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In the rectangle $ABCD$, the point $E$ is chosen on the side $AB$ such that the area of the triangle $ADE$ is equal to the area of the triangle $BCE$. Find the length of $AE$, given that $AB = 20$ and $AD = 12$. | nvidia |
A circle is inscribed in a square, and then a smaller square is inscribed in the circle. The ratio of the area of the smaller square to the area of the larger square is 1/8. What is the ratio of the area of the circle to the area of the larger square? | nvidia |
Find the maximum value of $\cos^2 x - \sin^2 x + \sin x \cos x$ over all real numbers $x.$ | nvidia |
The function $g(x)$ satisfies
\[g\left(\frac{x}{y}\right) = \frac{g(x)}{y^2}\]for all positive real numbers $x$ and $y.$ If $g(5) = 10,$ find $g(10).$ | nvidia |
Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2+ax+b=0$ has solutions $a$ and $b$. Find the ordered pair $(a,b).$ | nvidia |
Find the number of integer values of $c$ that satisfy the inequality $(x^2 + cx + 1)(x^2 - cx + 1) \ge 0$ for all integer values of $x$. | nvidia |
Find all values of $a$ for which the points $(0,0,0),$ $(1,a,0),$ $(0,1,a),$ and $(a,0,1)$ are coplanar. | nvidia |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum?
A circle passes through the points $(1, 7)$ and $(13, 16)$. Find the coordinates of the center of the circle for which the radius is min... | nvidia |
Let $A$ be a set of six numbers such that $A = \{2, 3, 4, 5, 6, n\}$, where $n$ is a positive integer greater than $6$. Determine the smallest value of $n$ such that the sum of the smallest prime factor of each number in $A$ is equal to $15$. | nvidia |
Compute the number of ways to choose 3 elements from the set {1, 2,..., 15} such that the sum of the chosen elements is a multiple of 3. | nvidia |
Let $x$, $y$ and $z$ be three positive integers such that
$\text{lcm}(x,y)=1620$
and
$\gcd(z,x)=36$, $\gcd(z,y)=72$.
Find $\text{lcm}(x,y,z)$. | nvidia |
Let $a,b,c$ be three real numbers such that $a$ is the $x$-coordinate of the vertex of the parabola $y=x^2+bx+c$, $b$ is the $x$-coordinate of the vertex of the parabola $y=x^2+ax+c$, and $c$ is the $x$-coordinate of the vertex of the parabola $y=x^2+ax+b$. Find $a+b+c$. | nvidia |
Compute the sum of the roots of the equation \[x\sqrt{x} - 6x + 7\sqrt{x} - 1 = 0,\]given that all of the roots are real and nonnegative. | nvidia |
Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$. | nvidia |
There are values $A$ and $B$ such that
\[\frac{Bx-11}{x^2-7x+10}=\frac{A}{x-2}+\frac{3}{x-5}.\]Find $A+B$. | nvidia |
Problem:
Consider a sequence of transformations $T_n$ defined on the complex plane, where $T_n(z) = z^n + 1$. Let $z_0$ be a fixed point such that $T_m(z_0) = z_0$ for some positive integer $m$. Find the smallest positive integer $k$ such that $T_{km}(z_0) = z_0$ for all positive integers $m$. | nvidia |
The vertices of a cube are labeled with 8 different integers such that the sum of the numbers at the endpoints of each edge is the same. Find the smallest possible sum of the labels. | nvidia |
Find the largest positive integer $n$ such that
$$\sum_{k=0}^n\arctan\frac{1}{k^2+k+1}\leq\frac{\pi}{4}+\arctan\frac{1}{100}.$$ | nvidia |
Find all real values of $x$ that satisfy the equation
\[
\cos(x) = \frac{x^2 - 1}{x^2 + 1} \, ,
\]
such that $|x| < \pi$. | nvidia |
Find the least positive integer $n$ such that $\tau(n)$ is a prime number and $\tau(n+1)$ is a perfect square. | nvidia |
Consider the parallelogram with vertices $(10,45)$, $(10,114)$, $(28,153)$, and $(28,84)$. A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| nvidia |
Let $a$, $b$, $c$ be the lengths of the sides of a right triangle, where $a$ and $b$ are the legs and $c$ is the hypotenuse. Find the number of possible choices of $a$ and $b$ such that $a+b+c=25$ and $a<b<c$. | nvidia |
If $x$ is a real number such that $\frac{1}{3}x+\frac{1}{4}x+\frac{1}{5}x=5$, what is the value of $\frac{4}{3}x$? | nvidia |
What is the maximum number of regions that can be formed by drawing 3 different parabolas and 2 different straight lines on the same plane, such that no three curves intersect at the same point? | nvidia |
Find the maximum value of
\[\frac{x^2}{y + 1} + \frac{y^2}{x + 1}\]for real numbers $x > 0$ and $y > 0$ such that $x + y = 1.$ | nvidia |
In the equation
$$7x^2-kx+2=0$$
determine $k$ such that the sum of the roots be equal to $3.25$. | nvidia |
Find the area of the triangle with vertices $(-2,3),$ $(5,1),$ and $(8,6),$ given that it is inscribed in a circle with center $(2,4)$ and radius $5$. | nvidia |
For positive integers $n$, define $S_n$ to be the minimum value of the sum
\[\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},\]where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $n$ for which $S_n$ is also an integer. | nvidia |
The diagonals of a convex quadrilateral $ABCD$ meet at point $E$ inside the quadrilateral. Prove that if $AB=AD$ and $[ACE]=[BCE]$, then $AE+BE=CE+DE$. | nvidia |
Find the smallest positive integer $k$ such that $2^k \cdot 5^k$ is a factor of $100!$. | nvidia |
Find the equation of the circle that passes through the points (2,3), (4,5), and (6,7). | nvidia |
What is the $23^{\mathrm{rd}}$ non-square positive integer that is divisible by exactly 4 distinct positive integers? | nvidia |
The perimeter of the largest rectangle that can fit inside a circle of radius 5 is what? | nvidia |
In the equation $2x^2+bx-6=0$, the product of the roots is $-3$. Determine the value of $b$ such that the sum of the cubes of the roots is equal to $-9$. | nvidia |
How many different 5-digit positive integers with exactly two 3's and no 0's exist? | nvidia |
Let $a$, $b$ and $c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is the least positive integer such that $a^c\equiv 1\pmod b$. Let $S$ be the sum of the first $c$ positive multiples of $a$. Determine the remainder when $S$ is divided by $b$. | nvidia |
Find the sum of all positive divisors of $70$ that are also divisors of $21$. | nvidia |
Find the sum of the numbers that are divisible by 5 between 20 and 50, inclusive. | nvidia |
Let $B$ be the set of all numbers which can be represented as the product of two consecutive positive integers. What is the smallest number that is not a multiple of any number in $B$? | nvidia |
Let $n$ be a positive integer which can be expressed as a product of two composite numbers that differ by one. For example, since $35 = 5 \cdot 7$, and $6 \cdot 8$ and $12 \cdot 13$ are not products of two prime numbers, it follows that $35$ is the only such number $n$. Determine whether there exist infinitely many suc... | nvidia |
Consider the function $f(x)=x^3+x^2+x+1$. Find the smallest prime number $p$ such that there exists a positive integer $n$ for which $f(n)$ is not relatively prime to $f(n+p)$. | nvidia |
Find the value of $\frac{2}{5}+\frac{3}{8}$, given that the sum of the two fractions is equal to the area of a rectangle with a length of 3 and a width of $\frac{7}{10}$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) f(y) - f(xy) = x + y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | nvidia |
Find all real numbers $x$ such that $x^3 + x^2 + x + 1 = 0.$ Enter the solutions, separated by commas. | nvidia |
The solutions $x$ and $y$ of the equation
$$x^2+y^2=k$$
are such that $e^x$ and $e^y$ are both within $\epsilon$ of 3 for some positive $\epsilon<0.01$.
What is the largest possible value of $k$? | nvidia |
Given that $A=(0,9)$ and $B=(0,12)$, and points $A'$ and $B'$ are on the line $y=x$, determine the angle between $\overline{AA'}$ and $\overline{BB'}$ if the intersection point $C$ is $(2,8)$. | nvidia |
Find the constant $a$ such that the quadratic expression $x^2+ax+1$ can be factored as the product of two binomials with integer coefficients. | nvidia |
Find the smallest positive integer n such that (n+1)! - n! is divisible by 4 and 17. | nvidia |
Let's consider the problem of finding the maximum value of the expression:
$$\frac{\lfloor x \rfloor \cdot \lfloor y \rfloor}{x+y},$$
where $x$ and $y$ are real numbers satisfying $2 < x < 5$ and $3 < y < 6$. Here, $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
This problem requires anal... | nvidia |
Let $f(x)$ be a polynomial such that $f(2+\sqrt{7}) = 1$. If it is known that $f(x)$ is a polynomial of degree $n$ with integer coefficients, and if all the natural numbers less than $11$ are also roots of this polynomial, determine $n$. | nvidia |
Determine the value of $c$ such that the inequality $x^2+4x+c \geq 0$ has exactly one integer solution. | nvidia |
Determine the maximum possible value of the sum
\[\frac{2a}{b} + \frac{b}{c} + \frac{c}{2a},\]where $a,$ $b,$ and $c$ are positive real numbers such that $a^2 + b^2 + c^2 = 1.$ | nvidia |
Let $x$ be the smallest positive real number such that $\lfloor x^n \rfloor + \lfloor x^{n+2} \rfloor$ is a multiple of 7 for every positive integer $n$. Compute the value of $\lfloor x \rfloor$. | nvidia |
Let $f(x)$ be a function such that $f(0) = 1$ and
\[f(xy) = f \left( \frac{x+y}{2} \right) + |x - y|\]for all real numbers $x$ and $y.$ Find $f(x).$ | nvidia |
Solve for the sum of all possible values of $x$ when $2^{x^2+5x+6}=4^{x+3}$. | nvidia |
If $a$, $b$ are nonzero real numbers such that $a^2+b^2=8ab$, find the value of $\left|\frac{a+b}{a-b}\right|$. | nvidia |
If $a$, $b$, and $c$ are integers such that $a<b<c$ and if the average of the three numbers $5a$, $5b$, and $5c$ is 15, determine the value of $b$. | 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 |
In triangle $ABC$, $A'$, $B'$, and $C'$ are the midpoints of sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ intersect at points $D$, $E$, and $F$, respectively, and that $\frac{AD}{DA'}+\frac{BE}{EB'}+\frac{CF}{FC'}=37$, find the ratio of the area of triangle $DEF$ to the area of triangle ... | nvidia |
For any integer $n>2$, the number of prime numbers greater than $n!+n$ and less than $n!+2n$ is:
$\text{(A) } 0\quad\qquad \text{(B) } 1\quad\\ \text{(C) } \frac{n-1}{2} \text{ for n odd, } \frac{n}{2} \text{ for n even}\quad\\ \text{(D) } n-2\quad \text{(E) } n-1$ | nvidia |
Given that the graph of the equation
$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
is a hyperbola where $a > b,$ and given that the angle between the asymptotes is $60^\circ,$ find $\frac{a}{b}$ if the hyperbola passes through the point $(4,0).$ | nvidia |
Simplify the expression $\sqrt[8]{x^3} \cdot \sqrt[4]{x^2}$ for any positive real number $x$. | nvidia |
Let $a$ be a positive integer. Find the smallest integer $n$ such that for all integers $x$ the expression $a^n+x^2$ is divisible by $a$. | nvidia |
Consider the infinite sequence consisting of positive integers, where every number in the sequence is divisible by $3$ and the sequence starts with $4$ numbers: $9$, $12$, $15$, and $18$. Determine how many numbers are in this sequence that have exactly $4$ digits. | nvidia |
Let $g(x)=3x-2$. Find the product of all $x$ that satisfy the equation $g^{-1}(x)=g(x^2)$. | nvidia |
A regular octagon with sides of length $8$ is inscribed in a circle. The vertices of the octagon are labeled in a counterclockwise order, with $A$ being the vertex at the top of the octagon. Eight segments are chosen, each starting at one of the vertices of the octagon and extending to a point on the circle. Each segme... | nvidia |
Determine x such that (8-x)/(2+x)=1/3 | nvidia |
Find the sum of all values of x such that x^3 = -8. | nvidia |
Let $S(n)$ denote the sum of the digits of a positive integer $n$. Find the number of positive integers $n$ that satisfy the condition $S(n)+S(n^2)+S(n^3)=2008$.
For example, if $n=2$, then $S(2)+S(2^2)+S(2^3)=2+4+8=14$, so $n=2$ does not satisfy the condition.
If $n=11$, then $S(11)+S(11^2)+S(11^3)=2+4+6=12$, so $n=... | nvidia |
Find the number of three-digit numbers whose digits are all distinct and form a geometric progression. | nvidia |
Let $m$ and $n$ be positive integers with $\gcd(m,n)=1$. If $\text{lcm}(m,n)+mn=72$, find the smallest possible value of $mn$. | nvidia |
Find the smallest possible perimeter of a triangle with integer side lengths, where the sum of the squares of the side lengths is a perfect square. | nvidia |
Determine the number of positive real solutions to the system of equations:
$$\begin{cases}
x^{10}+7x^9+14x^8+1729x^7-1379x^6=0 \\
x^7+5x^6-22x^5+63x^4-23x^3-81x^2+90x-19=0
\end{cases}$$
Consider using methods like elimination or substitution to simplify the system, and then analyze the resulting equation(s) to deter... | nvidia |
Let $g(n)$ be the sum of the prime factors of $3^n$ that are greater than $n$. What is the smallest positive integer $n$ such that $g(n) > n^2$? | nvidia |
Let $f(n)$ be the product of all the positive divisors of the integer $n$. How many integers $i$ exist such that $1 \le i \le 2010$ and $f(i) = i^{\sqrt{i}}$? | nvidia |
Find the sum of the real roots of the equation $2x^3 + 3x^2 - 6x - 2 = 0$ given that one of the roots is a complex number. | nvidia |
Let \(g\) be a function defined on the set of positive integers that takes positive integer values. The following information about \(g\) is given:
\[\begin{array}{ll}g(1)=3,&g(2g(1))=15,\\ g(g(1))=5,&g(4g(1))=9.\\ g(2)=1,&g(3g(1))=7,\\ g(3)=2,&g(5g(1))=11.\end{array}\]
Find \(g(9)\). | nvidia |
Find all distinct ordered pairs $(a, b)$ of positive integers such that $ab+1$ is a perfect square and $a^2+b^2$ is a perfect cube. | nvidia |
Let $\circledast$ be a binary operation on the set of positive real numbers $R_+$ such that for any $a, b \in R_+$:
$$a \circledast b = \sqrt{ab}$$
Given that the expression $(2 \circledast 5) \circledast (3 \circledast 4) = \sqrt{x}$, find the value of $x$. | nvidia |
Let $P(x)$ be a polynomial of degree 2 with integer coefficients such that $P(1) = P(2) = P(3) = 0$. Determine the sum of all possible values of $P(4)$. | nvidia |
Find the number of three-digit numbers whose digits are in strictly increasing order, such that the tens digit is a prime number. | nvidia |
In the equation
$$x^3+ax^2+bx+c=0$$
determine the sum of the coefficients a, b, and c, knowing that the sum of the roots of the equation is 1, the sum of the products of the roots taken two at a time is -3, and the product of the roots is 2. | 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 |
What is the smallest positive integer that has exactly 12 positive divisors and is also a perfect square? | nvidia |
Given a function $g(x)$ that satisfies the equation $g(x)g(y) = g(x+y)$ for all real numbers $x$ and $y$, if $g(2) = 3$, find $g(-4)$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $2x + 5y < 45.$ Find the maximum value of
\[xy (45 - 2x - 5y).\] | nvidia |
The sum of the squares of two consecutive positive integers is 481. What is the product of the two integers? | nvidia |
Determine the complex number $z$ satisfying the equation $2z-3\bar{z}=-2-30i$. Note that $\bar{z}$ denotes the conjugate of $z$. | nvidia |
Find the value of $x$ such that the four-digit number $\underline{x15x}$ is divisible by 18. | nvidia |
Let $g(x)$ be a function such that $g(x+1)-g(x)=x^2-x$ for every real number $x$. Determine $g(5)-g(0)$. | nvidia |
The sum of the digits of a three-digit number is 12, and the sum of the digits of its square is 18. Determine the number. | nvidia |
Determine the value of $k$ such that the sum of the squares of the roots of the equation $x^2 + kx + 2 = 0$ is equal to $6$. | nvidia |
Find the largest integer that is divisible by 5 and has exactly 3 digits. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx}.\] | nvidia |
Find the maximum possible value of the product
\[\left(a^2 + \frac{b^2}{9}\right)\left(\frac{b^2}{9} + \frac{c^2}{81}\right)\left(\frac{c^2}{81} + \frac{a^2}{729}\right),\]where $a,$ $b,$ and $c$ are positive real numbers. | nvidia |
In triangle $ABC$, points $D$ and $E$ are on sides $AB$ and $AC$ respectively, such that $BD:AD = 4:3$ and $CE:AE = 3:2$. If the area of triangle $ABC$ is 240 square cm, find the area of triangle $ADE$. | nvidia |
An equilateral triangle $ABC$ is inscribed in a circle with radius $1$. The distance from the center of the circle to the midpoint of the side $BC$ is $\frac{1}{2}$. Find the length of the side $AB$. | nvidia |
Let $p(x)=x^3-2x^2-7x+14$. For how many real numbers $r$ does there exist a polynomial $q(x)$ with integer coefficients such that $p(x)=(x-r)q(x)$? | nvidia |
In the coordinate plane, there are two circles, one with the equation $x^2 + y^2 = 100$ and the other with the equation $(x-3)^2 + (y-4)^2 = 36$. Find the length of the line segment that is tangent to both circles. | nvidia |
Find the smallest positive integer $x$ which is greater than $1$ and relatively prime to $120$ (recall that relatively prime means that the GCD of $x$ and $120$ is $1$) | nvidia |
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