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There are nine points evenly distributed on the circumference of a circle: $A_1$, $A_2$, $A_3$, $B_1$, $B_2$, $B_3$, $C_1$, $C_2$, $C_3$, where the $A_i$'s, the $B_i$'s, and the $C_i$'s form equilateral triangles. A line segment joining two points forms a side of one of these three triangles if the remaining vertex of ... | nvidia |
Let $f(x)$ be a function such that
\[f\left(\frac{n}{2}\right) = (-1)^n\]
for all positive integers $n.$ Find the value of the expression
\[\sum_{i=0}^{\infty} f\left(\frac{i}{2}\right) x^i.\] | nvidia |
For positive integers $n$, let $g(n)$ return the smallest positive integer $m$ such that $\frac{1}{m}$ has exactly $2n$ digits after the decimal point. How many positive integer divisors does $g(1005)$ have? | nvidia |
What is the largest integer $n$ such that $n^2 \leq 12$? | nvidia |
A circle with center at the origin has a tangent line that intersects the x-axis at point (5,0) and the y-axis at point (0,b). Determine the radius of the circle. | nvidia |
Let $p,q,r,s,t,u,v$ and $w$ be distinct elements in the set of all the positive divisors of 96. Find the maximum possible value of the sum $pq+rs+tu+vw$. | nvidia |
Let $a$ and $b$ be two positive real numbers such that $b$ varies inversely with the square root of $a$. If $b=4$ when $a=9$, find the value of $b$ when $a=16$. | nvidia |
Given that a circle has a central angle of 25 degrees. Determine the degree measure of the corresponding inscribed angle with its vertex on the circle and sides intersecting the sides of the central angle. | nvidia |
Determine the smallest positive integer $n$ such that $\arctan(n)$ exceeds $\pi/3$. | nvidia |
Determine the number of positive integer divisors of $3^6 \cdot 11^4$ that are also multiples of 121. | nvidia |
Determine the number of ways to color the vertices of a regular pentagon with 5 different colors, such that no two adjacent vertices have the same color. | nvidia |
Find the value of $b$ that satisfies the equation $161_{b}+134_{b}=315_{b}$. | nvidia |
Compute the sum of all the two-digit positive integers that have the same units and tens digits. | nvidia |
Find the smallest positive integer $n$ such that the product of its digits is equal to the number of distinct positive factors of $n$. | nvidia |
In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that $\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92$, find $\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}$.
| nvidia |
Find the smallest possible value of $n$ such that there exist at least 1000 different $n$-digit binary strings where no two consecutive digits are the same. | nvidia |
Let $f(x)$ be a monic cubic polynomial with real coefficients such that
$$\lvert f(2x) \rvert \geq \lvert f(x) \rvert$$
for all real numbers $x.$ Find the product of the zeros of $f(x).$ | nvidia |
Given three complex numbers $x+yi$, $m+ni$, and $p+qi$ such that $n = -q$ and the product of the three numbers is real, find $x + p$. | nvidia |
The real numbers $a$, $b$, $c$, and $d$ are such that
$$\gcd(a,b)=\gcd(b,c)=\gcd(c,d)=\gcd(d,a)=\gcd(a,c)=5$$
and $\gcd(a,b,c,d)=1$.
Find the smallest possible value of $\lceil a\rceil\lceil b\rceil\lceil c\rceil\lceil d\rceil$. | nvidia |
What is the smallest positive integer $n$ such that the expression $\dfrac{n+7}{n^2-19n+55}$ has a positive denominator? | nvidia |
The vertices of a triangle are located at points $(a, 0)$, $(b, c)$, and $(0, d)$, where $a, b, c,$ and $d$ are positive integers. If the area of the triangle is equal to $\frac{7}{15}$, find the number of possible values for $d$. | nvidia |
In a certain right triangle, the ratio of the length of the hypotenuse to the sum of the lengths of the legs is $7/10$. If the cotangent of one of the acute angles is $a/b$, where $a$ and $b$ are relatively prime positive integers, find $a+b$. | nvidia |
Find the number of positive integers $n$ such that when $n$ is divided by $13$, the remainder is $5$, and when $n$ is divided by $15$, the remainder is $8$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c=2$. Find the minimum value of
\[g(a,b,c) = \left(1-\frac{a}{b+c}\right)^2 + \left(1-\frac{b}{c+a}\right)^2 + \left(1-\frac{c}{a+b}\right)^2.\] | nvidia |
Find the smallest positive integer that can be expressed as the sum of three different positive integers, each of which is a divisor of the original number. | nvidia |
Find the value of $x$ such that $\log_{64} (3x+2) = 1/3$. | nvidia |
Compute the largest power of $7$ that divides the product of all positive integers whose digits sum to $100$. | nvidia |
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12},$ find all possible values for $x.$
(Enter your answer as a comma-separated list.) | nvidia |
Find the value of $a$ such that the product of the roots of the equation $x^3 + 2x^2 + ax - 1 = 0$ is equal to the sum of their squares. | nvidia |
Consider the set of all planes passing through the origin in 3-dimensional space. For each such plane, compute the distance between the point (1, 2, 3) and the plane. Determine the maximum distance that can be achieved. | 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 |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)-... | nvidia |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | nvidia |
Find the number of four-digit numbers that have a symmetric structure, meaning the first and fourth digits are the same, and the second and third digits are the same. | nvidia |
Determine the value of $b$ such that the graph of $y=\frac{x^2-4}{x^2+bx+6}$ has a horizontal asymptote at $y=1$. | nvidia |
Find the sum of all possible values of $a$ in the equation $a\sqrt{a}=\frac{5}{2}\sqrt{b}$, where $a$ and $b$ are positive real numbers and $b$ is the product of the roots of the quadratic equation $2x^2+3x-5=0$. | nvidia |
Given that $a,$ $b,$ $c$ are distinct complex numbers such that
$$\frac{a}{1-b}=\frac{b}{1-c}=\frac{c}{1-a}=k,$$
find the sum of all possible values of $k$.
Another problem:
Let $x,$ $y,$ $z$ be distinct real numbers such that
$$x^2 + y^2 + z^2 = x + y + z.$$
Find the maximum value of $xyz$. | nvidia |
If the point $(2,5)$ is on the graph of $y=f(x)$, and $k(x)=(f(x))^3+2f(x)$ for all $x$, then there is one point that must be on the graph of $y=k(x)$. What is the product of the coordinates of that point? | nvidia |
Find the largest possible value of $k$ for which the expression $\sqrt{4x^2+3x+k}$ is defined for all real numbers $x$. | 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 |
Let $a$ be a real number, $0<a<\frac{\pi}{2}$, and let $w$ be a complex number such that
\[w + w^2 = \tan a.\]
Find $|w|$. | nvidia |
In a triangle, the line through the midpoint of a side and parallel to another side intersects the third side at a point that divides it into segments with a ratio of 3:2. If the length of the side opposite the vertex is 12 cm, find the length of the other two sides of the triangle. | nvidia |
Find the positive value of $x$ such that $2\log_{x}5-\log_{x}3=3$. | nvidia |
In a rectangular garden, two pathways of 18 meters and 42 meters long are constructed to connect two opposite corners. Find the area of the smallest possible square that can be formed using parts of these pathways as its sides, such that the remaining parts of the pathways can be removed without disconnecting the corne... | nvidia |
Let $a$, $b$, and $c$ be positive integers such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$ and $\text{lcm}(a,b)=100$. Find the number of possible triples $(a,b,c)$. | nvidia |
Find the positive integer $m$ such that $$\arcsin\frac {1}{3} + \arcsin\frac {1}{4} + \arcsin\frac {1}{5} + \arcsin\frac {1}{m} = \frac {\pi}{2}.$$ | nvidia |
Let $a_1, a_2, \dots, a_{2024}$ be integers such that every two integers among them are coprime. Prove that there exists an integer $k$ such that $a_k$ divides $$\sum_{i=1}^{2024} a_i.$$ | nvidia |
The line $y = 3$ intersects the graph of $y = 4x^2 + x - 1$ at the points $A$ and $B$. The distance between $A$ and $B$ can be written as $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m - n$. | nvidia |
The degree of the polynomial $f(z)$ is two, and the degree of the polynomial $g(z)$ is less than two.
Determine the degree of the polynomial $f(z)\cdot g(z)$. | nvidia |
Given a triangle $ABC$ with $\angle{BAC}=60^\circ$, $\angle{CBA}=45^\circ$, and $\angle{ACB}=75^\circ$, let $D$ be the point on $\overline{AC}$ such that $\angle{ABD}=40^\circ$. If $\overline{CE}$ is perpendicular to $\overline{BD}$, find $\angle{BCE}$. | nvidia |
How many positive integers $n$ satisfy $200 < n^2 < 900$? | nvidia |
Determine the value of $x$ such that the difference between the reciprocals of $x$ and $2x$ is equal to 0.25. | nvidia |
What is the least positive integer with exactly five distinct positive factors? | nvidia |
Find the value of $a$ and $b$ such that $a^{2004}+b^{2004}$ is divisible by 10. | nvidia |
What is the smallest whole number that must be a factor of the sum of any four consecutive positive integers? | nvidia |
Let $x$ and $y$ be positive real numbers such that $[x]+[y]=3$.
Evaluate $\left[\frac{x+y}{2}\right]$. | nvidia |
Find the intersection point of the plane $x+y+z=0$ and the line that passes through the points $P(2,3,4)$ and $Q(5,6,7)$. | nvidia |
Let $P(x)$ be a polynomial with real coefficients such that
$P(x^2) = x^4+ax^2+1$
for all real numbers $x$. Find $P(16) + P(-1)$. | nvidia |
Find the value of $x$ that satisfies the equation:
$$\frac{2^{2x}}{2^{x+1}} = 3^x \cdot 2^{-1}$$ | nvidia |
The roots $x_1$, $x_2$ and $x_3$ of the equation
$$x^3-7x^2+11x-a=0$$
are such that
$\ds\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1}=5$.
Determine $a$. | nvidia |
Find the area of the triangle bounded by the $x$-axis and the lines $x+2y=5$ and $2x-3y=-7$. | nvidia |
What is the minimum number of guests that must be present in a party so that it is guaranteed that there are at least 2 people with their birthday in the same month? | nvidia |
If $x$ and $y$ are real numbers such that $x < y$ and $\lceil{x}\rceil - \lfloor{y}\rfloor = 2$, find the range of all possible values of $x + y$. Express your answer using interval notation. | nvidia |
Let $f(x)$ be a function defined on the set of positive integers such that
$$f(x) = \begin{cases} x^2 &\quad x\text{ is a perfect square} \\ f(y) + x - y &\quad \text{otherwise} \end{cases}$$
where $y$ is the largest perfect square less than or equal to $x.$ Find the set of all positive integers $x$ such that $f(x) =... | nvidia |
In a convex polygon with n sides, a diagonal is a line segment that connects two non-adjacent vertices. Prove that the number of diagonals in a convex polygon with n sides is given by the formula: (n(n-3))/2. Use this formula to find the number of diagonals in a convex polygon with 10 sides. | nvidia |
Let $f(x)$ be a function defined for all positive integers satisfying the conditions $f(x) > 0$ for all $x > 0$ and
\[f(x + y) = \sqrt{f(xy) - 1}\]for all $x, y > 0.$ Determine $f(2010).$ | 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 |
Let $S$ be the sum of the cubes of the first 10 positive integers and let $T$ be the sum of the cubes of the first 10 negative integers. Determine the value of $S + T$. | nvidia |
Find the sum of all values of $x \in [0, 2\pi]$ that satisfy
\[\sin^2 x \cdot \cos x = \frac{\cos x}{\cos^2 x + 1}.\] | nvidia |
Consider the expression $\left(\frac{1}{7^m}\right)^n+\left(7^n\right)^m$, where $m$ and $n$ are integers. For what values of $m$ and $n$ will this expression be equal to 2? | nvidia |
In the equation
$$ax^2-5x+c=0$$
determine $c$ such that the sum of the roots be equal to $2a$. | nvidia |
The vertices of a convex pentagon are the five complex numbers $z$ that satisfy
\[|z|^5 + |z|^3 + 1 = |z^5 + z^3 + 1|.\]
Prove that the pentagon is a regular pentagon. | 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 |
Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. | nvidia |
Find the number of positive integers less than 100 that are divisible by 3 but not by 7. | nvidia |
A square is inscribed in a semicircle with radius $r$, such that one vertex touches the semicircle's diameter and the opposite vertex lies on the semicircle's arc. Find the side length of the square as a function of $r$. | nvidia |
Find the sum of all coefficients in the expansion of $(x+y+z)^5$. | nvidia |
Let $a_n$ be a sequence of positive real numbers such that $a_n < 1$ for all $n$. Find the value of $b > 1$ such that the infinite series
$$1 + a_1 + ba_1^2 + b^2a_1^3 + \dotsb + ba_2^2 + b^2a_2^3 + \dotsb + b^2a_3^3 + b^3a_3^4 + \dotsb = 7.$$
(Note: Assume that the series converges.) | nvidia |
A circle and a square are inscribed in a larger square, such that the circle is tangent to the sides of the larger square and the square is inscribed in the circle. If the side length of the larger square is 12 cm, find the maximum possible difference between the areas of the circle and the inscribed square. | nvidia |
In a triangle, the lengths of the three sides are consecutive integers. The perimeter of the triangle is 18 units. The length of the longest side is twice the length of the shortest side. Find the area of the triangle. | nvidia |
How many 6-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 6 exactly once, such that the number contains at least one pair of consecutive digits that are coprime (i.e., have no common factors other than 1)? | nvidia |
Find the largest integer $n$ such that $n^3+2n^2+3n+1$ is a perfect square. | nvidia |
Find the smallest positive integer $n$ that satisfies the congruence $2n \equiv 5 \pmod{11}$. | nvidia |
How many four-digit numbers are there that have exactly two digits that are the same? | nvidia |
Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$? | nvidia |
What is the smallest positive integer $n$ such that the product of the terms in the sequence $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots, \frac{n-1}{n}, \frac{n}{n+1}$ is less than $\frac{1}{100}$? | nvidia |
A circle is inscribed in a regular hexagon. The area of the circle is 150 square centimeters. What percentage of the area of the hexagon is the area of the circle? | nvidia |
For all positive integers $n$, let $\nabla n$ be the product of all digits of $n$. Find the smallest positive integer $n$ such that $\nabla n = 100$. | nvidia |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real who... | nvidia |
Prove or disprove: The product of the floors of the square roots of the positive divisors of 1000 is equal to the product of the floors of the cube roots of the positive divisors of 1000. | nvidia |
Determine the sum of all the digits in the decimal representation of 1/13, given that it is a repeating decimal with a cycle of 6 digits. | nvidia |
Let $f(x)$ be a function defined on the real numbers, and let $a$ be a real number. Define $f^1(x) = f(x)$ and, for each positive integer $n$, define $f^{n+1}(x) = f(f^n(x))$. For what value of $a$ is $f^3(x) = x$ for all $x$, given that $f(x) = ax + (1-a) \cdot (-2)$? | nvidia |
Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^3 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite". | nvidia |
Find the sum of all positive integers less than 100 that are perfect cubes of prime numbers. | nvidia |
Find the smallest number that has exactly five proper divisors. | nvidia |
A portion of the graph of $f(x)=ax^2+bx+c$ is shown below. The distance between grid lines on the graph is $1$ unit.
What is the value of $a+b+2c$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-... | nvidia |
Find the maximum value of the expression $$\sqrt{x+2}\cdot\sqrt{3x+1}\cdot\sqrt{7-x}\cdot\sqrt{5-3x},$$ where $x$ is a real number. | nvidia |
A sequence $a_n$ has the property that $a_{n+1}=a_n+e^{-n}$ for all positive integers $n$. If $a_1=2$, what is $\lim_{n\to\infty} a_n$? | nvidia |
In the Cartesian plane, consider the set of points $(x,y)$ that satisfy the equation
$$\sqrt{x^2+y^2}+\sqrt{(x-3)^2+(y-4)^2}=7.$$
Determine the shape formed by this set of points and find the area enclosed by it. | nvidia |
Consider the set of all polynomials of degree 3 with real coefficients that have all roots equal to -1 or 1 and have the leading coefficient equal to 1.
Find the number of polynomials in this set, which have at least one root equal to -1. | nvidia |
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