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Find the smallest positive integer $k$ such that the three fractions $\frac{4}{7}$, $\frac{2}{5}$, and $\frac{3}{k}$ have a common denominator. | nvidia |
Let $\alpha$ and $\beta$ be real numbers, $\alpha\neq\beta$. Find the number of functions $f:\mathbb R \to \mathbb R$ such that for every real number $x$, $$f(f(x)-x)=\alpha x+\beta.$$ | nvidia |
Find the number of functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[f(x - f(y)) = x - y\]for all real numbers $x$ and $y.$ | nvidia |
Determine the number of prime positive integers that are factors of the sum of the squares of the first 100 natural numbers. | nvidia |
Find the greatest positive integer that is divisible by 10 and 15 and whose square is less than 6000. | nvidia |
Let $a$ and $b$ be positive integers such that $a + b = 100.$ Find the maximum value of $a \cdot b \cdot (a - b).$ | nvidia |
In triangle $PQR$, $S$ is on $\overline{PQ}$ such that $PS:SQ = 2:5$, and $T$ is on $\overline{QR}$ such that $QT:TR = 2:5$. If lines $ST$ and $PR$ intersect at $U$, then find $\frac{ST}{TU}$. | nvidia |
Find the number of positive integers $x$ such that $x^3 - 2x^2 - 5x + 6$ is a perfect square. | nvidia |
Find the largest positive integer $n$ such that the greatest common divisor of $n$ and $20$ is $2$ and the product of the digits of $n$ is a multiple of $3$. | nvidia |
Let $P_n$ be the product of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $P_n$ is a perfect square. | nvidia |
A set of 8 distinct integers from the interval $[1,20]$ is chosen randomly. The probability that exactly one pair of numbers from the set sums up to 21 would be greater if the number $n$ had first been removed from the interval. What is the value of $n$? | nvidia |
Determine the sum of all positive integers that are not greater than 15 and that are relatively prime to 15. | nvidia |
The geometric mean of two positive integers is the square root of their product. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the geometric mean of $x$ and $y$ equal to $2^{15}$? | nvidia |
Let $\mathbb{F}$ be the set of all integer residues modulo $p$ where $p$ is a prime number.
A line graph on the 2-dimensional $\mathbb{F}$-grid is a set of all points $(x,y)\in\mathbb{F}^2$ that satisfy a modular equation of the form $ax+by\equiv c\pmod p$ where $a,b,c\in\mathbb{F}$ and $(a,b)\neq (0,0)$.
Find the sm... | nvidia |
A complex number $z$ is chosen at random from the set of all complex numbers such that $|z| = 5$. Find the probability that $|z + 3 - 4i| < 2$. | nvidia |
How many positive integers $n$ are there such that the cubic equation $x^3 - n^2x + 2n = 0$ has only integer roots for $x$? | nvidia |
Find the minimum value of the function
\[g(y) = |y - 3| + |2y + 1|\]for all real numbers $y$ and determine the set of points where this minimum is attained. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 6.$ Find the minimum value of
\[\frac{1}{a} + \frac{4}{b} + \frac{9}{c}.\] | nvidia |
Consider the numbers $x=2^{2001}-1,y=2^{2001}+1$, and the base-$b$ representations of these numbers for different $b$. What is the largest value of $b$ such that the base-$b$ representations of $x$ and $y$ have the same number of digits? | nvidia |
Determine the number of ways to arrange the letters of the word ABCDEFGH such that no letter is in its original position. | 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 |
A five-digit integer $M$ has the property that its decimal representation and the decimal representation of its cube both end with the same sequence of digits. Find the largest possible value of $M$. | nvidia |
Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Find the minimum value of
\[\frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x}.\] | nvidia |
Find the number of digits that are not 0 or 1 in the binary representation of the sum of the base-8 representations of $163_{10}$ and $244_{10}$. | nvidia |
A box contains 20 cards, each with a number from 1 to 20. 5 cards are drawn at random. What is the probability that the sum of the numbers on the cards is even? | nvidia |
Determine $a$ such that the equation
$a^{log_a(7)} = 100$
has a valid solution. | nvidia |
Define a function $F(x,y,z) = x + y^z$. Find a positive value of $n$ such that the equation $F(n,n,2) = 85$ holds true. | nvidia |
What is the sum of the first 10 positive integers that are not divisible by 3? | nvidia |
In a right triangle with legs of length $a$ and $b$, where $a<b$, and hypotenuse of length $c$, the lengths of the sides form an arithmetic progression. What is the smallest possible value of $c$ such that $a$, $b$, and $c$ are all positive integers? | nvidia |
For all positive integers $n$, what is the smallest integer that always divides into the sum of $n$ and the square of $n$? | nvidia |
What is the smallest positive integer $n$ such that the sum of the squares of the first $n$ odd numbers exceeds 10,000? | nvidia |
The ages of three generations of a family are such that the sum of the ages of any two members equals the age of the third member multiplied by a fixed number. If the age of the middle generation is 30, determine the ages of the other two members and the fixed number. | nvidia |
In a convex quadrilateral, two of the interior angles are acute, and the other two are obtuse. If the sum of the measures of the acute angles is 120 degrees, what is the maximum possible measure of one of the obtuse angles? | nvidia |
A palindrome number is a number that remains the same when its digits are reversed. For example, 121 and 12321 are palindrome numbers.
How many 6-digit palindrome numbers are there, such that the number is a multiple of 3 and its first two digits are either 15, 24, or 27? | nvidia |
A circle is inscribed in an isosceles right triangle $ABC,$ where $AB=BC.$ The diameter of the circle is equal to the length of side $AC.$ Find the ratio of the area of triangle $ABC$ to the area of the circle. | nvidia |
How many three-digit positive integers exist, all of whose digits are 2's and/or 5's? | nvidia |
Let $n$ be a positive integer and $k$ be the number of solutions in nonnegative integers of the inequality $x^2+y^2+z^2<n$. Find the largest $n$ such that $k$ is not a multiple of $4$. | nvidia |
A number is called $\textit{digitally balanced}$ if the sum of its digits equals the sum of the digits of its square. Find the smallest two-digit number that is not digitally balanced. | nvidia |
The polynomial $P(z)=z^{28}-z^8-1$ has roots of unity as solutions. Consider a new polynomial
$Q(z)=z^{42}+az^{21}+bz^7+c$. Find the values of $a, b,$ and $c$ such that all the roots of $Q(z)$ that lie on the unit circle are also roots of $P(z)$. What is the sum of the values of $a, b,$ and $c$? | nvidia |
The blue parabola shown is the graph of the equation $y = ax^2 + bx + c$. The graph has two $x$-intercepts, $x_1$ and $x_2$, and the length of the segment connecting these intercepts is equal to the distance between the vertex of the parabola and the point $(0, 0)$. Find $a$. (Assume that the graph has the standard uni... | nvidia |
Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\] | nvidia |
Find the number of ways that 2050 can be expressed as the product of two integers, where one integer has three digits and the other has two digits. (Factorizations of the form $a\cdot b$ and $b\cdot a$ are considered the same). | nvidia |
The diagonals of a parallelogram intersect at the point $(2, 3)$. One vertex of the parallelogram is located at the point $(5, 7)$. Find the equation of the circle passing through all four vertices of the parallelogram. | nvidia |
Two different positive numbers are written on two different cards. The average of the two numbers is 12, and their product is 80. What is the positive difference between the two numbers? | nvidia |
Find the number of four-digit integers that are "cross-balanced", where the sum of the first and third digits equals the sum of the second and fourth digits. | nvidia |
Find the sum of all positive integers less than or equal to 24 that are not divisors of 24. | nvidia |
How many distinct positive integers are divisors of at least one of the numbers 24, 30, or 36? | nvidia |
A box contains a total of $n$ red and green balls, where $n$ is a positive integer greater than 1. If $n-1$ of the balls are randomly removed from the box, the probability that all the remaining balls are red is $\frac{2}{5}$. What is the smallest possible value of $n$? | nvidia |
Let $T$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that $x > 0$ and $\arg(x + yi) \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right].$ 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... | nvidia |
Find the equation of the directrix of the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ that passes through the point $(1,2)$. | nvidia |
Find the function $f(x)$ that satisfies the equation $f(x^3 + 2x) = x^2 + 3x - 1$ for all real numbers $x$. | nvidia |
Triangle $XZY$ is inscribed in a circle such that arc $XY$ measures 120 degrees. Point $W$ lies on segment $XZ$ such that angle $WYZ$ is 60 degrees. Find the ratio of the area of triangle $XWY$ to the area of triangle $WYZ$. | nvidia |
Let $f(x)$ be a real-valued function defined for all real numbers $x$. Suppose that for all real numbers $a$ and $b$, the equation
$$f(a) + f(b) = \frac{1}{2} f(a+b)$$
holds. If $f(1) = 1$, find the value of $f(2)$. | nvidia |
Let $n$ be a positive integer and consider a set of $2n+1$ points chosen at random inside or on a regular $n$-gon with side length $1$. Determine the smallest possible number $a$ such that it is always possible to select a pair of points from this set such that the distance between them is equal to or less than $a$. | nvidia |
Let $\mathbb{R}$ be the set of all real numbers. Define a binary operation $\odot$ on $\mathbb{R}$ by $a\odot b=ab+a+b$ for all $a, b\in \mathbb{R}$. Determine whether there exists a real number $x$ such that for any real number $y$, the equation $x\odot y=y$ holds. If such an $x$ exists, find it. | nvidia |
A sequence of numbers is defined as follows: $a_1=x$, and for each $n>1$, $a_n=\sqrt{a_{n-1}+a_1}$, where $x$ is a positive real number. Find the value of $x$ for which $a_{20}$ is equal to 10 times $a_1$. | nvidia |
Determine the sum of all real numbers $y$ satisfying
\[(y^2-6y+3)^{y^2-7y+3} = 1.\] | nvidia |
Given that the point $(9,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=\frac{f(2x)}2+2$. What is the sum of coordinates of that point? | nvidia |
The product of three different positive integers is equal to $7^3$. What is the sum of the three integers? | nvidia |
Determine the value of $x$ such that $\sqrt{x^2+3\cos(2x)}=2x-\sin(x)$. | nvidia |
Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $100$ with exactly three positive divisors. What is $m+n$? | nvidia |
The sum of the positive integers from $1$ to $k$ can be written as the sum of $k$ consecutive integers in exactly $5$ ways. What is the smallest such $k$? | nvidia |
For how many positive integers $n$ does $1+2+\cdots+n$ evenly divide $6n$? | nvidia |
Find the value of $x$ such that $\sqrt{x^2+2x+5}=3-x$. | nvidia |
Find the maximum value of
\[\frac{x^2 + 7}{\sqrt{x^2 + 3}}\]over all real numbers $x,$ given that $x^2 + 7 \leq 100.$ | nvidia |
In a series of seven coin tosses, what is the probability that the number of tails is at most two, given that the number of heads is at least five? | nvidia |
Find the smallest positive integer that is divisible by exactly five distinct prime numbers. | nvidia |
The numbers $1, 2, \ldots, n$ are written on $n$ different cards. The cards are then shuffled and placed in a row. What is the smallest value of $n$ such that the probability that the cards are in increasing order from left to right is less than 1 in 10? | nvidia |
In a right-angled triangle, the length of the hypotenuse is equal to the sum of the lengths of the other two sides. If the area of the triangle is 48 square cm, what is the length of the longest side of the triangle? | nvidia |
Find the sum of the infinite series $2+3\left(\dfrac{1}{2001}\right)+4\left(\dfrac{1}{2001}\right)^2+5\left(\dfrac{1}{2001}\right)^3+\cdots$. | nvidia |
Let $x$ and $y$ be positive real numbers. Find the minimum value of
\[\frac{x^3+y^3}{xy(x+y)}.\] | nvidia |
The sum of the digits of the number $7^m$, where $m$ is a positive integer greater than or equal to 10, is divisible by 9.
Prove that $m$ is divisible by 3. | nvidia |
In a circular sector of a radius of 50 units, two circles of radii 10 and 15 units are drawn such that they are tangent to the arc of the sector and to its radii, and each of the circles lies entirely inside the sector.
Find the area of the smallest triangle that can be formed by connecting three points, one from each ... | nvidia |
Find the value of $x$ such that $\left(\sqrt{x^3+2x^2-7x+1}\right)^4 = 256$. | nvidia |
Determine the number of positive integers $n$ for which $2^n$ is between 1,000 and 10,000. | nvidia |
Find all values of $a$ for which the product of the solutions of the equation $x^2 - 2ax + 3a^2 = 0$ equals the sum of the cubes of these solutions. | nvidia |
How many 4-letter words can be constructed from the letters $A$, $B$, $C$, $D$, and $E$ such that the number of consonants is equal to the number of vowels? | nvidia |
Find the smallest positive integer $n$ such that $\cos(2\pi/n) < 1/3$. | nvidia |
Given the cubic equation $y = ax^3 + bx^2 + cx + d$ with roots $-2$, $-1$, and $3$, the graph of the equation crosses the x-axis at these roots and has a y-intercept of $-12$. Find the product $abcd$.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=blac... | nvidia |
Find the four-digit integer $pqrs$ such that both $pqrs$ and $pqrs^3$ end in the same sequence of four digits, where $p$ is not zero. | nvidia |
In a triangle, the lengths of the three sides are positive integers and the perimeter is 120 units. What is the largest possible value of the length of the longest side such that the triangle is not isosceles? | nvidia |
The equation of the hyperbola is $\frac{x^2}{144} - \frac{y^2}{225} = 1.$ Find the positive value of $m$ such that the asymptotes of this hyperbola have the equations $y = \pm mx.$ | nvidia |
A positive integer $m$ is called a "cosmic" number if there exists a positive integer $b$ such that $m=\lfloor b\sqrt{m-b}\rfloor$. Determine the number of nonnegative integers $c$ for which there exist at least one cosmic number $m$ satisfying the condition $\lfloor m/3\rfloor=c$. | nvidia |
What is the sum of the fifth row of the Pascal's triangle with all even numbers removed? | nvidia |
There are $n$ coconuts and $n$ boxes; however, only one box has a size that can hold all $n$ coconuts. Find the minimum number of tries you have to make to find the box that can hold all coconuts. | nvidia |
What is the largest positive integer k such that there are exactly k percent of positive integers less than or equal to 2000 that have remainders equal to 1 when divided by 3? | nvidia |
Given that $\triangle XYZ$ is similar to $\triangle ABC,$ with side lengths $XY=3$, $YZ=4$, $XZ=5$, $AB=6$, and $BC=8$, determine the area of triangle $ABC$ if the area of triangle $XYZ$ is $6$ square units. | nvidia |
How many positive two-digit integers have an odd number of positive factors? | nvidia |
Determine the area of the quadrilateral with vertices $(-2,-1,3),$ $(3,1,2),$ $(7,6,4),$ and $(-1,1,1)$. | nvidia |
A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the numbe... | nvidia |
Find $x$ such that $(5,x)$ lies on the line through $(0,2)$ and $(3,1)$. | nvidia |
Determine the value of $a$ such that the equation $x^2 + ax + 2 = 0$ has two real roots with a product of 2, given that the sum of the roots is 3. | nvidia |
Problem:
Let $r > 2$ be a real number. Suppose a right circular cylinder with radius $r$ and height $2r$ is sliced in half along a plane that passes through the center of the cylinder and intersects one of the circular faces at two points $A$ and $B$, such that $\overarc{AB}$ measures $60^\circ$. If the area of one of... | nvidia |
The function $g(n)$ is defined on the positive integers such that $g(g(n)) = 2n + 1$ and $g(2n + 1) = 2n + 2$ for all positive integers $n.$ Find $g(2017).$ | nvidia |
Find the sum of all the even integers from 20 to 100, inclusive. | nvidia |
Two real numbers $x$ and $y$ satisfy $x^2+y^2=8$ and $x^4+y^4=16$. Compute $x^2y^2$. | 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 |
Let $P(x)$ be a polynomial such that $P(2) = 15$, $P(3) = 7$, and $P(4) = 19$. Determine the largest integer $n$ such that $\frac{P(x)-5}{x-2}$ is an integer for all integers $x$ with $2 \le x \le n$. | nvidia |
The sum of the fourth powers of four consecutive positive integers is 18721. What is the sum of the fifth powers of the four original integers? | nvidia |
The sum of the lengths of all the edges of a cube is 60 cm. Find the number of cubic centimeters in the volume of the cube. | nvidia |
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