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A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | nvidia |
Find the number of positive integers less than or equal to 100 that are either perfect squares or perfect cubes. | nvidia |
The roots of the polynomial
\[x^5+x^4+x^3+x^2+x+1\]
are all different. Three of them are complex conjugate pairs. Find the sum of all the real roots of this polynomial. | nvidia |
Find the number of possible values of $b$ such that the line $y=bx-1$ is tangent to the parabola $y=x^2-4x+3$. | nvidia |
Find the sum of the solutions of the equation: $\left| x-3 \right|+\left| x-5 \right|=4$ | nvidia |
Find the number that exceeds its reciprocal by $\ds\frac{5}{6}$. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Find the sum of the squares of all the numbers from $1$ to $20$ (inclusive) except for the squares of $4,6,8,10,12,14,16,18,20$. What is the difference between this sum and the sum of the squares of all the numbers from $1$ to $20$ (inclusive)? | nvidia |
Let $a$ and $b$ be real numbers such that
\[\frac{a + bi}{1 - 2i}\]is a real number. Find $\frac{a}{b}$. | nvidia |
Find the value of $x$ such that $(\sqrt[3]{x})^2 = \sqrt{x}$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + b = 1.$ Find set of all possible values of $\frac{1}{a} + \frac{1}{b}.$ | nvidia |
Find the sum of all the positive divisors of the number $2^3 \cdot 3^2 \cdot 5 \cdot 7$. | nvidia |
The roots $x_1$ and $x_2$ of the equation
$$x^2-3ax+a^2=0$$
are such that
$x_1^2+x_2^2=1.75$.
Determine $a$. | nvidia |
Find the positive difference between the sum of the first 30 positive multiples of 3 and the sum of the first 20 positive multiples of 5. | nvidia |
In the equation
$$x^2-3ax+a^2=0$$
determine $a$ such that the sum of the roots is equal to $2$. | nvidia |
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 1000^\circ.$ | nvidia |
Nine tiles are numbered $1, 2, 3, \cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
| nvidia |
Let $\triangle ABC$ be a right triangle with legs $AB = 6$ and $BC = 5$, and hypotenuse $AC$. Let $m$ be the number of positive integer divisors of the area of $\triangle ABC$. Find the number of positive integer divisors of the perimeter of $\triangle ABC$ in terms of $m$. | nvidia |
We are given a regular icosahedron whose edges are colored in red and blue such that no three edges of the same color meet at a vertex. Prove that the number of blue edges is equal to the number of red edges. | nvidia |
Suppose that $(u_n)$ is a sequence of real numbers satisfying
\[u_{n+2}=2u_{n+1}+u_n\]and that $u_3=9$ and $u_6=128$. What is $u_5$? | nvidia |
Find all real numbers $m$ for which there exists a nonzero, 2-dimensional vector $\mathbf{u}$ such that
\[\begin{pmatrix} 3 & 5 \\ 4 & 3 \end{pmatrix} \mathbf{u} = m \mathbf{u}.\]
Enter all the solutions, separated by commas. | nvidia |
What is the smallest positive integer that can be expressed as the sum of three squares in two different ways, and what are these two different ways of expressing it? | nvidia |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | nvidia |
There are 3 complex numbers $a+bi$, $c+di$, and $e+fi$. If $b=3$, $e=-a-c$, and the sum of the numbers is $2i$, find $d+f$. | nvidia |
Find the absolute value of the difference between the sum of the squares of the first 10 odd numbers and the sum of the squares of the first 10 even numbers. | nvidia |
For how many positive integers $n$ does $1+2+\cdots+n$ evenly divide $9n$? | nvidia |
A set of five positive real numbers has a mean of 10 and a range of 6. What is the minimum possible value of the sum of the squares of the five numbers? | nvidia |
Five points are chosen on a circle. In how many ways can three points be selected such that the triangle formed by joining these points contains the center of the circle? | nvidia |
Consider a sequence of circles where each circle has a radius twice as large as the previous circle in the sequence. Color each circle either red or blue so that no two consecutive circles have the same color. Determine the number of distinct sequences that can be formed using the first $20$ circles, ensuring that the ... | nvidia |
What is the smallest positive integer that has exactly $8$ positive divisors? | nvidia |
Find the number of integers $k$ that satisfy the inequality $(k-5)(k+2)>0$. | nvidia |
What is the smallest positive integer that has exactly three distinct prime factors, and can be expressed as the sum of two consecutive perfect squares? | nvidia |
The graph of a quadratic function $f(x)$ intersects the x-axis at two points $(a,0)$ and $(b,0)$ with $a\geq b$. The area of the region bounded by the graph of $f(x)$, the x-axis, and the line $x=a$ is equal to 10. The area of the region bounded by the graph of $f(x)$, the x-axis, and the line $x=b$ is equal to 5. Find... | nvidia |
Determine all possible integers that have exactly three positive divisors: 1, itself, and a third divisor. Find the smallest such integer. | nvidia |
How many ways can 1995 be expressed as the sum of two squares of positive integers, with the added restriction that one of the squares must be a perfect cube? | nvidia |
Find the sum of all values of $x$ for which the expression $\frac{x-3}{x^2-7x+12}$ is undefined. | nvidia |
Determine the number of distinct real solutions of the equation $g(g(g(g(g(x)))))=x$, where $g(x)=\frac{x}{x+1}$, for $x\neq -1$. | nvidia |
Find the number of ways to distribute 99 identical balls into four distinct boxes such that each box contains an odd number of balls. | nvidia |
Suppose that $a$ and $b$ are integers such that $$3b = 8 - 2a.$$How many of the first six positive integers must be divisors of $2b + 12$? | nvidia |
In the complex plane, let $A$, $B$, $C$, $D$, $E$, and $F$ be the vertices of a regular hexagon centered at the origin. Let $P$ be a point on the line passing through $A$ and $F$ such that $\frac{AP}{PF} = \frac{1}{3}$. Find the smallest positive value of the imaginary part of the complex number corresponding to point ... | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{1}{b}.\] | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2+b^2=50$. Find the maximum value of $\sqrt{ab}$. | 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 the value of $x$ that satisfies $e^{\sqrt{\log_2 x}}=5$. | nvidia |
Given that $2^{200}\equiv 1\pmod{500}$, find the last three digits of $2^{15000}$. | nvidia |
A right triangle with integer leg lengths is called "cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? | nvidia |
Find the area of the triangle with vertices $(2,3),$ $(5,-2),$ and $(8,1)$. | nvidia |
Let $x$ and $y$ be positive real numbers such that $x^2y + xy^2 = x + y.$ Find the minimum value of $\frac{x}{y} + \frac{y}{x}.$ | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 5.$ Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c}.\] | nvidia |
Find all real values of $x$ that satisfy $\frac{x^2+2x-3}{x^2+4x+4} \leq \frac{1}{2}$.
Note: This problem requires a different approach as it involves a quadratic inequality with a fraction, and the solution involves finding the values of $x$ that make the fraction less than or equal to $\frac{1}{2}$, rather than just... | nvidia |
Given the function $f(x)=3x^3+2$, find the value of $x$ so that $f^{-1}(x)=4$. | nvidia |
Let $\mathcal{S}$ be a set of non-zero complex numbers such that for every number $z$ in $\mathcal{S}$, the geometric mean of the set of values obtained by deleting $z$ from $\mathcal{S}$ is a real number. Given that $1 + 2i$ belongs to $\mathcal{S}$ and that $2002 + 3i$ is the complex number with the largest modulus i... | nvidia |
In a triangle $ABC$, the angle bisectors of $\angle BAC$ and $\angle ACB$ intersect at $I$. A point $M$ is chosen on the side $BC$ such that $\angle BAM = \angle CAM$. Prove that the line passing through $I$ and perpendicular to $MI$ is parallel to the side $AB$. | nvidia |
We call a set of three points on a line "good" if there is a point on the line that is closer to each of these three points than the points are to each other. If two of the points are on one side of a third point, then the set is not "good". Three points are chosen uniformly at random on a line segment. What is the pro... | nvidia |
Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians. | nvidia |
Consider a monic cubic polynomial of the form $x^3 + ax^2 + bx + 216$ that can be factored into $(x+n)^3 + m$, where $a$ and $b$ are constants and $n$ and $m$ are integers. What is the value of $a$ if $n$ and $m$ are the smallest positive integers that allow for this factorization? | nvidia |
Let $n$ be a positive integer. Find the sum of all numbers of the form $\frac{1}{k}$ where $k$ ranges from $-n$ to $n$, excluding $k = 0$. | nvidia |
Determine all real values of $x$ for which the equation \[\frac{x^2+x+2}{x-2} = \frac{x^3-2x^2-4x-1}{x^2-4}\]holds true. | nvidia |
Find the largest positive integer $m$ such that $m^3$ can be expressed as the difference of two squares and $5m + 2$ is a perfect cube. | nvidia |
A sequence of positive integers $a_1, a_2, \ldots$ has the property that for every $n \geq 1$, the number $a_n$ is the smallest positive integer that is not a proper divisor of $a_1 a_2 \cdots a_{n-1}$. How many different prime numbers must appear in the factorization of $a_{100}$? | nvidia |
Find the value of the sum
\[\sum_{n = 1}^{2700} \frac{1}{\sqrt{n + \sqrt{n^2 + 1}}}\]
This sum can be expressed as $a + b \sqrt{c}$, where $a, b,$ and $c$ are positive integers and $c$ is not divisible by the square of any prime. Determine $a + b + c$. | nvidia |
Given that a polynomial of degree n has n+1 distinct coefficients, find the sum of the coefficients that divide both 14 and 105. Recall that we consider the constant term of the polynomial to be its divisor. | nvidia |
Two positive integers $x$ and $y$ are chosen such that $x$ is the largest prime number less than $50$ and $y$ is the smallest positive integer with exactly four positive divisors. What is $x+y$? | nvidia |
The sum of two positive integers $x$ and $y$ is 1500. What is the largest possible value of $\text{lcm}(x,y)$? | nvidia |
Find the value of $x$ such that $\cos(x) = \frac{1}{2}$ and $x$ is in the third quadrant. | nvidia |
Let $x$ be a digit of a number $x \overline{2ab9}_{10}$, where $a, b \in [0, 9]$ and all the digits are different. Determine $x$ knowing that the number is divisible by 13. | nvidia |
What is the largest positive integer $n$ such that $n$ is a divisor of the least common multiple of any three consecutive positive integers? | nvidia |
The graph of $$x^2+y^2\equiv 1 \pmod{4}$$ consists of a set of points $(x, y)$ where $x$ and $y$ are integers modulo 4.
Find the number of points on the graph such that $x\neq 0$ and $y\neq 0$. | nvidia |
Let $f(x)=\ds\frac{2x-1}{x^2+1}$ and $g(x)=x^2-k$, where $k$ is a real number.
Determine all possible values of $k$ such that the equation $f(x)=g(x)$ has no solution. | nvidia |
Let $P$ be a point inside triangle $ABC$ such that $\angle APB = \angle APC$.
Prove that $$\frac{AP}{PD}=\frac{AC}{BC},$$
where $D$ is the intersection point of lines $BP$ and $AC$. | nvidia |
Let $S$ be the set of all real numbers of the form $a + b\sqrt{2},$ where $a$ and $b$ are integers. Consider the polynomial $p(x) = x^3 - 2x + 5.$ Find the probability that if $r$ is a randomly chosen element of $S,$ then $p(r)$ is also an element of $S.$ | nvidia |
Let $m$ and $n$ be relatively prime positive integers such that $m^n$ is divisible by $11$. Determine the smallest possible value of $m+n$. | nvidia |
The points $A(0,4)$ and $B(3,0)$ are the ends of the diameter of a circle. Find the distance from the center of the circle to the point $C(2,1)$. | nvidia |
Problem:
A coin is flipped repeatedly. The probability that the first $k$ flips are all heads is $P_k$. Find the smallest positive integer $n$ such that the probability that the first $n$ flips are all heads and the $(n+1)$th flip is tails is greater than $\frac{P_n}{2}$. | nvidia |
Find the largest value of $y$ that satisfies the equation $|y-2|=y-2+\sqrt{10-y}$. | nvidia |
Find the maximum value of $c$ such that the equation $\sqrt{(x-1)^2+(y-c)^2}+\sqrt{(x+1)^2+(y-2)^2}=4$ represents an ellipse. | nvidia |
Consider the point $O(0,0)$ on the complex plane and two points $A$ and $B$ on the unit circle such that the triangle $AOB$ has an area equal to $2$. Determine the number of positions of $A$ and $B$ fulfilling this condition. | nvidia |
Find the number of ordered pairs $(x,y)$ of real numbers such that
$$x^2 + y^2 = x^3 + y^3 = 1.$$ | nvidia |
A triangle is formed by the points $A(2,0)$, $B(0,4)$, and $C(0,0)$. A line with a slope of 3 intersects the positive $x$-axis at $D$ and the positive $y$-axis at $E$. The lines intersect at $F(1,3)$. What is the area of the shaded triangle $AFC$? | nvidia |
How many different positive integers can be formed using all the digits of the number 112233, if the number must start with 1 and the digits 2 must be together? | nvidia |
In the decimal system, a number $n$ has digits $x$, $y$, and $z$, such that $n=\overline{xyz}$. Consider the number $m$ obtained by rearranging the digits of $n$, such that $m=\overline{zyx}$. If $n$ and $m$ differ by 495, find the largest possible value of $n$.
Note: This problem requires a different approach than th... | nvidia |
In quadrilateral $ABCD,\ BC=8,\ CD=12,\ AD=10,$ and $m\angle A= m\angle B = 60^\circ.$ Given that the area of quadrilateral $ABCD$ is $x\sqrt{y}$, where $x$ and $y$ are positive integers, find $x+y$. | nvidia |
In a right triangle, the angle bisector of the right angle intersects the hypotenuse at a point that is 5 units from the vertex of the right angle. If the triangle is rotated 90 degrees clockwise around the vertex of the right angle, the bisector of the right angle in the rotated triangle intersects the hypotenuse of t... | nvidia |
Find all real numbers $q$ such that the equation
\[x^4 - 3qx^3 + 2x^2 - 3qx + 1 = 0\]
has at least two distinct positive real roots. | nvidia |
The vertices of a regular $n$-gon lie on the unit circle in the complex plane centered at the origin and are labeled $1,2,\ldots,n$. The function $f_n(z)$ is defined to be the area of the triangle formed by $z$, the center of the circle, and the midpoint of the side of the $n$-gon opposite to $z$. If $\sum_{k=1}^{2015... | nvidia |
Find the number of positive integers less than or equal to 100 that have exactly three distinct prime factors. | nvidia |
Find the maximum value of $x^2+y^2+z^2$ for positive real numbers $x,y,z$ satisfying the following equality:
\[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1.\] | nvidia |
What is the maximum number of consecutive positive integers that can be multiplied together before the product exceeds 1000? | nvidia |
Given a triangle with vertices $A(2,3)$, $B(7,-2)$, and $C(-1,1)$, find the equation of the perpendicular bisector of the side $BC$. | nvidia |
Positive integers $x$ and $y$ are such that $x\neq y$ and $xy=36$. Find the minimum value of $x+y$. | nvidia |
What is the largest possible value of the sum of the first five terms of an arithmetic sequence such that the sixth term is divisible by the sum of the first five terms? | 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 |
If $a$, $b$, and $c$ are positive integers such that $a^2b^3c^4=45\times 10^4$, compute the smallest possible value of $5a+3b+2c$. | nvidia |
A natural number can be written as $AB_8$, where $A$ and $B$ are the only digits of the number in base 8. The same number can also be expressed as $CB_4$, where $C$ and $B$ are the only digits of the number in base 4. If $A + B + C = 7$, find the number in decimal. | nvidia |
Given a positive integer $n$, find the largest number of fractions that can be formed from distinct sets of prime factors in the range $1$ to $n$, such that no fraction is greater than or equal to $\frac{n}{2}$. | nvidia |
Determine the sum of the digits of the number obtained when 783 is multiplied by 5, if the sum of the digits of the multiplicand is known to be 18. | nvidia |
Find the ratio of the area of square $A$ to the area of square $B$ if the side lengths of the squares are in the ratio of $3:5$. | 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 |
Given a rectangle $ABCD$ with $A = (0,0)$ and $C = (a,b)$, let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{CD}$, respectively. Let $P = (x,y)$ be a point inside the rectangle such that $\triangle MPA$ and $\triangle NPC$ are similar to each other. Find the value of $\frac{x}{a} + \frac{y}{b}$, ... | 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 |
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