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Find the number of ways to arrange 5 books, 2 of which are fiction, on a bookshelf, such that the fiction books are placed next to each other. | nvidia |
Problem:
In the diagram, the circles with centers $A$, $B$, and $C$ are mutually tangent. The radius of the circle with center $A$ is twice the radius of the circle with center $B$, and the radius of the circle with center $B$ is twice the radius of the circle with center $C$. If the area of the circle with center $C$ ... | nvidia |
Find the sum of all even numbers from 1 to 20 that are not divisible by 4. | nvidia |
Define a positive integer $k$ to be a factorial prefix if there is some positive integer $l$ such that the decimal representation of $l!$ starts with exactly $k$ digits. How many positive integers less than $2500$ are not factorial prefixes? | nvidia |
Find the product of all positive integral values of $n$ for which $n^2 + 2n + 1$ is a perfect square. | nvidia |
Find the least positive integer $n$ such that the decimal expansion of $\frac{1}{n}$ has exactly three repeating digits. | nvidia |
Let $a,$ $b,$ $c$ be positive real numbers such that $abc = 27.$ Find the minimum value of $2a + 3b + 5c.$ | nvidia |
Let $x$ be a real number such that $\lfloor x \rfloor = -3.$ Find the number of possible values of $\lfloor x^3 \rfloor$. | nvidia |
A rectangular prism has dimensions $a, b, $ and $c$, where $a$, $b$, and $c$ are positive integers with $a\le b\le c$. The volume of the prism is equal to the product of all integers from $1$ to $n$, for some positive integer $n$. What is the largest possible value of $n$ such that the surface area of the prism is less... | nvidia |
Determine the number of bases $b$, such that the representation of $5625_{10}$ in base $b$ contains only digits that are perfect squares. | nvidia |
Let $f(n)$ be the number of ways to arrange a deck of $n$ cards such that each card is either red or blue and there are no three adjacent cards of the same color. Determine the largest value of $n$ such that $f(n)$ is a perfect square. | nvidia |
Find the number of positive integers, including 1, that are divisors of both 60 and 105. | nvidia |
Let $a$ and $b$ be the roots of the polynomial $x^2+px+q$, where $p$ and $q$ are real numbers. Suppose that the sequence $(a^n+b^n)_{n=1}^\infty$ is periodic with period $k$. Determine the largest possible value of $k$. | nvidia |
Find the sum of the first ten distinct positive integer multiples of 5 that are greater than 15. | nvidia |
Determine all possible integer values for $A$, such that the number $5A312$ is divisible by 9. | nvidia |
Find the sum of the positive integer divisors of the least positive integer $n$ such that $$2^n + 3^n + 5^n + 7^n \equiv 0 \pmod{17}.$$ | nvidia |
In a circle with center $O$, $AB$ is a diameter and $C$ is a point on the circle such that $\angle AOC = 120^\circ$. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the perpendicular from $D$ to $AB$. Find the length of $DE$ in terms of the radius of the circle. | nvidia |
Let $a$ and $b$ be two positive real numbers. Define $x=a+\sqrt{a^2-b}$ and $y=a-\sqrt{a^2-b}$. Given that $x+y=ab$, find the value of $\sqrt{xy}$ in terms of $b$. | nvidia |
Someone noticed that $7! = 7 \cdot 8 \cdot 9 \cdot 10$. Find the largest positive integer $m$ for which $m!$ can be expressed as the product of $m - 2$ consecutive positive integers. | nvidia |
Find the smallest multiple of $7$ that is greater than the sum of its digits. | nvidia |
For real numbers $t$, the equation
$$\frac{(t^2+1)x}{t}+y=t^2-1$$
represents a family of lines. Show that there is a fixed point that lies on all the lines in this family, and find its coordinates. | nvidia |
Let $m$ represent the smallest integer that satisfies the following conditions:
* $\frac m3$ is a perfect square.
* $\frac m4$ is a perfect cube.
* $\frac m7$ is a perfect seventh.
Determine the smallest positive integer $k$ such that $m^k$ is divisible by 1000. | nvidia |
A regular polygon with an even number of sides has its vertices at the points $(x_i, y_i)$ in the Cartesian plane, for $i = 1, 2, \ldots, n$. The sum of the $x$-coordinates of the vertices equals $0$. Prove that the center of the polygon coincides with the centroid of the triangle formed by the midpoints of three conse... | nvidia |
Find the smallest positive integer $n$ such that $n^2$ is divisible by 4, 8, 16, and 32. | nvidia |
Compute: $55\times1212-15\times1212$ . | nvidia |
Given a triangle with angles $A$, $B$, and $C$, and opposite sides $a$, $b$, and $c$, find all values of $\theta$ for which the positive difference between the values of $a$ and $b$ equals the sum of the squares of the lengths of the altitudes from vertices $A$ and $B$. Enter all possible values of $\theta,$ separated... | nvidia |
Find the smallest positive integer $n$ such that $n^2 \equiv 1 \pmod{185}$, where $n$ is not congruent to $1$ or $-1$ modulo 185. | nvidia |
Let $g(x)$ be a function defined for all positive integers $x$, such that
$$g(x+1)-g(x)=3x+2.$$
Find the smallest possible value of $x$ for which $g(x+100)-g(x)$ is a multiple of 3. | nvidia |
Given that the polynomial $x^2+mx+25$ has only positive integer roots, find the sum of all distinct possibilities for $m$. | nvidia |
If $n$ is the smallest positive integer for which there exist positive real numbers $a$ and $b$ such that
\[(a + bi)^n = (a - bi)^n,\]compute $\frac{b}{a}.$ | nvidia |
Given positive real numbers $a$, $b$, and $c$ with $a \leq b \leq c$, find the minimum value of $\frac{a^3 + b^3 + c^3}{3abc}$ such that $a + b + c = 1$. | nvidia |
Simplify the expression
$$
\frac{~\frac{a+b}{b-a}~}{~\frac{c+d}{d-c}~}.
$$ | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(f(x) - y) = f(x) + f(f(y) - f(-x)) + x\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$ be the sum of all possible values of $f(3).$ Find $n \times s.$ | nvidia |
In a right triangle $DEF$ with right angle at $E$, a circle with diameter $EF$ intersects side $DF$ at point $G$. Given that the area of $\triangle DEF$ is $108$ and $DF = 18$, find the length of $EG$. | nvidia |
Find the only real number that can be expressed in the form \[(a + bi)^3 - 107i,\]where $i^2 = -1,$ and $a$ and $b$ are positive integers. | nvidia |
A regular hexagon is inscribed in a circle so that the vertices of the hexagon bisect the circle. The area of the circle is 36π square centimeters. What is the perimeter of the hexagon? | nvidia |
In a set of 100 consecutive integers, the sum of the numbers in the set is equal to 2000. Determine the range of the possible values of the median of the set, if the set is modified by removing any 20 of its elements. | nvidia |
How many positive integers $n$, where $3\leq n \leq 100$, have the property that when the digits of $46_n$ are reversed, the resulting number in base $n$ is a multiple of $11$? | nvidia |
Given that $\log_a{b} = x$ and $\log_b{c} = y$, find $n$ such that $a^n = c^{-2} \times \sqrt{\frac{b^{2x+1}}{a^3}}$. | nvidia |
Let $P$ be a point inside triangle $ABC$ such that the areas of triangles $PAB$, $PBC$, and $PCA$ are in the ratio $1:2:3$. Find the ratio of the area of triangle $PAB$ to the area of triangle $ABC$. | nvidia |
Let $a$ be a positive real number. Consider the expression
$\ds\sqrt[3]{\cos \frac{a}{3}}\cdot\sqrt[4]{\cos \frac{a}{4}}\cdots\sqrt[n]{\cos \frac{a}{n}}\cdots$.
Determine $a$ such that the given expression is equal to $\frac{1}{2}$. | nvidia |
Let $a$, $b$ be real numbers, with $b>0$, such that the complex number
$$z=\sqrt{2+b}+i\sqrt{a}$$
is such that the imaginary part of $z^2$ is equal to $a$. Determine all possible values of $a$. | nvidia |
Let $\diamondsuit (x)$ be the product of the digits of a positive integer $x$. $\mathcal{T}$ is the set of positive integers such that for all elements $n$ in $\mathcal{T}$, we have that $\diamondsuit (n)$ is a perfect square and $0\le n< 10^{5}$. If $t$ is the number of elements in $\mathcal{T}$, compute $\diamondsuit... | nvidia |
Find the sum of the squares of the roots of the equation $x^4 + 1 = 0$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 1.$ Find the minimum value of
\[(x + 2y)(y + 2z)(xz + 1).\] | nvidia |
Find all possible values of the function $g(x)$ that takes positive real numbers to real numbers, such that
\[xg(y) - yg(x) = g(xy) + \frac{x}{y}\]for all positive real numbers $x$ and $y.$ Enter all possible values of $g(10),$ separated by commas. | nvidia |
A sequence of real numbers $\{b_n\}$ has the property that for any positive integer $n$, the sum of the squares of the terms in the $n$th window of length 3 is constant. In other words,
$$b_n^2 + b_{n+1}^2 + b_{n+2}^2 = C$$for all $n \ge 1$, where $C$ is a constant. If the first two terms are $b_1 = 3$ and $b_2 = 4$, ... | nvidia |
Let $a,$ $b,$ $c,$ $x$ be four positive integers. We say that the quadruple $(a,b,c,x)$ is a superfine quadruple if $\sqrt{a}+\sqrt{b}+\sqrt{c}=x$. Prove that for any given three positive integers $a,$ $b,$ $c$ there exists at most one positive integer $x$ for which the quadruple $(a,b,c,x)$ is a superfine quadruple. | nvidia |
Determine the number of combinations of two letters from the English alphabet that form a palindrome. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that $g(x^2 + y^2) = g(x)g(y)$ for all real numbers $x$ and $y$. If $g(0) = 1$ and $g(1) = 2$, determine the number of possible values of $g(2)$. | nvidia |
Find the number of positive integers n ≤ 2000 such that $9(n + 1)$ is a cube of a prime number. | nvidia |
The points of intersection of the lines $y=ax+b$ and $y=cx+d$, where $a$ and $c$ are fixed constants and $b$ and $d$ are real parameters, are plotted. For what values of $a$ and $c$ do all the plotted points lie on a straight line? | nvidia |
Find the number of solutions to the equation $x^3 + x^2 + x + 1 = 0$ such that $x$ is a complex number with a real part greater than 0. | nvidia |
For all composite integers $n$, what is the largest integer that always divides into the difference between $n$ and the cube of $n$? | nvidia |
The product of two numbers is 240, and the sum of their reciprocals is 1/15. What is the value of the sum of the two numbers? | nvidia |
Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Find the smallest integer n such that the product of its prime factors is equal to 120. | nvidia |
Find the smallest positive integer $n$ such that the decimal representation of $\frac{1}{n}$ has a repetend that is a palindrome. | nvidia |
A square $ABCD$ is inscribed in a circle with radius $r$. An equilateral triangle $ABC$ is inscribed in the same circle, where point $A$ is shared between the square and the triangle. Find the side length of the square if the area of the triangle is $10\sqrt{3}$ and the distance between the center of the circle and poi... | nvidia |
Determine the smallest positive integer $x$ such that the product of the squares of its digits is divisible by $120$. | nvidia |
Find the smallest positive integer value of x such that the difference between (2x+1)! and (2x)! is greater than or equal to 50000. | nvidia |
Let $a$, $b$, and $c$ be the roots of the cubic polynomial $P(x)=x^3-x^2+7x-2$. Prove that
\[\frac{\sqrt{6a+1}-\sqrt{6b+1}}{\sqrt{6c+1}-1}+\frac{\sqrt{6b+1}-\sqrt{6c+1}}{\sqrt{6a+1}-1}+\frac{\sqrt{6c+1}-\sqrt{6a+1}}{\sqrt{6b+1}-1}=0.\] | nvidia |
Find the largest possible base that can be used for the number system where the following operations can be performed: $2345_{(base)}\div101_{(base)}\times11_{(base)}$. Express your answer in decimal. | nvidia |
Find all values of $k$ so that the domain of
\[b(x) = \frac{kx^2 + 2x - 5}{-5x^2 + 2x + k}\]is the set of all real numbers. | nvidia |
Find the sum of all the terms in the sequence $(-3)^{n}+3^{n}$, where $n$ ranges from $1$ to $6$. | nvidia |
Find the largest three-digit number that is not divisible by any of the first three prime numbers. | nvidia |
Three consecutive positive integers are such that the sum of the first two is a two-digit number whose digits are equal to the digits of the third number in reverse order. Find the third number. | nvidia |
Compute $1-2+3-4+ \dots -98+99$ . | nvidia |
What is the smallest positive integer $n$ such that $531n \equiv 1067n \pmod{24}?$ | nvidia |
In a convex quadrilateral, the diagonals intersect at a point inside the quadrilateral. Two circles are drawn such that each circle is tangent to three sides of the quadrilateral and the diagonals. Determine the maximum number of points of intersection between the two circles and the diagonals. | nvidia |
Find the sum of all positive integer solutions to $x \cdot (x+2) \cdot (x+4) < 1000.$ | nvidia |
Find the maximum value of $3^x - 2^x,$ over all real numbers $x.$ | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x^2 + y^2) = g(x)g(y) + x + y\]for all real numbers $x$ and $y.$
Let $a$ be the smallest possible value of $g(0),$ and let $b$ be the sum of all possible values of $g(1).$ Find $a + b.$ | nvidia |
There are two pairs $(x,y)$ of real numbers that satisfy the equation $x+y = 3xy = 4$. Given that the solutions $x$ are in the form $x = \frac{a \pm b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are positive integers and the expression is completely simplified, what is the value of $a + b + c + d$? | nvidia |
Given that $n$ is a positive integer, and given that $\mathop{\text{lcm}}[24,n]=72$ and $\mathop{\text{lcm}}[n,27]=108$, what is $n$?
Let $a$ and $b$ be positive integers such that $a$ is the product of two distinct prime numbers and $b$ is the product of two distinct prime numbers. If $\mathop{\text{lcm}}[a,n]=\matho... | nvidia |
Two non-congruent circles with centers at points $A$ and $B$ are tangent to the line $AB$ at points $C$ and $D$, respectively. The radius of the circle centered at $A$ is three times the radius of the circle centered at $B$. What is the ratio of the area of the larger circle to the area of the smaller circle? | nvidia |
Determine the number of ordered pairs $(m, n)$ where $m$ and $n$ are positive integers such that $mn = 180$, $m > n$, and $\gcd(m, n) = 1$. | nvidia |
Find the sum of all the positive integers from 1 to 100 that are not divisible by 3 or 5. | nvidia |
Find the surface area of the solid defined by the inequality $|x+y+z| \leq x^2+y^2+z^2 \leq 1$, where $x$, $y$, and $z$ are real numbers. | nvidia |
Find the value of $x$ such that $\sqrt{3x + 7} = 10$. | nvidia |
Simplify the expression $\frac{1+\sqrt{x}}{\sqrt{x}-1}$, where $x$ is a positive integer. Assume that the expression has been simplified to $\sqrt{a} + b$, where $a$ and $b$ are integers. Find the sum of $a$ and $b$ when $x = 2$. | nvidia |
Find the maximum value of the expression $|x+y+z|$ for real $x$, $y$, and $z$ satisfying $x^2+y^2+z^2=1$. | nvidia |
Find the sum of the digits of the largest three-digit integer $n$ such that $n$ is divisible by the sum of its digits. | nvidia |
Compute the sum of the binomial coefficients $\dbinom{505}{k}$ for all $k$ such that $k$ is a multiple of 3. | nvidia |
Determine all real values of $a$ such that the cubic polynomial $ax^3+6x^2+11x+6$ can be factored into the product of three binomials with integer coefficients. | nvidia |
Find the number of distinct real solutions of the equation $x^2 + ax + b = 0$, where $a$ and $b$ are real numbers such that $a + b = 1$ and $a - b = -3$. | nvidia |
Let $n$ be a positive integer with exactly five divisors. Determine the smallest value of $m$ for which $n^m$ has more than 20 divisors. | nvidia |
The vertices of an equilateral triangle are located on the boundaries of three sides of a cube, and each side of the triangle is tangent to one edge of the cube.
The length of a side of the cube is $3$.
Find the length of the side of the equilateral triangle. | nvidia |
In a regular polygon with n sides, each side length is equal to the sum of the squares of its adjacent angles (in degrees). If the sum of the interior angles of the polygon is 1500 degrees, find the perimeter of the polygon. | nvidia |
Find the product of all values of $y$ such that $|y-1| = |2y-6|$. | nvidia |
Find the sum of the $y$-coordinates of the solutions to the system of equations $x=|y^2-4y-3|$ and $x=\frac{5}{2}+y$. | nvidia |
Count the number of positive integers less than 100 that have only prime digits. | nvidia |
Let $S$ be the set of all numbers that can be represented as a sum of distinct powers of 10. Given that there are exactly 50 numbers in $S$ that have the digit 7 as their leftmost digit, find the number of elements of $S$ that have 3 as their leftmost digit. | nvidia |
Find all possible values of $k$ such that the quadratic equation $x^2 + kx + k = 0$ has roots that differ by a factor of 3. | nvidia |
The lengths of the sides of a triangle form an arithmetic progression. If the longest side has length 15, find the sum of the lengths of the two shorter sides, given that the perimeter of the triangle is 36. | 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 |
A positive integer $n$ is called good if there exists an integer $m$ such that $m^2-3m+2$ is divisible by $n$.
Find the number of good numbers in the range $1\le n\le 12$. | nvidia |
Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$. | nvidia |
A function machine has three possible outputs: $A,B,C.$ The input is a two-digit number. If the sum of the digits of the input number is odd, the output is $A.$ If the sum of the digits of the input number is even, but the number itself is odd, then the output is $B.$ In all other cases, the output is $C.$ What is the ... | nvidia |
Find the positive real value of $t$ that satisfies $|t+2i\sqrt{3}| |6-4i| = 26$. | nvidia |
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