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In the triangle $ABC$ the length of side $AB$ is equal to the greatest common divisor of the lengths of the other two sides. If $AC = 24$ and $BC = 63$, determine the positive value of $AB$ such that the area of triangle $ABC$ is equal to the least common multiple of the lengths of all three sides. | nvidia |
Find the value of $x$ such that $\sqrt{3x + 7} = 10$. | nvidia |
Find the sum of all values of $x$ such that $3^{x^2-2x-3} = 9^{x - 5}$. | nvidia |
Let $a_1, a_2, \dots, a_n$ be a sequence of integers such that $a_i = 0$ or $a_i = 1$ for each $i$. Find the maximum possible value of the sum
$$\sum_{1 \le i < j < k \le n} a_i a_j a_k$$
subject to the condition that the sum
$$\sum_{1 \le i < j \le n} a_i a_j$$
is equal to a given positive integer $m$. | nvidia |
Determine the value of $y$ that satisfies $\sqrt[4]{y\sqrt{y^2}}=2$. | nvidia |
Find the greatest natural number $n$ such that $n\leq 2008$ and $(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]$ is a perfect square.
| nvidia |
Find the determinant of the matrix
\[\begin{pmatrix} 2x^2-3y^2 & x^3+y^3 \\ xy & 2xy+5 \end{pmatrix},\]where \(x\) and \(y\) are positive integers such that \(x^2+y^2=13\). | nvidia |
Given a regular tetrahedron with vertices $V_1,V_2,V_3,$ and $V_4$, determine the side length of the largest cube that can be inscribed in the tetrahedron so that its vertices lie on the edges of the tetrahedron and the edges of the cube are parallel to the edges of the tetrahedron. | nvidia |
The sides of a hexagon are all parallel to the sides of a rectangle with sides of length 6 and 5. The hexagon shares a side with the rectangle. Prove that the perimeter of the hexagon is 22 units. | nvidia |
Determine the number of distinct ways to express the number 456 as the sum of j consecutive terms of a Fibonacci sequence, where j can take any positive integer value. | nvidia |
Let $A$ be a $3\times3$ matrix with integer entries, and let $x$ be a positive integer. It is known that $\det(A)=x^3$, and that the matrix $A$ can be factored as $A=BC$ where $B$ and $C$ are both $3\times3$ matrices with integer entries. Furthermore, it is given that $\det(B)=\det(C)=x$, and that $BC=CB$. Find the sma... | nvidia |
What is the smallest positive integer that is divisible by exactly 5 different positive integers? | nvidia |
Find the number of values of $b$ such that the circle $x^2 + y^2 = b^2$ intersects the parabola $y = x^2 + b$ at exactly two points. | nvidia |
Determine $n$ such that the expression $\frac{1}{3}\left(1+2^{2n+1}\right)$ is an integer. | nvidia |
Find the area of rhombus $PQRS$ given that the radii of the circles circumscribed around triangles $PQR$ and $PSR$ are $15$ and $30$, respectively. | nvidia |
Two positive numbers $a$ and $b$ are such that $\frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}} = \frac{4}{3}$ and $\sqrt{ab} = 3$. Determine $b$. | nvidia |
The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$?
| nvidia |
Consider a 3D cube with 8 vertices. The sum of the x-coordinates of these vertices is 12, the sum of the y-coordinates is 8, and the sum of the z-coordinates is 4. The midpoints of the edges of the cube form the vertices of a second polyhedron, a rhombic dodecahedron. Find the sum of the x, y, and z-coordinates of the ... | nvidia |
Bryson has $n$ boxes of paper clips, where $n$ is a positive integer with four digits. If he buys $m$ more boxes, he will have a number of boxes that is a perfect square, while if he loses $m$ boxes, he will have a number of boxes that is a perfect cube. What is the largest possible value of $n$ such that $m$ is less t... | nvidia |
Find the value of $x$ such that $\sin (\arctan x) = \frac{3}{5}$. | nvidia |
Three different numbers are in arithmetic progression. The difference between the two larger numbers is 5, and the difference between the two smaller numbers is 2. If the sum of the three numbers is 54, what is the value of the middle number? | nvidia |
How many positive four-digit integers have the property that their first two digits are the same as their last two digits in reverse order? | nvidia |
What is the smallest positive integer $n$ such that $n!$ is divisible by $2^{10}$? | nvidia |
Prove that the expression $$\binom{n}{k} \binom{n+1}{k+1} - \binom{n}{k+1} \binom{n+1}{k}$$ is always a perfect square for any positive integers $n$ and $k$. | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
Find the number of positive integers less than or equal to 100 that have exactly four positive divisors. | nvidia |
Let $x$ and $y$ be positive integers such that $x^2+y^2=100$. Find the minimum value of $|x-y|$. | nvidia |
A sequence of five digits is called "alternating" if the sum of the first and third digits equals the sum of the second and fourth digits. How many different five-digit numbers with non-zero leading digits can be formed such that the sequence of their digits is alternating? | nvidia |
Find the difference between the product of all the numbers from $-10$ to $10$ and the product of all the even numbers from $-10$ to $10$. | nvidia |
Given that the point $(4,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=3f(4x)+5$. What is the sum of the coordinates of that point? | nvidia |
Evaluate
\[\sum_{1 \le x < y < z} \frac{1}{7^x 11^y 13^z}.\](The sum is taken over all triples $(x,y,z)$ of positive integers such that $1 \le x < y < z.$) | nvidia |
The function $g(x)$ satisfies
\[g(x) \cdot g(y) = g(xy)\]for all positive real numbers $x$ and $y,$ and $g(2) = 3.$ Find $g(\sqrt{2}).$ | nvidia |
What is the maximum number of distinct triangles that can be formed by connecting 5 points on a circle, such that no three points are collinear? | nvidia |
Let $f(x)$ and $g(x)$ be two functions whose graphs intersect at the points $(0,1)$, $(2,3)$, $(5,7)$, and $(6,9)$. Find the point of intersection of the graphs of $y=f(x+1)$ and $y=g(2x)$ that has the largest $x$-coordinate. | nvidia |
If $A$ and $B$ are $3\times 3$ matrices such that $A^2=A$ and $B^2=B$, determine the number of possible values of $\det(AB)$, where $\det(AB)$ denotes the determinant of the matrix product $AB$. | nvidia |
Find the smallest positive integer $x$ such that
$\lceil x \cdot \frac{3}{5} \rceil + \lfloor x \cdot \frac{4}{7} \rfloor = 11$. | nvidia |
A parallelepiped has a rectangular base with dimensions $x$ and $2x$, and its height is equal to the length of the diagonal of the base. Determine the length of the diagonal of the parallelepiped. | nvidia |
The parabola $y = x^2$ and the circle $x^2 + y^2 = 2$ intersect at two points. Find the equation of the line that passes through these two points. | nvidia |
Let $P$ be a point inside triangle $ABC$ such that $\angle APB = \angle BPC = \angle CPA = \theta$. Find the maximum possible area of triangle $ABC$, given that $AP = 1$ and $\theta$ is a fixed angle. | nvidia |
Given that $(3,17)$ and $(9,-4)$ are the coordinates of two opposite vertices of a rectangle, find the sum of the $x$-coordinates of the other two vertices. | nvidia |
Find the number of ways to arrange the digits 1, 2, 3, 4, and 5 to form a 5-digit number such that the product of the digits in the even positions is equal to the product of the digits in the odd positions. | nvidia |
The sum of the terms in an infinite geometric series is 15, and the sum of their squares is 45. Find the first term.
A circle is inscribed in a triangle with sides 3, 4, and 5. The inradius of the circle is r. The sum of the squares of the radii of the circles drawn on the sides of the triangle is equal to 225. Fin... | nvidia |
Suppose that $f(x)$ is a cubic polynomial such that $f(0) = 7$ and the sum of the coefficients in $f(x)$ is 13. Determine $f(2)$. | nvidia |
What is the largest number that can be represented in base 5 using exactly 7 digits, such that the sum of its digits is equal to 15? | nvidia |
A positive integer with four digits $abcd$, where $a$ is not zero, is divisible by 50. What is the sum of all distinct possible values of the product $ac$? (Count each possible value of $ac$ only once, even if it results from multiple $a$, $c$ pairs.) | nvidia |
Find the number of positive integers not exceeding 150 that are divisible by 3 or 5 but not by both. | nvidia |
Find the smallest integer greater than 100 that has exactly three positive divisors, one of which is a perfect square. | nvidia |
Find the number of positive integers less than 30 that have exactly three distinct positive divisors. | nvidia |
Let $T$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[|x| + |y| = 1.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in T$ such that $z^n = -1.$ | nvidia |
Evaluate the smallest integer $k$ such that $\sqrt{32k}$ is an integer. | nvidia |
Find all positive integer values of $n$ for which the equation $x^2 - n^2x + n! = 0$ has integer solutions. | nvidia |
A deck of 52 cards is shuffled and two cards are drawn at random. What is the probability that the two cards are of the same suit? | nvidia |
How many three digit numbers are left if we exclude all three digit numbers in which there are exactly two digits the same, but these two digits are not adjacent? | nvidia |
A regular tetrahedron with side length $s$ is placed inside a cylindrical container with radius $r$ and height $h$. The container is then filled with a liquid, and the tetrahedron is removed. The volume of the liquid that is left in the container is $\frac{1}{3}$ of the volume of the container. Given that the tetrahedr... | nvidia |
What is the smallest positive integer $n$ such that the product of $1! \times 2! \times 3! \times \cdots \times n!$ is divisible by $1000$? | nvidia |
Find the sum of the largest one-digit prime, the largest two-digit prime, and the smallest three-digit prime that is a palindrome.
Note: A palindrome is a number that reads the same forwards and backwards. | nvidia |
Find the 1000th term of the sequence that consists of all positive integers that can be written as a sum of distinct perfect squares. | nvidia |
Let $a$, $b$, $c$ be three positive numbers such that $\sqrt[3]{abc}=5$, $\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}=2$, and $a+b+c=12$. Find $a^2+b^2+c^2$. | nvidia |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $\displaystyle {{m+n\pi}\over p}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m+n+p$.
Another problem... | nvidia |
Find all values of $x$ such that $\lceil x \rceil - x = 4.7$. | nvidia |
Among all convex polygons with perimeter 24 cm, find the one with the largest area and calculate its area. | nvidia |
At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$ | nvidia |
Let $S$ denote the set of all $x$ that satisfy
\[\sqrt[3]{x+\sqrt[3]{x+\sqrt[3]{x+\cdots}}}=2.\]Determine the sum of the squares of the elements of $S.$ | nvidia |
Let $C$ be a circle with center $O=(0,0)$ and radius $r=2$, and let $P$ be a point on the circle such that $PO=2$ and $OP$ makes an angle of $\theta$ with the positive $x$-axis. Find $\theta$ such that the area of the sector $OAP$, where $A=(2,0)$, is equal to $\frac{1}{4}$ of the area of the circle. | nvidia |
The function $f(x)=\sin x+\cos x$ takes on the value $a$ at $x=\frac{\pi}{4}$. Find $b$ such that $f(b)=0$ and $\frac{\pi}{4}<b<\frac{5\pi}{4}$. | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of
\[\frac{1}{a} + \frac{2}{b}.\] | nvidia |
There are three real numbers $x$ that are not in the domain of $$f(x) = \frac{1}{1+\frac{1}{1+\frac 1x}}.$$ What is the sum of those three numbers? | nvidia |
Find a cubic polynomial $q(x)$ such that its graph intersects the graph of the function $y=x^3$ at the points $(0,0),$ $(2,8),$ and $(4,64).$ | nvidia |
Let $P$ be a point inside a regular polygon of $n$ sides such that each vertex of the polygon is equally likely to be the closest vertex to $P$. Find the expected number of vertices of the polygon that are closer to $P$ than the center of the polygon. | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
Find the sum of the positive divisors of the number $2^4 \cdot 3^3 \cdot 5^2$. | nvidia |
Determine the value of $a$ such that the number $12345a678$ is divisible by $8$. | nvidia |
Find the value of $b$ such that the quadratic expression $x^2 + bx + 4$ can be factored into the product of two binomials with integer coefficients. | nvidia |
The sum of three different numbers is 67. The two larger numbers differ by 7 and the two smaller numbers differ by 3. What is the value of the largest number? | nvidia |
Find the sum of the positive divisors of 24. | nvidia |
Let $n$ be the smallest positive integer such that $2^n\equiv 1 \pmod{15}$. Compute $2^n+1 \pmod{3}$. | nvidia |
Find a nonzero $q$ such that $qx^2-2x-6=0$ has two solutions, one of which is three times the other. | nvidia |
An ellipse is centered at the origin and has the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$. A triangle is formed by the center of the ellipse and two vertices of the ellipse. If the two vertices are chosen such that the area of the triangle is maximized, what is the length of the base of the triangle? | nvidia |
Determine the largest positive integer n such that
\[\sqrt[n]{\frac{\sin^n x + \cos^n x}{2}} \leq \frac{1}{n}\]
holds for all real x. | nvidia |
Find the sum of the squares of the positive integer divisors of 91. | nvidia |
Find the value of $x$ in the expression $\dbinom{x}{2}$, given that $x$ is a positive integer and $\dbinom{x}{2} = 21$. | nvidia |
The sum of the lengths of all the edges of a rectangular prism is 84 cm. Find the number of cubic centimeters in the volume of the prism. | nvidia |
Find the length of the side of a square inscribed in a circle, if the area of the circle is $4\pi$ square units. | nvidia |
Find the value of $x$ such that \[\frac{\Gamma(x+1)\Gamma(2)}{\Gamma(x+3)} = \frac{1}{5}.\] | nvidia |
Find the number of terms in the sequence of positive integers that are multiples of 3 and have a sum of 150. | nvidia |
In a parallelogram $ABCD$, the point $E$ with coordinates $(x, y)$ is chosen inside the parallelogram such that the four small triangles $AEB$, $AEC$, $BED$, and $CED$ all have equal areas. If the vertices of the parallelogram are $A(2, 3)$, $B(4, 6)$, $C(6, 3)$, and $D(4, 0)$, find the value of $x + 2y$. | nvidia |
Consider a regular $n$-sided polygon with $n$ even, and let $k$ be a positive integer less than $n$ that is relatively prime to $n$. A $k$-diagonal of the polygon is a line segment connecting two vertices that are $k$ vertices apart. How many non-overlapping $k$-diagonals can be drawn in the polygon such that no two di... | nvidia |
Given a non-zero digit $a$, let $S$ be the set of all positive integers whose digits are all equal to $a$, and let $T$ be the set of all positive integers whose digits are all equal to $a$ or $0$. Determine the number of different possible values of $a$ such that all the elements of $S$ are factors of some element of $... | nvidia |
The decimal representation of $a/b,$ where $a$ and $b$ are relatively prime positive integers and $a < b,$ contains the digits $3, 7$, and $2$ consecutively, and in that order. Find the smallest value of $b$ for which this is possible. | nvidia |
Find the number of divisors of 315 that are also perfect squares. | nvidia |
How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5? | nvidia |
What is the smallest positive integer $n$ such that $3n$ is a perfect square and $2n$ is a perfect cube? | nvidia |
Let
$$f(x)=x^3-px+q$$
be a polynomial with real coefficients such that the sum of the squares of its roots is equal to the product of its roots. Find the sum of the coefficients of $f(x)$. | nvidia |
Find the number of real solutions of the equation
\[\frac{x}{50} = \cos x.\] | nvidia |
Find the number of real values of $c$ such that the equation $\left|\frac12-ci\right| = \frac34$ has exactly two complex solutions. | nvidia |
Find the number of real numbers $x$ such that \[\lfloor x^3\rfloor+\lceil e^x\rceil = \lfloor x^3+e^x\rfloor.\] | nvidia |
Let \(S\) be the set of all points \((x,y)\) in the coordinate plane such that \(|x+y|+|x-y|\leq 4\) and \(y\geq x\).
Find the largest possible area of a circle whose center lies on the line \(y=x\) and intersects or touches \(S\). | nvidia |
Parabola has its focus at $(2,1)$. Given that it passes through the point $(-1,3)$ and its vertex is on the line $y=x$, its equation can be written in the form
$$y-k=\frac{1}{4p}(x-h)^2$$
where $p,h,k$ are constants and $p>0$. Find $p+k$. | nvidia |
Problem:
Find the smallest positive integer $b$ for which $x^2+bx+2008$ factors into a product of two binomials, each having integer coefficients.
New Problem:
Find the smallest positive integer $k$ such that the equation $x^2-kx+2008=0$ has two distinct integer solutions. | nvidia |
In a regular octahedron, each face is colored with one of eight different colors such that no two faces that share an edge have the same color. A vertex is chosen at random, and a sequence of faces is selected by moving in a clockwise direction around the vertex, starting with the top face. If the colors of the first f... | nvidia |
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