a list of those options that are always true. | nvidia |
What is the smallest whole number that can be added to the sum of any three consecutive positive integers such that the resulting sum is always divisible by 9? | nvidia |
Find the largest positive integer $m$ such that
\[\sin^m x \cdot \cos^m x \le \frac{1}{2m}\]for all real numbers $x.$ | nvidia |
What is the smallest number that can be expressed as the sum of two different prime numbers in two different ways? | nvidia |
Find the area of the triangle formed by the points $(1,-1,2)$, $(-2,2,1)$, and $(-1,-1,3)$. | nvidia |
Let $a$ be a number such that the units digit of $a^2$ is 5. How many distinct three-digit positive integers $b$ are such that the units digit of $ab$ is 5? | nvidia |
Find all positive integer values of $n$ such that $2^n$ is a factor of $3^n+1$. | nvidia |
Prove that the product of the first $n$ terms of the sequence $\left(1+\frac{1}{n}\right)$ converges to $e$ as $n$ approaches infinity. | nvidia |
Find the values of $x$ such that $x^2 + 15x -54$ is a perfect square. | nvidia |
Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
The expression $\left( 1+i \right) ^{2005}=a+bi$ where $a$ and $b$ are real numbers, $i^{2}=-1$. Compute the smallest positive integer $n$ such that both $a$ and $b$ are divisible by $2^{n}$. | nvidia |
How many four-digit numbers are there whose digits are consecutive integers, and the sum of the middle two digits is 9? | nvidia |
Find the smallest integer $n$ such that $5^n > 10^6$. | nvidia |
How many non-negative integers can be formed using all the digits of the number 123456789 exactly once, such that the resulting number is divisible by 27? | nvidia |
The sum of the squares of the three positive integers is equal to four times the product of the smallest two integers, and the largest integer is twice the sum of the other two. Find the sum of all possible values of the product of the three integers. | nvidia |
Determine the number of positive integers less than 250 that are not divisible by any of the numbers 2, 3, or 5. | nvidia |
Find the sum of all $x$ that satisfy the equation $\frac{1}{f^{-1}(x)}=f(2x)$, given that $f(x) = \frac{x+1}{x-1}$. | nvidia |
Find the smallest positive integer $b$ such that $x^6 + bx^3 + 1$ is always composite for all integers $x$. | nvidia |
Find all the possible positions of the center of a circle that passes through three non-collinear points chosen from a set of 16 equally spaced points on the circumference of a circle. | nvidia |
Find the value of the term in a geometric sequence that has the first term 3 and a common ratio of 2, knowing that the sum of the terms of the sequence is 93. | nvidia |
In the equation
$$4x^3-5x^2-x+1=0$$
determine $k$ such that the product of the roots be equal to unity. | nvidia |
What is the least positive integer that is divisible by three distinct primes? | nvidia |
Find the smallest positive integer n for which there exists a non-empty subset S of {1, 2, ..., n} such that the sum of the reciprocals of the elements of S is a positive integer. | nvidia |
In a quadrilateral $ABCD$ with $\angle A=\angle C=90^{\circ}$, the diagonals have the lengths $\sqrt{2}$ and $2$. Determine all the possible values of the area. | nvidia |
There are five cards placed face down on a table, and there is a number written on each card, so that one card has the number 0, one card has the number 1, and the remaining cards have the numbers -1, -2, and -3. Also, one of the cards has been marked with a circle, one card has been marked with a square, and the rema... | nvidia |
Given the system of equations:
$$\left\{\begin{array}{ll}
x^2+ax+b & =0\\
ax^2+bx+a& =0
\end{array}\right.$$
find $a+b$ if $a$ and $b$ are real and nonzero. | nvidia |
A triangle has two sides that are consecutive integers, and the length of its hypotenuse is √5 inches. What are the lengths of the other two sides? | nvidia |
Find the number of different positive integers that can be represented as a sum of two distinct members of the set $\{1, 2, 3, \ldots, 14, 15, 16 \}$. | nvidia |
What is the smallest positive integer $n$ such that $3n$ is a perfect square and $2n$ is a perfect cube? | nvidia |
The product of two positive integers is 1200. When the smaller integer is subtracted from the larger one, the result is 10. Determine the integers. | nvidia |
Let $a$ and $b$ be constants such that the equation
$$\frac{1}{x^2+ax+b}=\frac{x}{x^2+2ax+3b}$$holds for all real values of $x$ except the roots of the denominator. Find $b/a$. | nvidia |
In a Cartesian plane, a quadrilateral has vertices with $x$-coordinates $a$, $b$, $c$, and $d$. The $x$-coordinates of the midpoints of its diagonals are $1$ and $7$, respectively. Find the sum of the $x$-coordinates of the vertices of the quadrilateral. | nvidia |
Find all values of $x$ that satisfy the equation $|x+2|=3x-5$. Express your answers in simplest fractional form. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x^2+y^2+z^2=1$. Find the maximum value of
\[\frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy}.\] | nvidia |
A rectangular piece of paper is folded once along a straight line, and then a circle is drawn on the folded paper such that the center of the circle lies on the fold line. When the paper is unfolded, the circle appears as two identical circles that overlap. What is the maximum number of points of intersection that can ... | nvidia |
Let $S$ be the set of all positive integers less than or equal to $30$ that are divisible by $3$, and let $T$ be the set of all positive integers less than or equal to $30$ that are divisible by $2$. Determine the number of elements of the set $S\cap T$. | nvidia |
Let $a,$ $b,$ $c$ be real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the set of all possible values of $abc.$ | nvidia |
Compute the smallest positive integer $n$ such that $n + i,$ $(n + i)^2,$ and $(n + i)^3$ are the vertices of a triangle in the complex plane whose area is greater than 2015. | nvidia |
Find the sum of all integer solutions to the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{7}$ that also satisfy $x < y$. (Recall that the solutions to the equation may be positive or negative.) | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 1.$ Find the maximum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$ | nvidia |
Consider the set of matrices of the form
$$ M_{n}=\begin{bmatrix} m_{11} & m_{12} & \cdots & m_{1n} \\ m_{21} & m_{22} & \cdots & m_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ m_{n1} & m_{n2} & \cdots & m_{nn} \end{bmatrix}, $$
where $m_{ij}$ are integers and each row and column sums to a distinct integer power of 2. ... | nvidia |
The number 96 can be expressed as the product of two natural numbers in 5 different ways. Find all possible values of $m$ such that the number $6^m$ can be expressed as the product of two natural numbers in exactly 5 different ways. | nvidia |
Find the polynomial $q(x),$ with real coefficients, such that
\[q(x^2) - q(x^2 + 1) = [q(x)]^2 - 8\]for all real numbers $x.$ | nvidia |
The geometric mean of a prime number of consecutive integers is $y$. Find the product of the smallest and largest of the integers in terms of $y$. | nvidia |
A function $g$ has the property that $g(2x+3)=x^2-2x-3$ for all real numbers $x$. What is $g(7)$? | nvidia |
What is the greatest number of integers that can be placed along a number line such that any two numbers differ by at least 5 units and any three numbers differ by at least 10 units? | 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 |
Let $x$ and $y$ be positive real numbers such that $x^2 + y^2 = 4.$ Find the minimum value of
\[\frac{x}{y} + \frac{y}{x} + \frac{x^2}{y^2} + \frac{y^2}{x^2}.\] | nvidia |
Let $a$, $b$, $c$ and $d$ be positive integers such that $\left\lceil\sqrt{a}+\sqrt{b}\right\rceil=\left\lceil\sqrt{c}+\sqrt{d}\right\rceil$ and $a+b=c+d$. Determine all possible values of the sum $\left\lceil\sqrt{a}\right\rceil+\left\lceil\sqrt{b}\right\rceil+\left\lceil\sqrt{c}\right\rceil+\left\lceil\sqrt{d}\right\... | nvidia |
What is the difference between the sum of the squares of the first 12 positive integers and the sum of the cubes of the first 8 positive integers? | nvidia |
Let $\theta$ be a real number, and let $z$ be a complex number such that
$$z^3 + \frac{1}{z^3} = 2\cos\theta.$$
Find the maximum possible value of $|z|$ in terms of $\theta$. | nvidia |
Determine the largest integer $n$ such that $2^n$ is a divisor of $200!$. | nvidia |
Let $g$ be a function defined on positive integers, such that $g(mn) = g(m) + g(n)$ for all positive integers $m$ and $n$. It is also known that $g(n) = 0$ for prime $n$. Determine the value of $g(2005)$. | nvidia |
Find the number of distinct positive integers that can be expressed as the product of two relatively prime positive integers, one of which divides 8400 and the other divides 7560. | nvidia |
The sum of the squares of the distances from a fixed point inside an equilateral triangle to its three vertices is equal to 18. Determine the largest possible value of the distance between the fixed point and a side of the triangle. | nvidia |
Find the number of positive integers $b$ less than $8$ such that the equation $2^bx + y = 2$ has integer solutions in $x$ and $y$. | nvidia |
Which of the following quantities is the largest? (Write $A$, $B$, or $C$.) \[ A.\ \ \frac{2006}{2005}+\frac{2006}{2007} \qquad B.\ \ \frac{2006}{2007}+\frac{2008}{2007} \qquad C.\ \ \frac{2007}{2006}+\frac{2007}{2008} \] | nvidia |
A triangle is formed by three points (A, B, C) on the coordinate plane such that the lengths of the sides AB, BC, and CA are all integers. What is the smallest possible perimeter of such a triangle that does not contain the origin (0, 0) as one of its vertices? | nvidia |
Find the smallest positive integer $z$ such that the product $x\cdot z$ is a perfect square, where $x = 8 \cdot 27 \cdot 32$. | nvidia |
Determine the smallest number that can be expressed as a sum of three distinct prime numbers. | nvidia |
Let's define a function f(n) that returns the largest prime factor of n. Determine the minimum value for n>1 such that the equality
f(n)+f(n+1)+f(n+2)=12 holds true. | nvidia |
The sides of a triangle are in the ratio of $2:x:3$, and the perimeter is $18$ units. Find the value of $x$ if the triangle is a right triangle. | nvidia |
Let $a$, $b$, $c$ be positive integers such that $a+b+c=12$ and $a<b<c$. Determine the maximum value of $a^a \cdot b^b \cdot c^c$. | nvidia |
Find all 5-digit numbers of the form $ab,\!cd5$ that are divisible by 15, where $a, b, c,$ and $d$ are distinct digits. What is the minimum value of $a+b+c+d$? | nvidia |
Let $P(x)$ be a polynomial with integer coefficients such that
$P(1)=2$
$P(2)=4$
$P(3)=6$
$P(4)=8$
Prove that there exists an integer $k$ such that
$P(x)=2x+k(x-1)(x-2)(x-3)(x-4)$
holds for all real numbers $x$. | nvidia |
Determine the largest positive two-digit number that is divisible by neither 3 nor 7, but the sum of its digits is divisible by 5. | nvidia |
In the figure, $AB$ is a diameter of the circle, $CD$ is a chord parallel to $AB$, and $AC$ intersects $BD$ at $E$, with $\angle AED = \alpha$. The ratio of the area of $\triangle CDE$ to that of $\triangle ABE$ is $k$. Prove that the ratio of the perimeter of $\triangle CDE$ to that of $\triangle ABE$ is $2k\sin\alpha... | nvidia |
Let $A$ be a point on the circle $x^2+y^2=1$ such that the distance from $A$ to the point $(\sqrt 2,0)$ is $\sqrt 2$.
Find $|OA|^2$ where $O$ is the center of the circle. | nvidia |
Find the number of functions $f : \mathbb{N} \to \mathbb{N}$ such that
\[f(x+y)f(x-y)+xy=f(x)f(y)\]for all positive integers $x$ and $y$ such that $x>y$. | nvidia |
Consider the set of all positive integers from 1 to 100 that are not divisible by 3 or 7. Let N be the number of elements in this set. Find N. | nvidia |
Find the number of ordered pairs $(a,b)$ of complex numbers such that
\[a^3 b^5 = a^7 b^2 = 1.\] | nvidia |
Given that $m$ is an integer and $0 < 3m < 40$, what is the sum of all possible integer values of $m$? | nvidia |
The graphs of the two quadratic equations $y=x^2+ax+b$ and $y=x^2-ax+b$ intersect at points $(c,d)$ and $(e,f)$, where $c$ and $e$ are $x$-coordinates of these points. Determine the value of $a$ such that the product $c \cdot e$ is equal to $b$. | nvidia |
The graphs of the equation $y^2=x^3-4x$ and $x^2+2y^2-8x-4y=0$ intersect at three points $A,B,C.$ Find the area of the triangle $ABC.$ | nvidia |
What is the largest four-digit number that can be formed using only the digits from the first five rows of Pascal's triangle, with the restriction that no digit can be repeated more than twice? | nvidia |
Find the sum of all $x$ that satisfy the equation $\frac{-9x}{x^2-1} = \frac{2x}{x+1} - \frac{6}{x-1}.$ | nvidia |
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