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What is the sum of all integer values $n$ for which $\binom{20}{n}+\binom{20}{10}=\binom{21}{11}$? | nvidia |
A $\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$.
How many $5$-digit palindromes are there such that the sum of their digits is equal to $10$? | nvidia |
Let $a$ and $b$ be two real numbers such that
$0 < a < b < 1$. The numbers $a$ and $b$ satisfy the following
property: when $a$ and $b$ are expressed as decimals in base 10,
with no leading zero, the digits of $a$ are a permutation of the
digits of $b$. Determine the smallest $b - a$. | nvidia |
Find the number of positive integers less than or equal to 1000 that are not divisible by any of the numbers 15, 20, or 25. | nvidia |
Find the minimum value of $x^2 + y^2 + z^2$ such that $x$, $y$, and $z$ are positive real numbers satisfying $xy + yz + zx = 1$ and $x + y + z = 2$. | nvidia |
Let f(z)=z^2+az+b, where a and b are complex numbers. A sequence F_n (starting from F_0) of iterates of f is defined by the recurrence relation F_n=f(F_(n-1)). It is given that for every complex number z, there is real number m such that F_m(z)=F_(m+k)(z) for some positive integer k. Determine a. | nvidia |
Find all values of $a$ such that the following system has a solution $(x, y)$ in real numbers:
\begin{align*}
x^y + y^x &= a^a, \\
a^{x + y} &= (xy)^a.
\end{align*} | nvidia |
Find the smallest positive integer n, such that n and 2^n have exactly 5 common divisors. | nvidia |
The function $f(x)$ takes on real values for all real $x$ except $x=1$ and $x=-1$, where it has vertical asymptotes. Its graph also has a horizontal asymptote at $y=2$. If $k$ is a randomly selected real number, what is the probability that the equation $f(x)=k$ has at least one solution that is not in the interval $(-... | nvidia |
A bookshelf contains 12 shelves, and the average number of books on each shelf is 8. However, the shelves are arranged in a circular manner, and it's known that the number of books on each shelf is directly proportional to the shelf's distance from the center of the circle. What is the maximum number of books that can ... | nvidia |
Let $P_1$, $P_2$, $P_3$, $P_4$ be four distinct points in the plane, such that the segments $\overline{P_1P_2}$, $\overline{P_2P_3}$, $\overline{P_3P_4}$, and $\overline{P_4P_1}$ form a convex quadrilateral. At each step, a particle, initially at $P_1$, moves to one of the two adjacent points with equal probability. Le... | nvidia |
The average of two 2-digit positive integers is equal to the decimal number obtained by writing one of the two-digit integers before the decimal point and the other two-digit integer after the decimal point. What is the smaller of the two integers? | nvidia |
Find the number of ordered 12-tuples $(b_1, b_2, b_3, \dots, b_{12})$ of integers, such that the cube of any number in the 12-tuple is equal to the product of the other 11 numbers. | nvidia |
Determine all possible integer values of $a$ such that there exists a pair of real numbers $(b,c)$ satisfying the equations $a+b^2 = c^2$ and $ab - bc = \frac{c^2}{2}$. | nvidia |
Circle $A$ has its center at $(3, -4)$ and a radius of $8$ units. What is the sum of the $x$-coordinates of the two points on circle $A$ that are also on the $x$-axis? | nvidia |
Let $x$, $y$ and $z$ be positive integers such that $xy-x$, $xy-y$, and $xy-z$ are all prime numbers. Find the largest possible value of $xy$. | nvidia |
Find the sum of all real numbers $c$ such that there exists a circle centered at $C = (0,c)$ with exactly one chord $\overline{AB}$ that is tangent to the parabola $y = x^2$. | nvidia |
How many four-digit numbers are there that have at least two consecutive digits that are the same? | nvidia |
Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x.$ | nvidia |
Determine the largest positive integer $m$ such that the product of the first $m$ prime numbers can be expressed as the product of $m + 2$ consecutive even positive integers. | nvidia |
Let $p(x)$ be a polynomial such that
$$p(x) p(y) = p(x) + p(y) + p(xy) - 2,$$for all real numbers $x$ and $y$. Find $p(2)$. | nvidia |
Compute the sum of the squares of all the divisors of 24. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=25$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=7$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
Let $r$ be a positive constant. Prove that if the incircle of triangle $ABC$ touches $AB$ and $AC$ at points $P$ and $Q$ respectively, then
\[\frac{1}{AP} + \frac{1}{AQ} + \frac{1}{BC}\]is constant, and find this constant in terms of $r.$ | nvidia |
Find the sum of all odd positive integers that can be represented in the form $4k^2+3$, where $k$ is a positive integer, and $4k^2+3<500$. | nvidia |
In the equation
$$7x^2-kx+3=0$$
determine $k$ such that the difference of the roots be equal to unity. | nvidia |
The sum of the first 12 positive odd integers is also the sum of three consecutive odd integers. What is the middle number of these three odd integers? | nvidia |
Determine the smallest positive integer $m$ such that $2^m$ contains only the digits 2, 4, 5, 6, and 8, and $m$ itself contains the digit 3. | nvidia |
A function $f(x)$ is defined as $f(x) = |x - 3| + 2$. Find the sum of all distinct numbers $x$ such that $f(f(f(x))) = 4$. | nvidia |
Find the positive real value of $s$ that satisfies $|s-3i\sqrt{2}| |4+5i| = 35$. | 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 |
The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | nvidia |
Find the number of positive integers x that satisfy the equation:
(1 + x)^(1 + x) = 4x^(4 - x)
Determine the values of x for which the equation is true. | nvidia |
Let $n \ge 4$ be an integer and let $T = \{4,5,6,\ldots,n\}$. Find the smallest value of $n$ such that for every partition of $T$ into three subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $a+b = c$. | nvidia |
Find the smallest positive integer that is both an integer power of 13 and is not a palindrome. | nvidia |
Determine the integer $x$, $-100 \le x \le 100$, such that $3x \equiv 7 \pmod{17}$. | nvidia |
Find all real values of $x$ that satisfy $\frac{1}{x+1} + \frac{3}{x+7} \ge \frac23.$ (Give your answer in interval notation.) | nvidia |
Find all real numbers $a$ such that the graphs of the functions $y = e^{x + a}$ and $y = e^{1 - x}$ are symmetric with respect to the line $y = x$. | nvidia |
Find the sum of all odd, positive integers less than 91. | nvidia |
Find the sum of the real solutions of: $|x^2-1|=3|x-1|$. | nvidia |
The solutions $p$ and $q$ of the equation
$x^2+2ax+13=0$ are such that
$\frac{1}{p}+\frac{1}{q}=3$.
Determine $a$. | nvidia |
Find the positive real number(s) $y$ such that $\frac{1}{3}\left( 2y^2+5\right) = \left( y^2-30y-15\right)\left( y^2+20y+10\right)$. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a+b+c=1.$ Find the maximum value of
\[\frac{ab}{a+b} + \frac{bc}{b+c} + \frac{ca}{c+a}.\] | nvidia |
A box contains 12 marbles, some of which are blue and the rest of which are green. When two marbles are drawn at random at the same time, the probability that both marbles are blue is $\frac{1}{20}$. How many marbles in the box are blue? | nvidia |
How many positive integers less than or equal to 100 can be represented as the sum of two perfect cubes in exactly two ways? | 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 |
Find the sum of all divisors of $144$ that also divide $75$, including both positive and negative divisors. | nvidia |
Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $1001$, then what is the sum of the numbers written on the faces? | nvidia |
Consider a polynomial function $f(x)=ax^3+bx^2+cx+d$. It is known that the function has three distinct roots $r_1$, $r_2$, and $r_3$. Moreover, it is given that the product of the roots taken two at a time is equal to the sum of the roots (i.e., $r_1r_2+r_1r_3+r_2r_3=r_1+r_2+r_3$). Find $a/b$. | nvidia |
The function $f(x)$ satisfies
\[f(xy) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(0) \neq 0.$ Find $f(10).$ | nvidia |
Find the values of $a$ for which the inequality
\[\frac{x^2+ax+1}{x^2+2x+2} < 1\]
holds true for all real numbers $x.$ | nvidia |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | nvidia |
Find the range of all possible values of $m$ such that the graphs of the functions $f(x)=mx-2$ and $g(x)=m^2x+m-1$ intersect on the x-axis. | nvidia |
Find the value of $x$ such that $\sqrt{x+ 7} = 9$. | nvidia |
Given the equation $\frac{x^2-7x-18}{x^2+3x-4}=x-5$, find the sum of the squares of its solutions. | nvidia |
A rectangular chessboard has a certain number of squares colored in such a way that no two adjacent squares have the same color. Two squares are adjacent if they share a common side. Suppose that the number of colored squares is the maximum possible for the given chessboard. How many ways are there to color the squares... | nvidia |
Determine the number of distinct triangles whose vertices are lattice points and whose sides are parallel to the coordinate axes, such that the perimeter of the triangle is equal to 30. | nvidia |
Point $P$ lies on the circle $x^2+y^2=16$ and is 5 units from the point $(0,0)$. Find the sum of all possible x-coordinates that satisfy the given conditions. | nvidia |
Determine $a$, $b$, and $c$ in the equation $y = ax^2 + bx + c$ so that the graph of the parabola passes through the points $(1,3)$, $(4,6)$, and has a minimum at the point $(5,4)$. Find the sum of the coefficients $a + b + c$. | nvidia |
Let $f(x)$ be a polynomial of degree 2006 with real coefficients, and let its roots be $r_1,$ $r_2,$ $\dots,$ $r_{2006}.$ There are exactly 1006 distinct values among
\[|r_1|, |r_2|, \dots, |r_{2006}|.\]What is the minimum number of real roots that $f(x)$ can have? | nvidia |
What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 9?
Another problem inspired by this one:
Find the smallest positive integer $n$ such that $\frac{1}{n}$ is a repeating decimal with a repeating cycle of length 3, and $n$ contains the digit 7. | nvidia |
A five-digit number $\underline{a2a4a}$ is divisible by 9. Determine the value of $a$ such that when the digits of the number are reversed, the resulting number is also divisible by 9. | nvidia |
Compute the product of all the roots of the equation $(x^2+2x+1)(x^2-4x+4)-(x^2+2x+1)(x^2-6x+9)=0$. | nvidia |
Determine the equation of the line that passes through the midpoint of the segment joining the points $(2,2)$ and $(-1,-1)$ and is perpendicular to the line connecting these two points. | nvidia |
A square $ABCD$ is inscribed in a semicircle with diameter $d$ so that vertex $A$ lies on the diameter and vertices $B$ and $C$ lie on the semicircle. It is given that the perimeter of the square is equal to $d$. Find the ratio of the area of the square to the area of the semicircle. | nvidia |
Find the sum of the first 5 terms of the geometric sequence with common ratio $r$ such that the sum of the first 3 terms is 21 and the sum of the first 6 terms is 91. | nvidia |
How many positive integers less than 1000 can be formed using only the digits 3, 4, and 7, such that no digit is repeated? | nvidia |
Find the smallest positive integer that has exactly five distinct prime factors. | nvidia |
Let $P$ be a regular 12-sided polygon whose vertices are distinct lattice points in the coordinate plane. The area of $P$ can be expressed as $a\sqrt{b} + c\sqrt{d}$, where $a, b, c,$ and $d$ are positive integers and $b$ and $d$ are square-free. Find the smallest possible value of $a + b + c + d$. | nvidia |
What is the largest positive integer that is guaranteed to divide the sum of the first n terms of any geometric sequence whose terms are positive integers and whose common ratio is greater than 1? | 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 |
The positive integers $m$ and $n$ are such that the decimal $0.070707\ldots$ can be represented as the fraction $\frac{8m+11}{6n+9}$. Determine the smallest possible value of the sum $m+n$. | nvidia |
The sum of the squares of two consecutive positive integers is 365. What is the larger integer? | nvidia |
Find the sum of all the real solutions of $\ds x^6-4x^3+1=0$ and determine if the equation has any complex solutions. | nvidia |
Given a positive integer $m$, it can be shown that every complex number of the form $p+qi$, where $p$ and $q$ are integers, can be uniquely expressed in the "base" $-m+2i$ using the integers $1,2,\ldots,m^2$ as "digits." That is, the equation\[p+qi=b_n(-m+2i)^n+b_{n-1}(-m+2i)^{n-1}+\cdots +b_1(-m+2i)+b_0\]is true for a... | nvidia |
What is the smallest positive integer $n$ such that $n^{n^n}$ ends in the digits 246? | nvidia |
A certain polynomial $P(x)$ has the property that whenever $x$ is a positive integer, $P(x)$ is divisible by $x$. What is the smallest positive integer $n$ such that $P(5n)$ is always divisible by 5? | nvidia |
Find the number of ordered pairs $(A, B)$ of subsets of a set with $n$ elements, such that the symmetric difference of $A$ and $B$ has exactly $k$ elements, and the intersection of $A$ and $B$ has exactly $m$ elements. | nvidia |
Find the sum of the coordinates of the midpoint of the segment with endpoints $(4, 8)$ and $(-2, 3)$. | nvidia |
Let $n \ge 2$ be an integer and let $T = \{1, 2, 3, \ldots, n\}$. Find the smallest value of $n$ such that for every partition of $T$ into three subsets, at least one of the subsets contains integers $x$, $y$, and $z$ (not necessarily distinct) such that $x + y = z$. | nvidia |
Find the value of $x$ such that $0.\overline{x234} = \frac{9}{11}$. | nvidia |
A portion of a sphere is cut out by two planes, one of which passes through the center of the sphere. Given that the portion has volume 17 cubic centimeters, and that the distance between the two planes is 3 centimeters, determine the radius of the sphere. | nvidia |
A number $n$ has the property that when it is divided by each of $7$, $8$, and $9$, the remainder is $6$ more than the divisor. What is the largest possible positive integer value of $n$ less than $1000$? | nvidia |
If $0.7=\frac{y}{250+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers. | nvidia |
Let $a$ and $b$ be relatively prime positive integers such that $\frac{a}{b}$ is not equal to $\frac{c}{d}$ for any integers $c$ and $d$ such that $0<c<a$ and $0<d<b.$ Find the smallest value of $b,$ given that $|7a-19b|=1.$ | nvidia |
Let $g$ be a function defined on the positive integers, such that $g(x+y) = g(x)g(y)$ for all positive integers $x$ and $y.$ Given $g(5) = 3$ and $g(7) = 2$, find $g(12)$. | nvidia |
In the equation
$$\frac{1}{\sin 2x} + \frac{1}{\sin 4x} = \frac{1}{\sin x},$$
find the smallest positive value of $x$ that satisfies the equation. | nvidia |
Find the number of trailing zeros in the sum of all positive integers from 1 to 1000, inclusive. | nvidia |
Let $a$ be a positive integer. Define the function $f(a)$ to be the smallest positive integer $b$ such that $b^2 \geq a$. Find the value of $f(2136)$. | nvidia |
Find the point on the graph of the equation $y=4x$ that is equidistant from the points $C(-4,0)$ and $D(0,6)$. | nvidia |
In the complex plane, find the number of complex numbers of the form $a+bi$ where $a$ and $b$ are integers, such that the distance from $a+bi$ to $2+3i$ is equal to 5. | nvidia |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | nvidia |
Find the number of distinct right triangles with integer side lengths and a hypotenuse of length 25. | nvidia |
Determine the number of integers $n$ such that the decimal representation of $n^2$ consists of only 2 digits. | nvidia |
The number $a_n$ is defined as the sum of the absolute values of the entries in the $n$th row of a modified Pascal's triangle where each entry is the difference between the corresponding entries of the original Pascal's triangle and its reflection across the vertical axis of symmetry. For example, the first few rows of... | nvidia |
What is the largest possible value of $a$ such that $a^2$ is the area of a rectangle with integer side lengths, one of which is even and the other odd? | nvidia |
Given a cube with an integral edge length $n$, a sphere is inscribed in the cube such that it touches the faces of the cube at their centers. A plane parallel to one of the faces of the cube intersects both the cube and the sphere, creating a cross-section. How many ordered pairs $(n, d)$ exist, where $d$ is the distan... | 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 |
What is the largest positive integer n such that the decimal representation of 2^n contains no repeating digits? | nvidia |
A triangle with sides $a$, $b$, and $c$ is inscribed in a circle. The angles opposite sides $a$ and $b$ are bisected, and the points where the bisectors intersect the circle are connected to form a smaller triangle. If the area of the larger triangle is 40, find the area of the smaller triangle. | nvidia |
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