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Find the product of the positive divisors of 48 that are multiples of 3. | nvidia |
The vertices of a triangle lie on the intersection of the circle with equation $(x-2)^2+y^2=4$ and the curves $y=x^2+ax+b$ and $y=x^2+cx+d$.
Given that the area of the triangle is $\frac{1}{2}$, find the value of $ad-bc$.
(Note: The problem is not solvable using the approach of the original problem, as it involves fin... | nvidia |
Find the matrix $A$ such that $A^2=\begin{pmatrix}1 & 3\\0 & 1\end{pmatrix}$. | nvidia |
Given positive integers $a$, $b$, and $c$ such that $a + b + c = 180$, find the maximum possible value of $\frac{\sin a \sin b \sin c}{\sin \left(\frac{a}{2}\right) \sin \left(\frac{b}{2}\right) \sin \left(\frac{c}{2}\right)}$. | nvidia |
Determine the value of $a$ such that the polynomial $x^3 - ax^2 + 2x + 3$ has the sum of its coefficients equal to its value at $x=1$. | nvidia |
Find the value of $k$ such that the equation $\tan(3x) = k$ has exactly one solution in the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$. | nvidia |
What is the smallest positive integer $n$ such that the decimal representation of $\frac{1}{n}$ has a repeating block of digits of length 4? | nvidia |
Two circles of radius 1 are externally tangent at point P. Two points M and N are chosen on the larger arc of each circle, respectively, such that the line segment MN is tangent to both circles and intersects the line joining the centers of the two circles at a point Q. Find the length of QP if $\angle MPN$ measures 12... | nvidia |
Find the number of nonnegative integers less than 1000 that are divisible by 3 but not by 9. | nvidia |
How many positive integers $n$ satisfy $200 < n^2 < 900$? | nvidia |
The number of increasing sequences of positive integers $a_1 < a_2 < a_3 < \cdots < a_n$ such that $a_i \leq 2i$ for all $1 \leq i \leq n$ is equal to the number of ways to arrange $n$ non-overlapping dominoes on a $2n \times 1$ rectangular grid, where each domino covers two cells and no two dominoes can share a cell. ... | nvidia |
In the triangle ABC with vertices A(1, 2), B(3, 4), and C(5, 6), the segment MN is the median of the side BC. Determine the coordinates of the point N such that the area of the quadrilateral AMBC is twice the area of the triangle ANC. | nvidia |
Find the coordinates of the point that divides the line segment joining the points $(2,3)$ and $(7,9)$ into three equal parts. | nvidia |
The sum of the first three positive integers that are not prime numbers is equal to 13 times the smallest positive integer that is not a prime number. Find the sum of the digits of these three integers. | nvidia |
Find the smallest positive integer n such that the sum of the prime factors of n is equal to 10. | nvidia |
Find all three-digit numbers that are divisible by 11 and whose cubes end with the digits 08. | nvidia |
A line through the point $(2, 0)$ is such that the total area of the triangle formed by the line, the $x$-axis, and the $y$-axis is $3\sqrt{2}$. Determine the slope of this line. | nvidia |
Determine the largest real number $x$ such that
$$\lfloor x\rfloor+2\{x\}=12,$$
where $\{x\}$ represents the fractional part of $x$. | nvidia |
Find the number of divisors of $2007$ that are divisible by $3$ but not by $9$, given that its prime factorization is $3^2 \cdot 13 \cdot 17$. | nvidia |
Find the function $g(x)$ defined for all positive real numbers, given that
$g(xy)=\sqrt{g(x+y)+1}$ for all $x>0$ and $y>0$,
and $g(2009)=2$. | nvidia |
Given two positive integers $x$ and $y$ such that $\frac{1}{x} + \frac{1}{y} = \frac{3}{4}$ and $xy = 108$, find the value of $x+y$. | nvidia |
A sequence of real numbers $b_1, b_2, b_3, \ldots$ is chosen so that $b_n = 2b_{n - 1} - 3b_{n - 2}$ for each $n \ge 3.$ The sum of the squares of the first $100$ terms of this sequence is $1337$, and the sum of the cubes of the first $100$ terms is $23456$. What is the smallest integer $k$ such that the sequence is pe... | nvidia |
Let $P(x)$ be a polynomial of degree $n$ with integer coefficients such that it has at least one root $r_i$ of multiplicity $m_i$, where $m_i \geq 2$, for $i = 1, 2, \ldots, k$. Find the maximum possible number of distinct rational roots that $P(x)$ can have if it has exactly $k$ repeated roots $r_1, r_2, \ldots, r_k$. | nvidia |
In the equation $x^2-px+q=0$, the sum of the roots is 2 times the product of the roots. Determine the value of $p$ in terms of $q$. | nvidia |
Find the absolute value of the sum of the solutions of $x^2+7x-2=0$. | nvidia |
The $8$ vertices of a cube are each marked with a number from $1$ to $8$. The sum of the numbers on each face of the cube is $18$. Find the sum of the numbers on the vertices of a diagonal of the cube. | nvidia |
Let $a\circ b=a^2+b^2$ for all real numbers $a$ and $b$. Find $x$ such that $x\circ 2=x\circ 5$. | nvidia |
The function $g(n)$ is defined on the positive integers such that $g(g(n)) = 3n$ and $g(2n + 1) = 2n + 2$ for all positive integers $n.$ Find $g(2001).$ | nvidia |
Let $f(x)$ be a polynomial of degree $2023$ such that the sum of the squares of its roots is $2864$, and the sum of the cubes of its roots is $4997$. Determine the sum of the roots of $f(x)$. | nvidia |
Let $a$ and $b$ be positive real numbers such that each of the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ has real roots. Find the smallest possible value of $a + b.$ | nvidia |
Count the number of positive integers $n$ such that there exist positive integers $a$, $b$, $c$ satisfying the equation
$n=a^2+b^2-c^2$
and also the inequality $3c<a+b$. | nvidia |
Consider a set of strings of length $n$ consisting only of 0's and 1's, with the restriction that any two adjacent digits cannot be 1. Determine the number of strings in this set that end with a 1, for $n = 10$. | nvidia |
Determine the number of positive integers $a$ less than $24$ such that the congruence $ax\equiv 5\pmod{24}$ has a solution in $x$. | nvidia |
Determine the number of positive real solutions to the equation $\frac{x^5-7x^4-14x^3-1729x^2+1379x}{x^5+x^4+x^3+x^2+x+1}=0$. | nvidia |
In a triangle with side lengths $a, b,$ and $c$, where $a<b<c$, it is known that the largest side is twice the smallest side, and the sum of the measures of the two smaller angles is $60^\circ$. Find the sine of the largest angle. | nvidia |
Find the largest integral value of $x$ such that the inequality $\frac{x}{7} + \frac{x}{11} < 1$ holds true. | nvidia |
Find all positive real numbers $x$ that satisfy
\[x \sqrt{20 - x} + \sqrt{20x - x^3} \ge 20.\] | nvidia |
Let $x$ and $y$ be relatively prime positive integers such that $\dfrac xy=\dfrac1{\sqrt{2}+1}+\dfrac1{2+\sqrt{3}}+\dfrac1{\sqrt{5}+2}+\dfrac1{3+\sqrt{7}}+\cdots$, where the denominators are the sums of consecutive integers and square roots of consecutive odd numbers. Compute $x+y$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a^2 + b^2 = 1.$ Find the set of all possible values of $\frac{1}{a^2} + \frac{1}{b^2}.$ | nvidia |
When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively.
When the sum of the three integers is divided by $24$, what is the remainder? | nvidia |
Let a, b, and c be positive integers such that
$\frac{(a-b)^2}{ab}+\frac{(b-c)^2}{bc}+\frac{(c-a)^2}{ca}=4$.
Find a relationship between a, b, and c. | nvidia |
A positive integer has a base $b$ representation of $ABCD$ and its base $b+2$ representation is $DCBA$. What is the smallest possible value of $b$ such that the integer has exactly four digits in both representations? | nvidia |
The product of two distinct positive integers is 1007. One of the integers is greater than twice the square root of 1007, while the other is less than 2/3 of the cube root of 1007. What are these two integers? | nvidia |
Given that the point $(1,-\sqrt{3})$ in rectangular coordinates has the polar coordinates $(r,\theta)=(2,\frac{5\pi}{3})$, determine the rectangular coordinates of the point that has polar coordinates $(r,\theta)=(2,\frac{7\pi}{6})$. | nvidia |
The sum of two positive integers is 80 and their difference is 18. What is the value of the positive difference of the cubes of the integers? | nvidia |
A circle with center $O$ and radius $r$ is given. We want to place $n$ congruent circles with radius $r/2$ inside the larger circle so that each of the smaller circles touches the larger circle and two of its neighbors, and the sum of the areas of the smaller circles is equal to the area of the larger circle. For how m... | nvidia |
Find the value of $x$ such that $\sin(x+30^\circ) = \sin(x-20^\circ)$. | nvidia |
Determine $a$ such that the equation $x^2-4x+y^2-4y+z^2=a$ represents a sphere tangent to the plane $x-y+2z=0$ at the point $(1,1,1)$. | nvidia |
Let $a,$ $b,$ $c$ be distinct real numbers such that
$$
abc \left( \frac{1}{b-c} + \frac{1}{c-a} + \frac{1}{a-b} \right) = (a+b+c)^2 .
$$
Find all possible values of
$$
a^2b^2 + b^2c^2 + c^2a^2 .
$$
Enter all the possible values, separated by commas. | nvidia |
What is the sum of all positive integers less than 100 that are not multiples of 3 or 7? | nvidia |
In a triangle with vertices at (3,4), (5,6), and (9,10), find the length of the altitude from the vertex (5,6) to the side connecting the other two vertices. | nvidia |
Suppose that $x$ and $y$ are positive real numbers such that $x^2-xy+2y^2=8$. Then the maximum possible value of $x^2+xy+2y^2$ can be expressed in simplest form as $\frac{a + b \sqrt{c}}{d},$ where $a,$ $b,$ $c,$ $d$ are positive integers. Find $a + b + c + d.$ | nvidia |
Find the value of $k$ such that the expression $\frac{x^2+2x+k}{x^2-4x+5}$ is equal to 1 when $x=3$. | nvidia |
What is the sum of the first five positive integers that are both even and multiples of 5? | nvidia |
A line is defined by the equation \(\frac{x-2}{3}=\frac{y+1}{2}=\frac{z}{4}\), and a point \(A(3,1,0)\) is given. Find the distance from point \(A\) to the line such that the segment connecting \(A\) and the point on the line is perpendicular to the direction vector of the line. | nvidia |
Let $a$ and $b$ be real numbers such that $$3\left(a^2+\frac{1}{a^2}\right)+7\left(b^2+\frac{1}{b^2}\right)=19.$$ Find $ab+\frac{1}{ab}$. | nvidia |
A sequence of positive integers $a_1, a_2, a_3, ... , a_{2021}$ satisfies the conditions: (1) for every integer $i$ from $1$ to $2021$, $a_i > i$; (2) for every pair of distinct integers $i, j$ from $1$ to $2021$, $(a_i - i)(a_j - j)$ is divisible by $i + j$; and (3) the sum of the squares of these 2021 numbers is divi... | nvidia |
Find the perimeter of the triangle with vertices $(2,3),$ $(5,6),$ and $(8,4),$ given that the line passing through the first two vertices has a slope of $3/2.$ | nvidia |
Let $ABCD$ be a parallelogram, and let $P$ and $Q$ be points on the lines $AC$ and $BD$, respectively, such that $AP=\frac{1}{3}AC$ and $BQ=\frac{2}{3}BD.$ Find the ratio of the area of the quadrilateral $APQD$ to the area of the parallelogram $ABCD$. | nvidia |
What is the largest integer $n$ such that the sum of the squares of its digits is less than $n$? | nvidia |
What is the smallest positive integer that can be expressed as the sum of two different perfect squares in exactly three different ways? | nvidia |
Find the number of positive factors of the number 24 that are also perfect squares. | 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 value of $x$ such that $|x+5|+|x-3|=|x^2-6x+9|.$ | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that
\[\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} = 3.\]Find the maximum value of $abc.$ | nvidia |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | nvidia |
How many integers $M$ less than $2000$ can be expressed as the sum of $k$ consecutive even positive integers from exactly 3 values of $k\ge 1$? | nvidia |
At a certain university, there are 5 faculties, each with a different number of departments. If each department has an equal number of students, and every student is a member of exactly two departments from different faculties, determine the smallest number of students that can satisfy these conditions. | nvidia |
What is the sum of all the positive integer divisors of 24 that are also prime numbers? | 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 |
Find the largest integer n such that 3^n is a factor of 7776. | nvidia |
Find the smallest positive integer n that can be represented as a sum of two consecutive integers in base a and as a sum of three consecutive integers in base b, where a and b are any bases larger than 2. | nvidia |
What is the greatest integer less than or equal to\[\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\]
$\textbf{(A) }80\qquad \textbf{(B) }81 \qquad \textbf{(C) }96 \qquad \textbf{(D) }97 \qquad \textbf{(E) }625\qquad$
| nvidia |
Find the value of $\cos\alpha$ for which the sum of the two roots of the equation $\cos^2\alpha x^2 - 3x + 1 = 0$ is a maximum. | nvidia |
The arithmetic mean of an odd number of consecutive odd integers is $y$. Find the sum of the smallest and largest of the integers in terms of $y$. | nvidia |
Given a square with side length 10, four semicircles are constructed on each side of the square such that they are tangent to one another at their endpoints. A rectangular hole is cut out from the center of the square, leaving a thin border around the edges. As the size of the hole increases, the border's dimensions de... | nvidia |
Let $a$ and $b$ be the roots of the quadratic equation $x^2-mx+n=0$ where $m$ and $n$ are integers. Given that $a^3+b^3=-7$ and $a^7+b^7=47$, find the value of $a+b$. | nvidia |
If the function $f(x)$ satisfies
$f(x+y)=f(x)+f(y)+1$ for all real numbers $x$ and $y$, determine all possible values of $f(0)$.
If there is any condition or constraint on the function $f(x)$, state it. | nvidia |
Determine $a$ such that the polynomial
$$P(x)=ax^3-7x^2-ax-7$$
is divisible by
$$x^2-1$$
when
$$P(-2)=0$$
holds. | nvidia |
Let $g$ be a function from the positive integers to the positive integers, such that $g(a \times b) = g(a) + g(b)$ for all positive integers $a$ and $b.$ If $g(3) = 4$ and $g(6) = 6,$ determine the number of possible values of $g(12)$ and find their sum. | nvidia |
The vertices of a triangle are A(1, 2), B(4, 6), and C(7, 9). Find the sum of the coordinates of the centroid of the triangle. | nvidia |
The sum of the three numbers is 500. If the first number is 200, and the value of the second number is twice the value of the third number, find the value of the third number. | nvidia |
Determine the value of x such that the equation $\cos(\arctan x) = \frac{1}{2}$ holds. | nvidia |
Let a positive real-valued function $g$ have the properties that $g(x) = g(-x)$, and that its graph is a straight line segment on the interval $[-a,a]$. It is also known that
$g(g(x)) = 3x^2 + 4$, and that $g(7) = 29$.
Determine the smallest $x$ for which $g(g(x))=300$. | nvidia |
Determine the sum of the coefficients of the polynomial
$$(x^2-4x+2)(x^2+5x+1).$$ | nvidia |
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\mbox{th}}$ term of this sequence.
Another problem inspired by this one:
A student is playing a game where she has to write down all the positive integers that... | nvidia |
Let $A=\begin{pmatrix} a & 1 \\ b & 0 \end{pmatrix}$ be a matrix such that its inverse is equal to itself. Determine all possible values of $a$. | nvidia |
Consider a sphere with radius 1, and a cylinder with radius 2 and height 4 that sits on a table and has its axis of symmetry passing through the center of the sphere. Find the radius of the largest cone, symmetric about the same axis, that can be placed on top of the cylinder such that it touches the sphere but does no... | nvidia |
Find the value of the expression
$$\frac{1}{2^{11}}+\frac{1}{2^{12}}+\frac{1}{2^{13}}+\frac{1}{2^{14}}+\frac{1}{2^{15}}+\frac{1}{2^{16}}\pmod{13}.$$
Hint: Use the fact that $2^{11} \equiv 1 \pmod{13}$ and the fact that the sum of a geometric series is given by $\frac{a(1-r^n)}{1-r}$. | nvidia |
Find the number of positive integers $n \le 1000$ that can be expressed in the form
\[\lceil x \rceil + \lceil 2x \rceil + \lceil 3x \rceil = n\]for some real number $x$, where $\lceil x \rceil$ denotes the smallest integer greater than or equal to $x$. | nvidia |
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
$\textbf{(A) } 324 \qquad \textbf{... | nvidia |
Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ | nvidia |
Find the sum of the two smallest one-digit composite numbers and the smallest two-digit composite number. | nvidia |
Given a triangle with side lengths 18, 16, and x, find the length of x such that the absolute value of the difference of the squares of the two known sides is equal to twice the square of the third side. | nvidia |
What is the smallest integer that can be expressed as the sum of two distinct perfect squares in two different ways? | nvidia |
Determine the number of distinct, natural-number factors of $n$ that are also multiples of $k$, given that $n=2^3 \cdot 3^2 \cdot 5$ and $k=2^2 \cdot 3$. | nvidia |
Find a nonzero $q$ such that $qx^2+20x+5=0$ has only one solution. | nvidia |
Given that $0\le x_3 \le x_2 \le x_1\le 1$ and $(1-x_1)^2+(x_1-x_2)^2+(x_2-x_3)^2+x_3^2=\frac{1}{4},$ find $x_1.$
Another problem inspired by this one:
Let $x_1, x_2, x_3, x_4$ be four non-negative real numbers such that $x_1+x_2+x_3+x_4=2$ and $x_1x_2+x_2x_3+x_3x_4+x_4x_1=1$. Determine the maximum value of $x_1+x_3$... | nvidia |
Let $f(x)$ be a function that maps each positive integer $x$ to the smallest integer $y$ such that $x^2+y^2$ is divisible by 4. Determine the sum of all possible values of $f(4)+f(5)+f(6)+f(7)$. | nvidia |
The function $g(x)=\frac{1}{\pi}\arctan x$ is defined for all real numbers $x$. The number $k$ is the smallest positive integer such that the equation $k\cdot g(x\cdot g(x))=x$ has infinitely many real solutions. What is $k$? | nvidia |
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