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Determine the value of $x$ such that $\sqrt[3]{32} < x < \sqrt[3]{125}$. | nvidia |
Let $f$ be a function taking the positive integers to the positive integers, such that
(i) $f$ is increasing (i.e. $f(n + 1) > f(n)$ for all positive integers $n$)
(ii) $f(mn) = f(m) f(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $f(m) = n$ or $f(n) = m.$
Find the sum of al... | nvidia |
Find the distance between the center of the ellipse with equation $\frac{x^2}{4} + \frac{y^2}{9} = 1$ and the point $(2,3)$ that lies on the ellipse. | nvidia |
Find the smallest positive integer that can be expressed as the sum of three distinct perfect squares. | nvidia |
Let $P$ be a point inside a regular hexagon $ABCDEF$ such that the triangle $PAB$ is equilateral. How many points $P$ are there? | nvidia |
Consider a 4x4 grid with randomly colored black and white squares, such that each row and each column contain exactly two black squares. How many possible ways can we color this grid such that no two adjacent squares have the same color? | nvidia |
Find the number of positive divisors of 2002. | nvidia |
Let $a^2=\frac{16}{44}$ and $b^2=\frac{(2+\sqrt{5})^2}{11}$, where $a$ is a negative real number and $b$ is a positive real number. If $(a+b)^3$ can be expressed in the simplified form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are positive integers, what is the value of the sum $x+y+z$? | nvidia |
In the equation
$$3x^2+4kx+3=0$$
determine $k$ such that the product of the roots be equal to 2.
Write another problem inspired by this one:
The roots $x_1$ and $x_2$ of the equation
$$x^2+2ax-2=0$$
are such that
$x_1^2+x_2^2=12$.
Determine $a$. | nvidia |
A set of five positive integers includes two integers $a$ and $b$ each occurring twice and a fifth integer $c$. The sum of all the integers is 175, and the product of the two distinct integers occurring twice is 160. What are the three integers $a$, $b$, and $c$? | nvidia |
Find the value of $x$ that satisfies the equation $\sin x + \cos x = \sqrt{2}$, where $x$ is in radians. | nvidia |
Find the number of positive integers less than or equal to 100 that can be expressed as the product of a perfect square and a perfect cube. | nvidia |
How many different convex quadrilaterals can be formed using 8 points on a circle, such that each vertex is one of the 8 points and no three points are collinear? | nvidia |
Given that a real number $x$ satisfies \[\sqrt{64-x^2}-\sqrt{36-x^2}=4.\]Find the value of $\sqrt{64-x^2}+\sqrt{36-x^2}$. | nvidia |
Find the maximum value of $$\ds\frac{yz-xy-zx}{xy-zy-xz}$$
under the condition that $$\ds\frac{x+y}{x+z}=\frac{5}{3}.$$ | nvidia |
A rectangular prism is cut by a plane in such a way that the cross-section is a triangle with vertices on the vertices of the prism. The dimensions of the prism are 5, 7, and 11. How many unit cubes of the prism have interiors lying entirely on one side of the plane? | nvidia |
Find the area of a rectangle with a length that is three times its width, and the width is 2/3 of the side length of a square with a perimeter of 12 cm. | nvidia |
Given the polynomial equation:
\[P(x) = \prod_{k=1}^{n}(x-a_k),\]
where $\{a_k\}_{k=1}^{n}$ is a set of real numbers, determine the number of solutions to the equation $\frac{P(x)}{P(1-x)} = 0$. Consider that some of the roots of $P(x)$ might also be roots of $P(1-x)$. | nvidia |
What is the sum of all positive integer values of $n$ such that $\frac{n+18}{n}$ is an integer?
Problem:
Find the largest value of $n$ such that the product of all positive integers less than or equal to $n$ is divisible by $n+18$. | nvidia |
Find a number that is a factor of every sum of four consecutive even integers. | nvidia |
A point $(x,y)$ is randomly chosen from the triangle with vertices $(0,0),(1,0),$ and $(1,1)$ such that $x+y\neq 1$. What is the probability that $\tan x > \tan y$? | nvidia |
Find all values of $x$ that satisfy the inequality $\frac{x^2-4x+3}{x+1}\geq 1$. Express your answer in interval notation. | nvidia |
Find the smallest positive integer $m$ such that $3m \equiv 5m \pmod{7}$ and $4m \equiv 6m \pmod{11}$. | nvidia |
Let $x$ be a real number such that $\lceil x \rceil = n,$ where $n$ is a positive integer. Suppose that $x$ is not an integer, i.e., $x \neq n$. Determine the number of possible values of $\lfloor x^2 \rfloor.$ | nvidia |
A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$ for some integers $a$ and $b$. | nvidia |
Find the equation of the line that is perpendicular to the line $y=3x+5$ and passes through the point $(2,6)$.
Express the equation of the line in the form $y=mx+b$, where $m$ and $b$ are constants. | nvidia |
The circle centered at $(1,2)$ and with radius $3$ intersects the circle centered at $(1,-3)$ and with radius $\sqrt{8}$ at two points $C$ and $D$. Find $(CD)^2$. | nvidia |
A sequence $\{u_n\}$ is defined by the initial condition $u_1=6$ and the recurrence relation $u_{n+1}=u_n^2-3u_n+2$. Determine the positive integers $m$ and $n$ such that $u_m=u_n^2$. | nvidia |
Find all values of $a$ and $b$ so that the function $f(x) = \sqrt{ax^2 + bx + c}$ has a domain that is the set of all real numbers, where $a$, $b$, and $c$ are constants. | nvidia |
Given that $a$ and $b$ are complex numbers such that $|a|=2$ and $|b|=\sqrt{26}$, find the value of $t$ that satisfies $ab = t-2i$.
Another problem inspired by this one:
Let $z$ be a complex number such that $|z|=5$ and $|z^2+z|=13$. Find the value of $\text{Re}(z)$. | nvidia |
Let
$x=0.\overline{001002003004...}$
Determine the sum of the digits of the integer part of
$1000\,x$. | nvidia |
Two lines that are not parallel to each other intersect two parallel lines in points $A$, $B$, $C$, and $D$, respectively, so that $A$ and $C$ lie on one of the parallel lines. If $AB=x$ and $CD=y$, and if $ABCD$ is a trapezoid with $AB$ and $CD$ as bases, find the ratio of the area of triangle $ABD$ to the area of tri... | nvidia |
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2=1$ and $x+y+z=\sqrt{2}$. Let $n=\max\{xy,yz,xz\}$. Find the smallest possible value of $n$. | nvidia |
There exists a positive real number $y$ such that $ \sin (\arccos (y)) = y^2 $. Find the value of $y^3$. | nvidia |
Let $a$ and $b$ be nonzero real numbers such that
\[(3 + 4i)(a + bi)\]is a real number. Find $\frac{a}{b}.$ | nvidia |
Let $x,$ $y,$ $z$ be real numbers, all greater than 2, so that
\[\frac{x^2}{y+z-2}+\frac{y^2}{z+x-2}+\frac{z^2}{x+y-2}=\frac{3}{2}.\]
Prove that $x=y=z=3.$ | nvidia |
Find the number of integers between 100 and 900 that can be expressed as the sum of two perfect cubes. | nvidia |
Find the value of $s$ such that the point $(s,2)$ lies on the line passing through $(0,1)$ and $(-6,0)$. | nvidia |
A square has a side length that is 20% larger than its diagonal. What percent of the area of the square is equal to the area of a triangle formed by connecting the midpoints of the square's sides? | nvidia |
The graph of the function $f(x)$ consists of three line segments with positive slopes, such that $f(0)=0$, $f(2)=1$, $f(4)=2$, $f(6)=3$, and $f(8)=4$. For which values of $x$ is the value of $f(f(x))$ equal to the value of $f(2x)$? | nvidia |
The complex numbers $x$, $y$, and $z$ are the vertices of a right-angled triangle with hypotenuse 24 and are such that $x+y+z=0$. Find the smallest possible value of $|xy+yz+zx|$. | nvidia |
Let $S$ be the sum of all positive divisors of $22$ that are not divisors of $33$. Determine the number of positive integers $n$ such that $22 < n < 33$ and $S$ is divisible by $n$. | nvidia |
A company's stock price starts at $ \$50 $ and increases by $ 10\% $ every quarter for the next two years. However, at the end of each year, the company pays a dividend of $ \$5 $ per share, which reduces the stock price. What is the price of the stock, in dollars, at the end of the two-year period? | nvidia |
Given an ellipse with semi-major axis $a$ and semi-minor axis $b$, find the largest value of $t$ such that the area of the trapezoid formed by the tangent lines to the ellipse at points $(a\cos t, b\sin t)$ and $(a\cos (t + \pi), b\sin (t + \pi))$ is equal to half the area of the ellipse. | nvidia |
Let $f(x)$ be a real-valued function such that $f(xy)=f(x)+f(y)$ and $f(x^2)+f(x)=0$ for all $x\neq 0$. Determine the value of $f(123)$. | nvidia |
In trapezoid $ABCD$, the diagonals $AC$ and $BD$ intersect at $O$. Given that $\angle AOB=120^\circ$ and $AB+CD=2AD$, find the minimum value of $\frac{AB+CD}{BC}$. | nvidia |
Given that $\frac{a}{b} > \frac{x}{y}$, where $x$ and $y$ are positive integers, and $a$ and $b$ are relatively prime, find the smallest integer value of $y$ such that there exists a unique integer solution $x$ satisfying the inequality when $a = 7$ and $b = 9$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) f(y) - f(xy) = x^2 + y^2\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \times s.$ | nvidia |
Let $S$ be a set of 9 non-zero complex numbers such that the sum of any 3 distinct numbers in $S$ is a real number. Find the smallest possible value of the product of the absolute values of the elements of $S$. | nvidia |
In the figure, square $ABCD$ has side length $6$ cm and semicircles with diameters $AD$ and $BC$ are drawn inside the square. If the area of the shaded region formed by the two semicircles is equal to the area of the square, find the radius of the circle inscribed in the square.
Good luck solving it! | nvidia |
Determine the number of positive integers $b$ less than $24$ such that the congruence $bx\equiv 1\pmod{24}$ has a solution in $x$. | nvidia |
A solid rectangular block with dimensions $n \times m \times p$ is formed by gluing together smaller $1 \times 1 \times 1$ cubes. A number is assigned to each of the $n \times m \times p$ unit cubes, with the number in each cube being the sum of the numbers in the three adjacent cubes that it shares a face with. All th... | nvidia |
Find the largest value of $n$ such that $5x^2+nx+48$ can be factored as the product of two linear factors with integer coefficients.
Problem:
Find the largest value of $k$ such that the quadratic equation $x^2+kx+5$ has two real solutions that are in geometric progression. | nvidia |
Let $P(x)$ be a polynomial such that $P(x^2+1)=x^4+4x^2$. Find $P(12)$. | nvidia |
The function $f(x)$ takes a real number $x$ as input and returns the maximum possible integer $n$ such that the numbers $\{x\},$ $\lfloor x \rfloor,$ and $n$ form a geometric sequence in that order. If no such integer exists, the function returns $0$. Find the smallest real number $x$ such that $f(x) > 1$. | nvidia |
Let $n$ be a positive integer such that
$$(n-1)!(n+1)^2=n!(n^2-n+1).$$
Determine $n$. | nvidia |
Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1.$ Find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$ | nvidia |
What is the largest value of $x$ that satisfies the equation $\sqrt{2x}=4x$?
Consider the equation $\sqrt{x^2+1}=x^2$. Find the largest value of $x$ that satisfies this equation. | nvidia |
Find the sum of all the integers from 1 to 100 that are not divisible by 3 or 5. | nvidia |
Let $n$ be a positive integer with exactly five divisors. Determine the smallest value of $m$ for which $n^m$ has more than 20 divisors. | nvidia |
How many of the natural numbers from 1 to 600, inclusive, contain the digit 5 at least once? (The numbers 152 and 553 are two natural numbers that contain the digit 5 at least once, but 430 is not.) | nvidia |
Determine the range of values of $x$ such that the expression $\frac{x^2-5x+6}{x^2-7x+12}$ is positive. | nvidia |
Find the smallest positive integer $k$ such that $5^{2k} \equiv 1 \pmod{11}$ | nvidia |
How many of the positive integers less than or equal to 1000 can be expressed as the sum of two perfect cubes, where the two cubes may be positive, negative, or zero? | nvidia |
Determine the number of positive multiples of $5$ that have exactly three digits and end with the digit $0$, such that the sum of their digits is divisible by $5$. | nvidia |
Let $y = f(x)$ be a periodic function whose graph is shown below, where the period is $T=8$. Note that the graph is symmetric about $x=0$, $y=0$, and the line $y=x$. The equation $f(f(f(x)))=1$ has 16 solutions in $[-8,8]$. How many of them satisfy $f(x)>1$? [asy]
size(150);
real ticklen=3;
real tickspace=2;
real... | nvidia |
Find all integers $n$ such that $n^2+n+1$ is a perfect square. What is the sum of the squares of these integers? | nvidia |
A quadrilateral is inscribed in a circle with center at the origin, and has vertices at $(3,4)$, $(-3,4)$, $(-5,-12)$, and $(x,y)$. Given that the diagonals of the quadrilateral are perpendicular, find $x+y$. | nvidia |
The equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$ has infinitely many solutions in positive integers. Find the smallest value of $z$ such that there are exactly 5 distinct pairs of positive integers $(x,y)$ satisfying the equation. | nvidia |
Let $y_1,$ $y_2,$ $y_3$ be positive real numbers such that $y_1 + y_2 + y_3 = 20.$ Find the largest possible value of
\[\frac{1}{y_1} + \frac{1}{y_2} + \frac{1}{y_3}.\] | nvidia |
An ellipse and a hyperbola share a focus, and the center of the ellipse lies on one of the asymptotes of the hyperbola. The ellipse and hyperbola intersect at two points. Given that the equation of the hyperbola is $\frac{x^2}{16} - \frac{y^2}{9} = 1,$ find the area of the triangle formed by these two points and the ce... | nvidia |
In the expansion of $(1+x)^n$ using the binomial theorem, where $n$ is a positive integer and $x$ is a real number, what is the sum of the coefficients of the even-powered terms? | nvidia |
What is the largest prime factor of a positive integer $n$ that can be expressed as the sum of two distinct perfect squares? | 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 |
A complex number $z$ is such that $|z-4|=2$ and $|z+2|=4$. Find the product of the real part of $z$ and its imaginary part. | nvidia |
Find the minimum value of the expression $\frac{3}{x^2} + \frac{2}{x} + x$, where $x$ is a positive real number. | nvidia |
Let $a$, $b$, and $c$ be different positive integers less than 100,
such that for every integer $n>2$, the number $an^2+bn+c$ is not prime.
Do such numbers exist? | nvidia |
Determine $n$ such that the decimal representation of $\frac{1}{10^n-1}$ has a block of digits that repeats itself every $4$ digits. | nvidia |
Given that $p$ and $q$ are positive integers such that $p < q$ and $\frac{1}{p} - \frac{1}{q} = \frac{1}{pq}$, find the smallest possible value of $q$ if $pq$ is a perfect square. | nvidia |
Let $k$ be a real number such that $k > 1$ and
\[\sum_{n=1}^{\infty} \frac{5n-1}{k^n} = \frac{13}{4}.\]Find $k.$ | nvidia |
Find the number of distinct integers that can be expressed as the sum of two distinct elements from the set $\{1, 2, 4, 8, 16, 32\}$. | nvidia |
Let $T$ be the set of all integer ordered triples $(x, y, z)$ such that $x$, $y$, and $z$ are distinct and satisfy $0 \leq x$, $y$, $z \leq 4$. Three ordered triples are chosen independently and at random with replacement from the set $T$ and labeled $t_1$, $t_2$, and $t_3$ in the order they are chosen. The probability... | nvidia |
Determine the smallest positive integer that can be expressed as the sum of two cubes in two different ways, where the cubes are integers. | nvidia |
Let $x_1, x_2, \ldots, x_n$ be real numbers such that $x_1x_2\cdots x_n = 1$ and
\[|x_1|+|x_2|+\cdots+|x_n|\leq 10.\]
Find the largest possible value of $n$. | nvidia |
Consider a polynomial $P(x)$ of degree $n$, where $n$ is a positive integer. The coefficients of the polynomial are such that $a_k = a_{n-k}$ for all $k$. Prove that if all the roots of $P(x)$ are real, then $P(x)$ has at least one root in the interval $[-1, 1]$. | nvidia |
Consider two positive integers $m$ and $n$ such that $mn = 192$. Find the smallest possible value of $\operatorname{lcm}(m,n)$. | nvidia |
Palindromes are numbers that read the same backwards and forwards, like 5665. What is the least possible positive four-digit palindrome that is divisible by 3? | nvidia |
Let's say we have a sequence of squares of consecutive integers: $1^2$, $2^2$, $3^2$, $\ldots$. Find the value of $n$ such that the sum of the last 5 squares is equal to 825. | nvidia |
The product of three different positive integers is equal to $7^3$. What is the sum of the three integers? | nvidia |
The graph of $y=g(x)$ is shown below, with $1$ unit between grid lines. Assume $g(x)$ is defined only on the domain shown.
What is the sum of all integers $d$ for which the equation $g(x)=d$ has exactly $5$ solutions?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm... | nvidia |
In a certain triangle, the lengths of the three sides are consecutive integers, and the sum of the two shorter sides is equal to 27. Find the length of the longest side. | nvidia |
For how many positive integers $n$ is $100 \leq 2^n \leq 200$? | nvidia |
Let $p(x)$ be a polynomial such that
\[p(x^2 + 1) = p(x^2 + 2)\]
for all real numbers $x$. Find the smallest possible degree of $p(x)$. | nvidia |
If $n$ is an even integer, then find the largest integer that always divides the expression \[(n^2+n)(n^2-n)(n+3)(n-1).\] | nvidia |
If $m$ and $n$ are positive 3-digit integers such that $\gcd(m,n)=2$, what is the smallest possible value for $\mathop{\text{lcm}}[m,n]$? | nvidia |
Let \(p(x)\) be a monic polynomial with positive integer coefficients. The polynomial \(p(x)\) is divisible by \((x+1)^3\) and \(p(1)=9\). Find the smallest possible value of \(p(0)\). | nvidia |
Determine the number of positive integers less than or equal to $50$ that can be expressed as the difference of two perfect squares. | nvidia |
A quadratic function $f(x) = ax^2 + bx + c$ has two real roots. The product of the roots is $-2$ times the sum of the roots. Determine the value of $a$ if the graph of the function has a vertex at the point $(1, -1)$. | nvidia |
If $a$ and $b$ are positive integers such that $ab=2^{64}$ and $a$ has exactly $5$ digits more than $b$, determine the last two digits of $a$. | nvidia |
What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers? | nvidia |
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