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The points $0$, $z$, $iz$ and $(1+i)z$ form a square in the complex plane. Given that the length of the diagonal of this square is 6, determine the smallest possible perimeter of the square if the imaginary part of $z$ is positive and $\arg(z) \in (\frac{\pi}{4}, \frac{\pi}{2})$. | nvidia |
There are 2023 positive integers with both their sum and the sum of their reciprocals equal to 2024. Let y be one of these integers. Find the minimum value of y + 1/y. | nvidia |
Find all non-zero digits $a$ such that there are exactly 3 three-digit numbers whose digits are a permutation of 1, 2, and $a$, and are divisible by $a$. | nvidia |
Let $a$ and $b$ be two positive integers such that $\text{gcd}(a,b)=1$. Suppose that there exist integers $x$ and $y$ such that $ax+by=1$.
Determine the remainder when $(ab-b-a+1)x+(a+b-1)y$ is divided by $ab$. | nvidia |
Find the smallest positive integer n such that the product of its proper divisors (excluding n itself) is equal to n. | nvidia |
Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. | nvidia |
Find all positive real numbers $x$ that satisfy
\[x \sqrt{12 - x} + \sqrt{12x - x^3} \ge 12.\]Enter all solutions, separated by commas. | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 312^\circ.$ | nvidia |
Find the number of positive integers that are less than or equal to 24 and have a least prime factor of 3.
(Note: This problem requires a different approach than the original problem, which was a simple sum of factors. This problem involves analyzing the prime factorization of numbers and counting the numbers that mee... | nvidia |
Let $a$ and $b$ be two real numbers such that $|a|=|b|=1$ and $a+b=-1.$ Find the exact value of the expression $|ab-2|.$ | nvidia |
Find the only positive real number $x$ such that $\displaystyle \frac{x^2-2x+1}{4x} = \frac{2x+1}{x^2-4x+4}$, where $x \neq 2$. | nvidia |
In a triangle, the length of the hypotenuse is 10 cm. What is the greatest integer number of centimeters that the sum of the lengths of the other two sides could be? | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $a+b+c=3$ and
$$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=\frac{15}{7}.$$ Determine $abc$. | nvidia |
Find the minimum value of \[\frac{x^3}{y-2} + \frac{y^3}{x-2}\]for real numbers $x > 2$ and $y > 2$. | nvidia |
Let $f(x)$ be a function such that for any positive real numbers $x$ and $y$, the following holds:
\[f(xy) = f(x) + f(y).\]
It is also known that for all positive integers $n$, the value of $f(n)$ is a positive integer and $f(100) = 10.$
Find the smallest possible value of $\displaystyle \sum_{k=1}^{49} f(k).$ | nvidia |
Let $f(x)$ be a function such that $f(x) = \sin^2 x$ for $x < \pi/2$ and $f(x) = \cos^2 x$ for $x > \pi/2$. Determine all real values of $x$ such that $f(x) = f(\pi/2-x)$. | nvidia |
In a Cartesian coordinate system, consider the set of points that satisfy the inequality $x^2 + y^2 < 25$. Among the points $(1,3)$, $(4,2)$, $(3,4)$, and $(-2,5)$, which ones lie within the region defined by the inequality? Find the sum of the $x$-coordinates of these points. | nvidia |
Determine the value of $k$ such that the graphs of $y=2^x$ and $y=kx$ intersect at exactly two points. | nvidia |
Find the largest value of $x$ such that the equation
$\cos x + \cos 2x = 1$ holds true. | nvidia |
Suppose that $x$ and $y$ are integers such that $$2y = 15 - 3x.$$How many of the first five positive integers must be divisors of $3y + 9$? | nvidia |
Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. Prove that
$$\cos\left({\frac{\alpha}{3}}\right)\cos\left({\frac{\beta}{3}}\right)\cos\left({\frac{\gamma}{3}}\right)\leq\frac{1}{8}$$ | nvidia |
Six 6-sided dice are rolled. What is the probability that the number of dice showing even numbers and the number of dice showing odd numbers is equal? | 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 |
The smallest positive integer $n$ such that $\tan n$ is defined and the smallest positive integer $m$ such that $\tan (n + m)$ is undefined are such that $n + m$ has exactly $3$ distinct prime divisors.
Find $n$. | nvidia |
Find the sum of the reciprocals of the roots of $x^2-13x+4=0$. | nvidia |
Let $n$ be a positive integer. Find the largest power of $n$ that divides $\frac{n! \cdot (n + 1)!}{(n!)^2}$. | nvidia |
Find the value of $x$ in the geometric sequence $3, x, 12$ such that the sum of the terms is equal to 25. | nvidia |
Find the area of a triangle with vertices at (-2,1), (5,8), and (3,-4). | nvidia |
Find the number of positive real solutions of the equation $\left| x^2 - \frac{3}{2} \right| = 5$. | nvidia |
Let \(p\) be a prime number. Suppose that \(x\) and \(y\) are positive integers such that \(p^{x}-p^{y}=p^{2}+1\) and \(x-y\) is minimal. Prove that \(p=7\). | nvidia |
A fascinating number is a positive integer that is divisible by the product of its digits. What is the largest positive multiple of 5 that is not a fascinating number? | nvidia |
Given that $x$ is a positive integer, find the smallest value of $x$ such that $x^2+2x+1$ is a multiple of $7$. | nvidia |
A sequence of squares is formed by connecting the midpoints of the sides of a larger square. The resulting sequence of squares has side lengths that form a geometric series with common ratio $1/2$. If the area of the largest square is 16 square units, what is the sum of the areas of all the squares in the sequence? | nvidia |
What is the sum of the digits of the smallest positive integer $n$ such that $n$ is a multiple of 3 and $n$ can be expressed as the sum of three distinct perfect squares? | nvidia |
Find the number of positive integers $n \le 1000$ such that $15n$ is a perfect cube. | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x^2) + f(y^2) - f(x)f(y) \le -1\]
for all real numbers $x$ and $y$. | nvidia |
A right circular cone has a base with radius $300$ and height $200\sqrt{3}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $150$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $225\sqrt{5}.$... | nvidia |
Two positive integers are such that their sum is 57, and the sum of their squares is 1435. What is the product of these two integers? | nvidia |
The function $g$ has the property that, for any positive integers $m$ and $n$ such that $m + n$ is a perfect square, the equation
\[g(m) \cdot g(n) = \frac{1}{m + n}\]holds. What is $g(25)$? | nvidia |
Consider a function $g$ defined on the real numbers, satisfying the following properties:
1. $g(x + 5) = g(x)$ for all $x$
2. $g(x + 3) = -g(x)$ for all $x$
3. $g(0) = 1$
Find the least number of distinct values that $g(x)$ must take on the interval $-1000 \leq x \leq 1000$. | nvidia |
Find the sum of all positive integers that are one more or one less than a perfect square and can be expressed as the sum of two distinct perfect squares. | nvidia |
Let $f(x)$ be a function that satisfies the following conditions:
$f(x+y)=f(x)+f(y)$ for all real numbers $x$ and $y$,
$f(0)=0$,
$f(1)=1$, and
$f(2)=4$.
Find the value of $f(3)$. | nvidia |
In a given right triangle with legs of length 7 and 24, a circle is inscribed such that it is tangent to all three sides of the triangle. Determine the radius of this inscribed circle. | nvidia |
Let $z$ be a complex number such that $z^2 + z = -1$. Determine the sum of the squares of the real and imaginary parts of $z$. | nvidia |
Given the equation of a circle $(x-2)^2 + (y-3)^2 = 4$, find the sum of the $x$-coordinates of the points where the circle intersects the $x$-axis and the sum of the $y$-coordinates of the points where the circle intersects the $y$-axis. | nvidia |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | nvidia |
Find the smallest positive integer $k$ such that $\frac{k+25}{k-17}$ is equal to a terminating decimal.
In this new problem, we need to find the smallest positive integer $k$ such that $\frac{k+25}{k-17}$ can be expressed as a terminating decimal. To do this, we can use the same approach as in the original problem, wh... | nvidia |
A sphere is inscribed in a cube such that the center of the sphere coincides with the center of the cube.
The sphere touches the sides of the cube at the middle points.
The ratio of the volume of the sphere to the volume of the cube is $k$.
Find $\ds\sqrt[3]{\f{\sqrt[3]{5}\sqrt[3]{k}}{1+\sqrt[3]{4k}}}$ | nvidia |
If $f(x)=\cos x+i\sin x$ for all real numbers $x$ and $$\prod_{k=1}^{20}f(12k)=a+bi,$$where $a$ and $b$ are real numbers, then compute $a$. | nvidia |
The sum of the digits of a two-digit number is equal to the product of the digits of the same number. How many such numbers exist? | nvidia |
If $x + \frac{1}{x} = \sqrt{3}$, then find $x^{18}$.
Find all values of $x$ that satisfy the equation $\sin(x) + \frac{1}{\sin(x)} = \sqrt{3}$. | nvidia |
Find the number of $5\times 5$ arrays with entries 0, 1, or -1, such that the sum of the entries in each row is 1 and the sum of the entries in each column is -1, and no two adjacent entries (horizontally or vertically) have the same value. | nvidia |
Let $g$ be a function taking the positive integers to the positive integers, such that
(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n,$ and
(iii) if $m \neq n$ and $m^n = n^m,$ then $g(m) = n$ or $g(n) = m.$
Find the sum of al... | nvidia |
What is the maximum number of consecutive odd positive integers that can be multiplied together before the product exceeds 1000? | nvidia |
Let $w$ be the positive real number satisfying $\left(\frac{1}{5}\right)^{1/w} \cdot \left(\frac{1}{5}\right)^w = \frac{1}{25}$. Find $w$. | nvidia |
Find the number of positive integers less than or equal to 100 that have an odd number of positive integer divisors. | nvidia |
Compute the product of the squares of the solutions of the equation
$$x^3-2x^2+x-3=0$$ | nvidia |
Find three positive integers $a$, $b$, and $c$ such that the sum of the cubes of $a$ and $b$ is equal to 138, the sum of the cubes of $a$ and $c$ is equal to 98, and the product of $b$ and $c$ is equal to 120. What is the sum of $a$, $b$, and $c$? | nvidia |
Let $ S $ be the set of all sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? | nvidia |
In a rectangular room with dimensions 12 meters by 10 meters, a certain number of identical, non-overlapping, right-angled triangles can be placed such that the hypotenuse of each triangle is along the length of the room and one leg is along the width of the room. If the side lengths of the triangles are integers and t... | nvidia |
Find the value of the sum of all the coefficients in the expansion of $(1+\sqrt{3})^6$. | nvidia |
What is the sum of all real numbers x such that 2x + 3 = x^2 + 3/x? | nvidia |
The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? | nvidia |
Find the sum of all integral values of $d$ with $d\ge -20$ for which the equation $z=x^3-5x-d$ has exactly one real root. | nvidia |
Find the real number $x$ such that $\lfloor x \rfloor \cdot (x-\lfloor x \rfloor)=19$. Express $x$ as a decimal. | nvidia |
Find the sum of all the perfect squares between 1 and 20. | nvidia |
Find all integers $n$ such that $201n + 299m = 1$ for some integer $m$, and then determine the smallest value of $|n|$ that satisfies this equation. | nvidia |
Find the number of triples of nonnegative integers $(x,y,z)$ such that $x+y+z=12$ and $\sin(\frac{x\pi}{4})+\sin(\frac{y\pi}{4})+\sin(\frac{z\pi}{4})=0$. | nvidia |
Find the product of the solutions of the equation $x^3-6x^2+11x-6=0$. | nvidia |
A circle is inscribed in a rhombus with side length 2, such that one of the diagonals of the rhombus forms the diameter of the circle. What is the length of the other diagonal? | nvidia |
Find the sum of all the digits that appear after the decimal point when $\frac{1}{17}$ is expressed as a decimal. | nvidia |
Find the matrix $\mathbf{A}$ such that $\mathbf{A}^2 \mathbf{v} = \mathbf{v}$ for all vectors $\mathbf{v}.$ | nvidia |
Find the distance between the center of the circle with equation $x^2+y^2=8x-2y+5$ and the point $(4,3)$. | nvidia |
Let $P$ be a point on the line $2x+3y=1$. Find the point on the line $3x-2y=4$ that is closest to $P$. | 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 |
Given that $m$ is an integer and $0 < 2m < 20$, what is the sum of all possible integer values of $m$? | nvidia |
Let $m$ be a positive integer, and let $a_0, a_1, \dots , a_m$ be a sequence of real numbers such that $a_0 = 37$, $a_1 = 72$, $a_m=0$, and $$ a_{k+1} = a_{k-1} - \frac{3}{a_k} $$for $k = 1,
2, \dots, m-1$. Find $m$. | nvidia |
The product of the $x$-values that satisfy the equation $2=\frac{x^2+3x-4}{x-1}$ is equal to 15. Find the sum of these $x$-values. | nvidia |
The sum of the cubes of two consecutive integers is equal to 1575. Determine the product of the two integers. | nvidia |
Find the minimum value of the expression $x^2+y^2+8x-2y+25$ for real $x$ and $y$. | nvidia |
Find the value of x such that the area of the rectangle with sides (3 + x) and (8 - x) is equal to the area of a square with side 3√2. | nvidia |
There are 6 different cameras and 5 different smartphones that can be displayed in a store window in the following way: all the cameras are displayed on the left side of the window and all the smartphones are displayed on the right side. Determine the number of ways to arrange the items in the window if one specific ca... | nvidia |
The decimal expansion of $\frac{1}{9}$ is $0.\overline{1}$. Find the value of $k$ such that the decimal expansion of $\frac{1}{3^k}$ has a repeating block of length $3$. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$$\frac{1}{x_1}+\frac{1}{x_2}=\frac{1}{2}.$$
Determine $a$. | nvidia |
Given that $x$ is a positive integer, determine the number of terms in the expansion of $$\left(x^2+2x+1\right)\left(x^3+3x^2+x\right)-\left(x^3-2x^2+7x\right).$$ | nvidia |
What is the smallest value of the base $a$ for which the number $21000_{a}$ can be written as the sum of two perfect squares in exactly two different ways? | nvidia |
Inside a regular hexagon with side length 8, two congruent isosceles triangles are drawn such that they share one side and each has one vertex on a vertex of the hexagon. What is the side length of the largest regular hexagon that can be inscribed in the space inside the hexagon and outside of the triangles? | nvidia |
If $k$ is a positive integer, then let $g(k)$ be the product of the digits of $3^k$ that are to the left of the decimal point. What is the smallest positive integer $k$ such that $g(k) > 200$? | nvidia |
Find all values of $a$ such that $\lfloor a \rfloor + a = 23.7$. | nvidia |
For how many positive integral values of $a$ is it true that $x = 2$ is the only positive integer solution of the system of inequalities $$
\begin{cases}
2x>3x-3\\
3x-a>-6
\end{cases}
$$ | nvidia |
Determine all real values of $a$ such that there exists a positive integer $n$ satisfying $a^n + a^{n-1} + \dots + a + 1 = n!$. | nvidia |
Find the sum of all values of $x$ such that $x^3 = 64$. | nvidia |
Determine the number of distinct prime factors of the sum of all positive integers less than or equal to 555 that are divisible by 3. | nvidia |
Let $f(x) = \sin(x) + \cos(x)$. Determine all the integers $n$ such that $$\frac{f(n)}{f(n+1)}\leq \frac{1}{n}$$ | nvidia |
Determine the smallest positive integer $n$ such that the sum of its positive divisors is a perfect square. | nvidia |
A point $P$ lies on the curve $y = \sqrt[3]{x}$. Find the value of $a$ such that the equation of the tangent line to the curve at $P$ can be expressed in the form $y = ax + \sqrt[3]{x}$, where $a$ is a constant. | nvidia |
Find the number of positive integers $k$ less than 200 that have a corresponding integer $j$ divisible by 4, such that the roots of $x^2 - kx + j = 0$ are consecutive negative integers. | nvidia |
The function $f(x)$ satisfies
\[f(x + y) = f(x) f(y)\]for all real numbers $x$ and $y.$ Find all possible values of $f(0).$
Enter all the possible values, separated by commas. | nvidia |
Find the number of integers $n$ that satisfy
\[10 < n^2 < 99.\] | nvidia |
Three equal chords are drawn from a point on the circumference of a circle radius 20 so that each chord is tangent to two circles that lie entirely inside the circle. All three chords and the circle form a figure with four regions inside it. If the area of the region inside the circle and outside the three chords is $x... | nvidia |
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