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Given a right-angled triangle with vertices $A$, $B$, and $C$, where $AB=6$ and $BC=8$. The point $P$ lies on $AC$ such that $AP=3$ and $PC=4$. A circle is drawn with center $P$ and radius $r$. Find the value of $r$ such that the circle is externally tangent to a circle centered at $A$ with radius $2$ and internally ta... | nvidia |
Find the equation of the circle that passes through the points $(1,2)$, $(3,4)$, and $(5,6)$. | nvidia |
Let $g(x)$ be a continuous function such that $g(g(x)) = x$ for all real numbers $x$. Given that $g(3) = 7$ and $g(7) = 3$, find the sum of all real numbers $x$ that satisfy the equation $g(x) = x$. | nvidia |
Consider a 5-digit palindrome number, where the first and last digits are the same, and the second and fourth digits are the same. How many 5-digit palindrome numbers can be formed such that at least one of the digits is even? | nvidia |
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$.
| nvidia |
Let $g(n)$ be the sum of the absolute values of the differences between consecutive digits of $\frac{1}{7^n}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $g(n) > 5$? | nvidia |
A fair six-sided die is rolled four times. What is the probability that at least one of the rolls is a multiple of 3, given that the sum of all the rolls is even? | nvidia |
In a regular pentagon $ABCDE$ with side length $4$, a circle is inscribed in triangle $ABC$ such that the distance from its center to line $\overline{BC}$ is equal to $1$. Find the radius of the circle. | nvidia |
Three consecutive even integers are such that the sum of their squares exceeds twice their product by 50%. Determine the middle number. | nvidia |
Find the sum of all possible values of $m$ such that the equation $3x^2 - mx + 12 = 0$ has two distinct integer roots. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 1$ and
\[f(xy + f(x)) = xf(y) + f(x)\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f \left( \frac{1}{2} \right),$ and let $s$ be the sum of all possible values of $f \left( \frac{1}{2} \right).$ Find $n \times s.$ | nvidia |
For how many integer values of $x$ is the inequality $x^2 + ax + b \leq 0$ satisfied, where $a$ and $b$ are positive integers and the equation $x^2 + ax + b = 0$ has two distinct real roots? | nvidia |
What is the minimum number of sides a convex polygon can have such that the number of its diagonals is greater than the number of its sides? | nvidia |
Determine the number of positive integers $a$ less than $12$ such that the congruence $ax\equiv 1\pmod{12}$ has a solution in $x$. | nvidia |
Find all real numbers $m$ for which the equation $\sin (\arctan x)^m = \sin (\arccos x)^m$ has exactly two distinct solutions in the interval $[-1,1]$. | nvidia |
Determine the number of real monic polynomials of degree 4, with distinct integer roots, whose coefficients are all non-zero integers. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $xyz=1$. Find the minimum value of $x^2+y^2+z^2+xy+yz+zx$. | nvidia |
The polynomial $x^3 -ax^2 + bx -2010$ has three positive integer roots. What is the smallest possible value of $a$?
Write another problem inspired by this one:
A polynomial of degree $n$ has $n$ distinct positive integer roots, whose sum is equal to $S$. Determine the maximum possible value of $S$ such that the polyn... | nvidia |
In a unit square $ABCD$, a right-angled triangle $DEF$ is drawn so that no point of the triangle lies outside $ABCD$. Find the minimum possible area of such a triangle. | nvidia |
Let $a$ and $b$ be the roots of the equation $x^2-mx+n=0$. Given that the sum of squares of $a$ and $b$ is equal to $m^2-6$, find the value of $n$. | nvidia |
A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The prob... | nvidia |
In the parabola $y = x^2 + 4x + 5,$ find the $y$-coordinate of the vertex.
[asy]
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdow... | nvidia |
Problem:
Given that $a,$ $b,$ $c,$ $d,$ $e,$ $f$ are real numbers satisfying
$$\frac{(a-b)(c-d)(e-f)}{(b-c)(d-e)(f-a)} = 1,$$ find the value of
$$\frac{(a-c)(b-e)(d-f)}{(a-f)(b-d)(c-e)}.$$ | nvidia |
Find the remainder when the sum of the squares of the first 50 positive integers is divided by 123. | nvidia |
Find the number of integers that satisfy the inequality $\frac{x-3}{2} \ge \frac{x+1}{3}$ and also belong to the set of all odd integers between -5 and 7. | nvidia |
Let $A$ be a set of positive integers less than $101$ that are multiples of either $3$ or $7$, but not both at once.
Let $B$ be a set of positive integers less than $101$ that are multiples of both $3$ and $7$.
Determine the size of the set $C$, which consists of all positive integers less than $101$ that are not ele... | nvidia |
Given a parabola with vertex at $(1,2)$ and $y$-intercept of $5$, find the largest possible value of the sum of the $x$-intercepts. | nvidia |
Find the sum of all numbers $x$ satisfying $x+25/x = 10.$ | nvidia |
The circle centered at $(2,-1)$ and with radius $4$ intersects the circle centered at $(2,5)$ and with radius $\sqrt{10}$ at two points $A$ and $B$. Find $(AB)^2$. | nvidia |
The ratio of $x+2$ to $2x+2$ is equal to the ratio of $4x+3$ to $7x+3$. What is the product of all real x which satisfy this statement?
Another problem:
The ratio of the sum of the squares of two consecutive integers to the sum of the cubes of the same integers is equal to the ratio of the sum of the fourth powers o... | nvidia |
Find the sum of all the odd numbers from 1 through 99, inclusive, that are not divisible by 5. | nvidia |
What is the largest positive integer that is not the sum of a positive integral multiple of $42$ and a positive composite integer?
| nvidia |
What is the greatest whole number that must be a divisor of the sum of any three consecutive positive integers? | nvidia |
Let $C$ equal the number of four-digit numbers that are either odd or multiples of 5. Let $D$ equal the number of four-digit numbers that are both odd and multiples of 5. Find $C-D$. | nvidia |
Find the maximum value of $n$ for which there exists a square $ABCD$ with side length $1$ that can be partitioned into $n$ triangles whose areas are all less than or equal to $\frac{1}{100}$. | nvidia |
The sum of two positive integers plus their product is equal to 3 times the product of their sum and difference. The integers are distinct and both less than 12. What is the smaller of the two integers? | nvidia |
Find the number of positive integers less than 1000 that are divisible by 7, but not by 3 or 11. | nvidia |
If $f(x)=\cos x+i\sin x$ for all real numbers $x$ and $$\prod_{k=1}^{20}f(12k)=a+bi,$$where $a$ and $b$ are real numbers, then compute $a$. | nvidia |
Find the sum of all the positive divisors of 72, including 1 and 72. | nvidia |
Let $y$ be a positive real number. Find the minimum value of $2y^3 + 3y^{-2}$. | nvidia |
The graph of a function $y = f(x)$ is shown below. The graph consists of two line segments, one connecting the points $(0,0)$ and $(2,2)$, and the other connecting the points $(2,2)$ and $(4,0)$.
What is the value of the integral $\int_{0}^{4} f(x) dx$?
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklen... | nvidia |
Find all perfect cubes between 100 and 200 that can be expressed as the sum of two perfect squares. | nvidia |
The sequence $a_1$, $a_2$, $a_3$, $\ldots$ is defined recursively as follows:
$a_1=1$,
$a_2=4$,
$a_n=a_{n-1}+a_{n-2}$, for all $n>2$.
Find the largest $n$ such that $a_n<100$. | nvidia |
A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Determine the set of all possible values for the product of the remaining two zeros.
Consider the polynomial $(x-a)(x-b)(x-c)(x-d)$, where $a$ and $b$ are integers and $c$ and $d$ are complex... | nvidia |
Compute the number of distinct arrangements of 3 queens on a 5x5 chessboard such that no two queens are on the same row or column. | nvidia |
Compute the sum of the following infinite series:
$$\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)+\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)+\cdots$$ | nvidia |
Find the area of the triangle with vertices $(0,7,10),$ $(-1,6,6),$ and $(-4,9,6).$ | nvidia |
In a regular hexagon $ABCDEF$, a circle is inscribed such that it is tangent to each side of the hexagon at its midpoint. What is the ratio of the area of the circle to the area of the hexagon? | nvidia |
The decimal representation of $p/q,$ where $p$ and $q$ are relatively prime positive integers and $p < q,$ contains the digits $3, 7$, and $2$ consecutively, and in that order. Find the smallest value of $q$ for which this is possible. | nvidia |
Let $a$ and $b$ be two positive real numbers such that $a^2+b^2=1$ and $a^3+b^3=\frac{1}{2}$. Find the value of $\frac{b}{a}$. | nvidia |
Given that $p(x)$ is a polynomial with integer coefficients such that $p(14) = p(-6) = 0$ and $\frac{p(x)}{x^2 + x - 84}$ has a remainder of $7x - 32$, find the largest possible value of $|p(0)|$. | nvidia |
Find the maximum value of $x^2+y^2+z^2$ given that $A,$ $B,$ and $C$ are points in 3-dimensional space such that $A=(x,y,z),$ $B=(-x,y,z),$ and $C=(x,-y,z),$ and the area of triangle $ABC$ is 12 square units. | nvidia |
Given a positive real number $a$, find the range of the function
$$f(x)=\frac{x}{\sqrt{x^2-a^2}},$$
where $|x|>a$. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c=1$. Find the maximum value of
\[\frac{a^2b + b^2c + c^2a}{a^2 + b^2 + c^2}.\] | nvidia |
The set of all positive real numbers $x$ for which
\[\sum_{k=0}^\infty 2^kx^{2^k}=2017\]can be written as $\{a^n \mid n \in \mathbb{Z}\}$ for some real number $a$ and some set $\mathbb{Z} \subseteq \mathbb{Z}$. Find $a$. | nvidia |
Find the number of positive integers less than $10^8$ that are squares of primes, but not squares of cubes of primes. | nvidia |
Two numbers are independently selected from the set of positive integers less than or equal to 10. What is the probability that the difference of the squares of the two numbers is greater than their product? Express your answer as a common fraction. | nvidia |
A hexagon that has consecutive sides of lengths $a,b,c,d,e,f$ is inscribed in an ellipse. The point of tangency of the inscribed circle to the side of length $c$ divides that side into segments of length $m$ and $n$. If the sum of the lengths of the diagonals is $3(a+b+c+d+e+f)$, find $|m-n|$. | nvidia |
Find the integer $k$, $0 \le k \le 11$, such that \[k \equiv 8735 \pmod{12}.\] | nvidia |
Find the number of positive integers less than or equal to $1000$ that are multiples of $17$ but not of $34$. | nvidia |
A die is rolled repeatedly until the same number appears three consecutive times. Let $p$ be the probability that all three consecutive numbers are the same. Let $q$ be the probability that the first two consecutive numbers are the same and the third number is different. What is the value of $q/p$? | nvidia |
Find all positive integer values of $b$ such that the equation $x^3 + bx + 18 = 0$ has at least one integer solution for $x$. | nvidia |
How many positive integers $n$ less than 100 have a corresponding integer $m$ divisible by 3 such that the roots of $x^2-nx+m=0$ are consecutive positive integers? | nvidia |
Given $P$ is a set of all positive integers, let function $g : P \to P$ satisfies the following properties:
(i) First,
\[g(2x)=2g(x)\]for all $x \in P$.
(ii) Second,
\[g(x)+g(y)=g(xy)+1\]for all $x,y \in P$.
Let $m$ be the number of possible values of $g(1)$, and let $t$ be the sum of all possible values of $g(1)$. ... | nvidia |
Find the number of 3-digit numbers that are divisible by 4 and have no repeating digits. | nvidia |
Let $g$ be a function from the positive integers to the positive integers, such that:
(i) $g$ is an even function (i.e., $g(-n) = g(n)$ for all positive integers $n$);
(ii) $g(a+b) = g(a) + g(b)$ for all positive integers $a$ and $b$;
(iii) if $a \neq b$ and $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}$, where $a, b, c$... | nvidia |
What is the maximum number of consecutive positive integers that can be multiplied together before the product exceeds 1000? | nvidia |
Compute the value of $x$ such that
$\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\cdots\right)\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots\right)=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\cdots$. | nvidia |
Find the positive difference between the sum of the first 30 positive multiples of 3 and the sum of the first 20 positive multiples of 5. | nvidia |
Let $A$ be a set of 10 distinct integers, and let $B$ be a set of 5 distinct integers. How many ways are there to choose 5 integers from $A$ and 2 integers from $B$ such that the 5 integers from $A$ are in increasing order, the 2 integers from $B$ are in increasing order, and the 5 integers from $A$ are all less than t... | nvidia |
Let $b_n$ be the sum of the first $n$ terms of the sequence defined by $b_k = k\cos k$, where $k$ represents radian measure. Find the smallest positive integer $m$ such that $b_m > 0$ and $b_{m+1} < 0$. | nvidia |
Let $a_1=2,a_2=3$ be the first two terms of an infinite geometric sequence. Given that the $10$th term of the sequence can be written as $a_{10}=\frac{a+b\sqrt{c}}{d},$ where $a,b,c,d$ are positive integers and $c$ is not divisible by the square of any prime, find $a+b+c+d.$ | nvidia |
A triangle is formed by the points $A=(a,1)$, $B=(2a+1,a-1)$, and $C=(4,1)$. Find the value of $a$ such that the length of side $AB$ is equal to the length of side $AC$. | nvidia |
Let $P(x)$ be a polynomial with integer coefficients such that
$$P(\sin x) + P(\sin 2x) + P(\sin 3x) = 24 \sin x \sin 2x \sin 3x$$holds for all real numbers $x.$ Find the smallest possible degree of $P(x).$ | nvidia |
Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 3. If $\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. | nvidia |
There is only one positive integer $N$ such that there are exactly one thousand five-digit numbers (in base $10$) that start with $N$, have distinct digits, and are divisible by $7$. What is $N$? | nvidia |
A cat is tied to a point on the positive branch of the parabola $y^2=4x$ by a rope whose length is equal to the distance from the point to the $y$-axis. Find the maximum possible distance of the cat from the origin. | nvidia |
Using only the digits 7, 8 and 9, how many positive seven-digit integers can be made that are palindromes? | nvidia |
Find the smallest positive integer such that its decimal representation contains only 1's and 0's, and the sum of its digits is divisible by 7. | nvidia |
What is the number of different six-digit integers that can be formed using the digits 1, 2, 3, 4, 5, and 6, with the condition that the sum of the digits in each integer is exactly 21? | nvidia |
Find the number of integers between $100$ and $150$ that are divisible by neither $5$ nor $7$. | nvidia |
Find the smallest possible value of the sum of five consecutive positive integers, such that the sum is divisible by the middle integer. | nvidia |
Determine the value of $b$ such that the quadratic expression $bx^2 + 16x + 16$ can be expressed as the square of a binomial. | nvidia |
Let $P$ be a function that maps rational numbers to rational numbers. Find all functions $P(x)$ such that $P(x+y)P(x-y) = P(x)^2-P(y)^2$ for all rational numbers $x$ and $y$. | nvidia |
Let $a$ be a positive integer. Find the largest $a$ such that for some integer $n$, the number
$\underbrace{11\ldots 11}_{2n\text{ ones}}\cdot 11\ldots 11a$
formed by $2n$ ones, followed by a block of $n$ ones and then the digit $a$, is a perfect square. | nvidia |
Consider the equation $x^4+px^3+qx^2+rx+s=0$ with four non-real roots. The product of two of these roots is $5+2i$ and the sum of the other two roots is $2+3i,$ where $i^2 = -1.$ Find $q.$ | nvidia |
Let $f(x)$ be a function such that $f(ax+b)=cx^2+d$ for all real $x$. Given that $f(-3)=14$ and $f(6)=56$, determine $f(3)$. | nvidia |
A sequence of positive integers is defined recursively by $a_{n+1}= \lceil \frac{7}{4}a_n\rceil$ for $n\geq 1$. Given that $a_1=17$, find the smallest $n$ for which $a_n$ is divisible by 2016. | nvidia |
Find a positive integer $k$ such that the numbers $\sqrt{12+k}$, $\sqrt{96+k}$, and $\sqrt{216+k}$ are three consecutive terms of a geometric progression. | nvidia |
Find all real values of $b$ so that the polynomial
\[x^4 + 2x^3 + bx^2 + x + 1 = 0\]has at least one real root in the interval $(-1, 1)$. Enter all possible intervals of $b$ values. | nvidia |
What is the smallest number that can be formed using only the digits 1 and 5, such that the product of its digits is equal to 25? | nvidia |
Find the number of positive divisors of the product of 56 and 99. | nvidia |
Find the smallest positive integer that can be multiplied by $1458$ to obtain a perfect fifth power. | nvidia |
The sum of the first and last terms of an arithmetic sequence with 7 terms is 20. If the sum of all 7 terms is 63, find the value of the middle term. | nvidia |
Let $n$ be a positive integer such that the sum of the digits of $2^n$ is equal to 10. What is the smallest possible value of $n$? | nvidia |
There are $10$ monkeys and $5$ trees in the forest. The monkeys can play on the trees, but they are very picky. Each monkey will only play on a tree if it is with a certain subset of the other monkeys. Each monkey has a list of the subset of monkeys it is willing to play with on each tree. If a monkey is playing on a t... | nvidia |
A circle of radius 5 is centered at the origin and a circle of radius 2 is centered at (7,0). The circles intersect in two points. A point P is such that it is equidistant from the two intersection points and it lies on the circle centered at the origin. Find the distance of P from the origin. | nvidia |
Let $p$ be a prime number and $a,b,c$ be distinct positive integers. Suppose that there exists a function $f:\{1,2,3,\ldots\}\to\{1,2,3,\ldots\}$ such that for any $x$:
\begin{align*}
f(f(f(x)))&\equiv af(x)+bx+c\pmod p,\\
f(f(x))&\equiv bf(x)+cx+a\pmod p,\\
f(x)&\equiv cf(x)+ax+b\pmod p.
\end{align*}
Prove that $p=2... | nvidia |
Find the product of the real roots of $x^4 - 2x^2 + 2x + 1 = 0$, given that the sum of the squares of the real roots is equal to 5. | nvidia |
Let $A$ be the area of the right triangle whose legs are the roots of the equation $x^2-4x+3=0$ and $B$ be the area of the right triangle whose legs are the roots of the equation $x^2-2x+4=0$. Determine $\ds\frac{A}{B}$. | nvidia |
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