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A function $f$ has the property that $f(3x-1)=x^2+x+1$ for all real numbers $x$. What is $f(5)$? | nvidia |
Find the number of pairs of positive integers $(a, b)$ such that $a^2 - b^2$ is a multiple of 1111 and $a + b \leq 100$. | nvidia |
Find the sum of all positive divisors of $50$ that are also divisors of $15$. | nvidia |
Consider a Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, ... where every term after the first two is the sum of the two preceding ones.
Let $m$ and $n$ be positive integers such that $f_m$ is divisible by $f_n$.
Can it be true that n > m? | nvidia |
Let $\oplus$ be a binary operation on positive integers such that $a\oplus b$ is the largest integer less than or equal to $\sqrt{ab}$ for all positive integers $a$ and $b$. What is the value of $(7\oplus 3)\oplus 5$? | nvidia |
Find the area of the triangle with vertices $(3,-5),$ $(-2,0),$ and $(1,-6).$ | nvidia |
Let $x, y, z$ be three positive integers such that
\begin{align*}
xy&=12,\\
yz&=20,\\
xz&=15.
\end{align*}Find the value of $x^2 + y^2 + z^2$. | nvidia |
Let $A$ be the set of all positive integers less than $50$ that are not divisible by $4$, and let $B$ be the set of all positive integers less than $50$ that are not divisible by $3$. Determine the number of elements in the set $A \cap B$ that are one more than a multiple of $5$. | nvidia |
Determine the number of ordered 4-tuples $(x_1, x_2, x_3, x_4)$ of positive real numbers such that $x_1 < x_2 < x_3 < x_4$ and
$$\frac{1}{x_1}+\frac{2}{x_2}+\frac{3}{x_3}+\frac{4}{x_4} > \frac{2}{x_1+x_2}+\frac{3}{x_2+x_3}+\frac{4}{x_3+x_4}.$$ | nvidia |
Let $a$ and $b$ be two positive real numbers such that their arithmetic mean is 4 and their harmonic mean is 3. Determine the value of $|a-b|$. | nvidia |
Determine the sum of all the roots of the equation $$2x^4+5x^2+3=0$$ | nvidia |
If $a$, $b$, and $c$ are positive integers such that $a^2=c^b$, $b^2=a^c$, and $c^2=b^a$, find the smallest possible value of $a+b+c$. | nvidia |
Find the value of $x$ such that $\cos^{-1} x + \sin^{-1} x = \frac{\pi}{2}$ | nvidia |
A circle is inscribed in a triangle with sides of length 5, 6, and 7 units. Determine the radius of the circle such that it touches all three sides of the triangle. | nvidia |
Find the value of the 25th term of the arithmetic sequence $a, a+d, a+2d, \ldots$, given that the sum of the first 25 terms is 1250 and the sum of the first 50 terms is 4500. | nvidia |
Let $a$ be the number of positive multiples of $6$ that are less than $30$. Let $b$ be the number of positive integers that are less than $30$, and a multiple of $3$ and a multiple of $2$. Compute $(a - b)^3$. | nvidia |
In a triangle $ABC$ with acute angles, the sum of the squares of the lengths of the sides opposite to the vertices $A$ and $B$ equals $75$, and the sum of the squares of the lengths of the sides opposite to the vertices $B$ and $C$ equals $50$. Find the square of the length of the side opposite to the vertex $A$. | nvidia |
Given a square with sides equal to x, a circle is drawn inside it such that it is tangent to all four sides of the square. What is the maximum area of the circle that can be inscribed within the square? | nvidia |
Find the value of $x$ in base 5 such that $x_{5}+2x_{5}=113_{5}$. | nvidia |
Find the number of integer solutions to the inequality $x^2 + y^2 + z^2 < 25$ such that $x$, $y$, and $z$ are all positive and distinct. | nvidia |
Find the number of natural numbers between 200 and 400 that have a digit sum divisible by 7. | nvidia |
Given the points $(1, -2)$, $(3, 5)$, and $(5, k)$, find $k$ such that the area of the triangle formed by the points is equal to 12 square units. | nvidia |
Problem:
Find the sum of all the real coefficients of the expansion of ${(1+ix)}^{2009}$.
New Problem:
A polynomial $P(x)$ is defined as the product of all the monic quadratic factors of $x^4 + x^2 + 1$. Find the sum of the absolute values of all the coefficients of $P(x)$. | nvidia |
Find all real values of $x$ such that $\sin (\arccos (x)) = x^2$. | nvidia |
The roots of the cubic equation $x^3 + ax + b = 0$ are all real numbers. One of the roots is a prime number $p$. The sum of the squares of the roots is equal to $p^3$. Determine the value of $a$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the minimum value of $x^2 y^2 + y^2 z^2 + z^2 x^2.$ | nvidia |
Let $g(x)$ be a function that satisfies the following conditions:
* $g(x+y)=g(x)g(y)$ for all real numbers $x$ and $y$
* $g(2)=3$
If $\sum\limits_{i=0}^{100} g\left(\frac{i}{100}\right)=\alpha$, find $\alpha-\left[\alpha\right]$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) + g(y) + g(xy) = x^2 + y^2\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(1),$ and let $t$ be the sum of all possible values of $g(1).$ Find $m \times t.$ | nvidia |
A circle is inscribed in a regular polygon. The circle touches the sides of the polygon at the midpoints of each side. If the polygon has an odd number of sides, how many ways can the circle be colored with two colors, such that adjacent sides of the polygon (connected by a vertex) are touched by sections of the circle... | nvidia |
Find the smallest positive integer $n$ such that the sum of the reciprocals of its divisors is an integer. | nvidia |
A rectangle is inscribed in a circle of radius 5 inches, and one of its vertices coincides with the center of the circle. The angle formed by the two diagonals of the rectangle is 60 degrees. Find the area of the rectangle. | nvidia |
In a round-robin tournament, six teams play each other once. Determine the number of possible outcomes such that exactly three teams win exactly two matches. | nvidia |
The points $(2,4)$ and $(6,8)$ are the endpoints of a diameter of a circle. Find the sum of the coordinates of the point where the circle intersects the line $y=2x$. | nvidia |
Let $S$ denote the set of all positive integers $n$ such that $n$ and $2n+1$ have the same greatest common divisor. Find the sum of all possible values of the greatest common divisor of $n$ and $2n+1$ for all $n \in S$. | nvidia |
Find the sum of the coefficients of the polynomial obtained by multiplying the binomial $x+2$ by the trinomial $x^2+x+1$. | nvidia |
Determine the largest positive integer $k$ such that for any positive integer $n$, the sum of the digits of $n^k$ is always a multiple of 3. | nvidia |
Determine the smallest positive integer $x$ such that $11x \equiv 17 \pmod{29}$. | nvidia |
If $x$ is a positive number such that $(x+\frac{1}{x})^2=49$, what is the minimum possible value of $x^3+\frac{1}{x^3}$? | nvidia |
Given the function
\[g(x) = \left\{
\begin{array}{cl}
x-2 &\text{ if }x>2, \\
\dfrac{x}{2} &\text{ if }x \le 2,
\end{array}
\right.\]
find the fixed points of $g(g(x))$, that is, the values of $x$ such that $g(g(x))=x$. | nvidia |
Find a positive integer that is divisible by 25 and whose square root is a number between 10 and 10.2. | nvidia |
What is the number of distinct ways to arrange the letters of the word “astronauts” such that no two vowels are adjacent? | nvidia |
In a regular hexagon inscribed in a circle, a point is chosen on the circle so that the line segment connecting it to the center of the hexagon forms a right angle with the side of the hexagon it intersects. Find the ratio of the area of the triangle formed by the chosen point and the two vertices of the intersected si... | nvidia |
Let the four real numbers $a$, $b$, $c$, and $d$ be such that $\cos ax+b\cos 3ax+c\cos 5ax+d\cos 7ax=0$ for every $x$.
Prove that $a+b+c+d=0$. | nvidia |
Find the sum of all possible positive integer values of $c$ such that the quadratic equation $3x^2 + 7x + c = 0$ has rational roots. | nvidia |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is greater than 6? | nvidia |
Find the sum of all positive integers that are divisible by 3 and have exactly two digits. | nvidia |
Let $x$ and $y$ be real numbers such that $|x+y|=0$ and $|x-y|=2$. Determine $x^2+y^2$. | nvidia |
In the diagram, $ABCD$ is a rectangle. $M$ and $N$ are points on $AB$ and $DC$, respectively, such that $\angle MAN=42^\circ$ and $\angle MCD=61^\circ$. Find the measure of $\angle ANB$. | nvidia |
Problem:
Consider a sphere with radius 2 units, centered at the origin. Determine the volume of the set of points that are inside or within one unit of the sphere's surface. Express the volume as $\displaystyle {{m+n\pi}\over p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime. Find... | nvidia |
The perimeter of a rectangle is 48 cm. What is the maximum possible area of the rectangle, given that the lengths of its sides are positive integers? | nvidia |
What is the least positive integer that can be expressed as the sum of two different perfect squares in two distinct ways? | nvidia |
Find all real numbers $a$, $b$, and $c$ so that the expression $(a + bi)^2 - (a - bi)(c + bi)$ is a real number. Enter all the solutions, separated by commas. | nvidia |
Let $f(x)=x^4+ax^3+bx^2+cx+d$ be a polynomial with real coefficients. Suppose that $f(x)$ has four real roots: $r_1$, $r_2$, $r_3$, and $r_4$. If $r_1$ and $r_2$ are the roots of $x^2-2x-3=0$ and $r_3$ and $r_4$ are the roots of $x^2+2x-3=0$, find the value of $a$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(x)$ is defined for all $x$. Given that $f(\frac{a+b}{2}) = \frac{f(a)+f(b)}{2}$ and $f(0) = 0$, find the value of $f(2)$. | nvidia |
Find the smallest positive integer that can be expressed as the least common multiple of three different pairs of prime numbers, and determine which pairs of primes yield that value. | nvidia |
Determine the largest possible positive integer value of $k$ for which there exists a positive integer $m$ such that $4m^2 - 2m + 1$ is a factor of $12^k$. | nvidia |
In a right triangle with integer side lengths, the hypotenuse has a length of $h$ units and the area of the triangle is $k$ square units. Determine the value of $2h + 3k$. | nvidia |
Three consecutive integers are such that the sum of the smallest and the largest is equal to the square of the middle one. What is the smallest possible value of the smallest of these integers? | nvidia |
Solve for the sum of all possible values of $x$ when $3^{x^2+4x+4}=9^{x+2}$. | nvidia |
Find the sum of the roots of the polynomial $2x^3 + 3x^2 - 4x + 1$, given that one of the roots is a complex number. | nvidia |
Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence such that
\[a_n = \frac{1}{n-1} \left( \sum_{i=1}^{n-1} a_i \right)^2\]for all $n \ge 2.$
Determine the smallest value of $n$ such that $a_n > 100.$ | nvidia |
Find the value of $y$ such that the expression $\frac{y^2(y+2)-y^3}{y^2+2y+1}$ is equal to $-3$. | nvidia |
Find the number of possible values of $[x^3]$, given that $x$ is a real number and $[x]=7$, where $[x]$ denotes the greatest integer less than or equal to $x$. | nvidia |
Find the area of the region enclosed by the graph of $|x-60|+|y|=\left|\frac{x}{4}\right|.$
Find the area of the region enclosed by the graph of $\left|x-\frac{3}{2}\right|+\left|y-\frac{3}{2}\right|=|x-2y|.$ | nvidia |
Let $A$ and $B$ be two points on the $x$-axis and $y$-axis, respectively, such that $A$ is $5$ units away from the origin and $B$ is $3$ units away from the origin. Find the distance from the origin to the point on the line segment $AB$ that divides it in a $2:1$ ratio. | nvidia |
Find all real-valued functions $f$ such that for all positive real numbers $x$ and $y$, the following functional equation holds:
$$f\left(\frac{x+y}{xy}\right)=f(x)f(y).$$ | nvidia |
Let's define a function $F(n)$ as the number of integers less than or equal to $n$ that are either prime or can be represented as the product of two distinct primes. For example, $F(10) = 7$ because the integers that satisfy the condition are $2, 3, 5, 7, 6, 10, 8$.
Find $F(99)$. | nvidia |
Let $z_1=10+15i$ and $z_2=20+5i$. If $a$ and $b$ are real numbers such that $az_1+bz_2$ is a real number, determine the smallest possible value of $|a|+|b|$. | nvidia |
Let $a$ and $b$ be two positive integers such that their greatest common divisor is equal to 1. We call the pair $(a, b)$ friendly if $a$ is equal to the sum of the proper divisors of $b$ and $b$ is equal to the sum of the proper divisors of $a$. Find a pair of friendly numbers such that their sum is the smallest among... | nvidia |
Let $\mathbf{A}$ be a $3\times 3$ matrix such that $\mathbf{A}^T\mathbf{A}$ has all diagonal entries equal to 1. If $\det \mathbf{A} = 0$, show that at least one of the non-diagonal entries of $\mathbf{A}^T\mathbf{A}$ is equal to 1. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2+y^2=1$ and $x^2-y^2=\frac{1}{7}$. Find the value of $x-y$. | nvidia |
Consider a point $P$ on the plane such that its perpendicular distances to two given lines are $2k$ and $k$, respectively, for some positive constant $k$. Among all such points, find the line(s) that the point $P$ is always at a fixed distance to. | nvidia |
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? | nvidia |
Let $x$ and $y$ be positive real numbers such that $x > 3y$. What is the smallest possible value of $\frac{x+y}{x-y} + \frac{x-3y}{x+3y}$? | nvidia |
Find all values of $k$ such that the inequality
\[2x^2 - (k - 1) x - k + 5 > 0\]
holds for all real numbers $x$. | nvidia |
Three coplanar ellipses intersect in such a way that each pair of ellipses has exactly two intersection points, and no three ellipses intersect at the same point. What is the maximum number of points on the ellipses that a line passing through all three ellipses can touch? | nvidia |
Given that the roots of the equation \[px^4+qx^3+rx^2+sx+t=0\] are $\frac{1}{2}$, $-\frac{3}{4}$, $2i$, and $-2i$, determine $\frac{q-r}{p}$. | nvidia |
Let $z_1, z_2, z_3$ be three complex numbers that are $5$th, $6$th and $7$th roots of unity, respectively, such that $\arg(z_1z_2z_3)$ is a multiple of $\pi/3$. Find the least possible value of $|z_1+z_2+z_3|$. | nvidia |
Let $\alpha$ and $\beta$ be complex numbers such that $\alpha+\beta$, $\alpha\beta$, and $\alpha^2+\beta^2$ are all integers.
Prove that $\gcd(\alpha+\beta, \alpha\beta)=1$ if and only if $\alpha^2+\beta^2$ is a perfect square. | nvidia |
Factor the expression $x^4 + 4x^2 + 4$ into the product of two quadratic expressions with integer coefficients. Then, find the sum of the squares of all the coefficients in the factored expression. | nvidia |
Given that $x,$ $y,$ $z$ are real numbers such that $x^2 + y^2 + z^2 = 1,$ find the maximum value of $\sqrt{x} + \sqrt{y} + \sqrt{z}.$ | nvidia |
Find the real part of the complex number $(2\cos 3\theta + 2i\sin 3\theta)^5$, where $\theta$ is an acute angle such that $\sin 2\theta = \frac{1}{2}$. | nvidia |
Find all possible values of $p$ such that the equation
$$(x^2 + px + 1)^2 = (x^2 + 2x + 1)(x^2 + 4x + 4)$$
has at least one real solution. | nvidia |
Let $r$ be the common ratio of a geometric sequence whose terms are all positive integers. If $a, b, c$ are three consecutive terms of the sequence such that $\sqrt[3]{a}\cdot\sqrt{b}\cdot\sqrt[4]{c}$ is an integer, determine the smallest possible value of $r$. | nvidia |
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the ... | nvidia |
What is the product of all positive integer divisors of 108 that are also perfect squares? | nvidia |
Find the smallest four-digit positive integer that is divisible by 3, 5, and 11. | nvidia |
Given that the absolute value of the difference between the cube of a number and its reciprocal is equal to 23, determine the sum of all possible values of the number. | nvidia |
Let $x, y, z$, and $w$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}x^3+y^3&=&z^3+w^3&=&343,\\ xy+yz+zw&=&100.\end{array}\]
If $P=x+y+z+w$, determine the value of $\lceil P\rceil$. | nvidia |
Find the smallest positive integer $y$ which satisfies the congruence $27y+3\equiv 11\pmod {25}$. | nvidia |
Find all positive integers $k$ such that the expression $\frac{k^2+3}{k+2}$ is an integer, and determine the largest possible value of $k$. | nvidia |
A sequence of integers $(b_n)$ is defined recursively by $b_1=1$, $b_2=2$, and for $n\geq 3$, $b_n$ is the smallest positive integer that satisfies the condition: the product of the last $b_{n-1}$ terms in the sequence divides the product of the last $b_{n-2}$ terms. Find the smallest positive integer $k$ such that the... | nvidia |
Determine the largest positive integer less than 1000 that has the same number of digits in both base 2 and base 3 representations. | nvidia |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes? | nvidia |
Find the product of all values of $x$ for which the expression $\frac{x-2}{x^2+3x-10}$ is undefined. | nvidia |
Find all real $x$ such that \[\left\lfloor x \lfloor x \rfloor\right \rfloor = 49.\] | nvidia |
In a regular polygon, the sum of the interior angles is equal to twice the number of diagonals. Find the number of sides of this polygon. | nvidia |
Determine the smallest positive integer $a$ such that the sum of the reciprocals of the first $a$ positive integers exceeds 2. | nvidia |
Find the number of functions $f : \mathbb{Z} \to \mathbb{Z}$ such that
\[f(x + f(y)) = x + y + f(0)\]for all integers $x$ and $y$. | nvidia |
Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers. | nvidia |
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