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Let a and b be angles such that 0 < 2b < a < π, and let x be a complex number such that x(1+x+2sin(a)x²)=1 and |x| < 1.
Write a new problem similar to this one:
Let a and b be angles such that 0 < 2b < a < π, and let y be a complex number such that y(1+y+2cos(b)y²)=1 and |y| < 1. | nvidia |
Let $x,y,z$ be positive real numbers satisfying $x+y+z=1$ and $x^2+y^2+z^2=2xy+2yz+2zx$. Find the maximum value of $x^3+y^3+z^3$. | nvidia |
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that for each complex number $z$, $f(z)$ is equidistant from both $z$ and the origin. Given that $|a+bi|=8$, find $b^2.$ | nvidia |
Let $F(x)$ be a polynomial such that $F(6) = 15$ and\[\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}\]for $x \in \mathbb{R}$ such that both sides are defined. Find $F(12)$.
| nvidia |
Find the absolute value of the sum of the solutions of $x^2+7x+12=0$. | nvidia |
A number $x$ is equal to $7\cdot24\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube? | nvidia |
Let $c$ and $d$ be nonnegative real numbers such that
\[\cos (cx + d) = \cos 23x\]for all rational numbers $x.$ Find the smallest possible value of $c.$ | nvidia |
Find the smallest positive integer $n$ such that $2n+3$ is a multiple of 5 and $3n-2$ is a multiple of 4. | nvidia |
Find the sum of the digits in the base 6 representation of the number obtained by reversing the digits of $2143_{10}$. | nvidia |
Let $M$ be a point inside triangle $ABC$ such that the triangles $MAB$, $MBC$, and $MCA$ all have the same area. Determine the locus of points $M$ that satisfy this condition, and find the area of this locus. | nvidia |
The number $0.\overline{123}$ can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are positive integers. Determine $a+b$. | nvidia |
Given that
\[g(x) = \left\{
\begin{array}{cl}
\sin x & \text{if } x \le \pi/2, \\
\cos (x-\pi/2) & \text{if } x > \pi/2,
\end{array}
\right.\]
find all values of $x$ for which $g(x) = \frac{\sqrt{2}}{2}$. | nvidia |
Find the sum of all positive integer values of $a$ such that the equation $3x^2 - 7x + a = 0$ has two real roots. | nvidia |
Let $S$ denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when $S$ is divided by $1000$. | nvidia |
The figure shows a square inscribed in a circle, with the side length of the square being $s$. The radius of the circle is $r$. Find the area of the shaded region in terms of $s$ and $\pi$. | nvidia |
Determine the largest possible three-digit palindrome that is divisible by 11 and has all digits different from each other. | nvidia |
The sequence $b_1, b_2, \ldots$ is geometric with $b_1=b$ and common ratio $s,$ where $b$ and $s$ are positive integers. Given that $\log_{10} b_1+\log_{10} b_2+\cdots+\log_{10} b_{15} = 2010,$ find the number of possible ordered pairs $(b,s).$ | nvidia |
Problem:
Consider three pairs of complex numbers $(a_1,b_1)$, $(a_2,b_2)$, and $(a_3,b_3)$ that satisfy the system of equations:
$$\left\{
\begin{aligned}
a^2+b^2&=2017 \\
ab+ba &=2018
\end{aligned}
\right.$$
Find the product $(a_1-a_2)(a_2-a_3)(a_3-a_1)(b_1-b_2)(b_2-b_3)(b_3-b_1)$. | nvidia |
Let $A$ be the set of all positive integers less than or equal to $100$ that are not divisible by $2$ or $3$. Find the number of elements in $A$ that are also divisible by $7$. | nvidia |
Find the largest possible value of $k$ for which the expression $\sqrt{4x^2+3x+k}$ is defined for all real numbers $x$. | nvidia |
The function $g(x)$ satisfies
\[g(xy) = \frac{g(x)}{y^2}\]for all positive real numbers $x$ and $y.$ If $g(20) = 15,$ find $g(25).$ | nvidia |
Find the smallest integer $n$ such that $n^2>56789$. | nvidia |
Consider the set of all possible sums of two or more distinct elements from the set {1,2,3,5,11}. What is the number of unique sums that can be obtained? | nvidia |
Find the number of positive integer factors of 324 that are perfect squares. | nvidia |
Determine the sum of the series $\sum_{k=2}^{\infty} \frac{2k-1}{k(k-1)}$ if it is known that it converges. | nvidia |
Let $x$ and $y$ be two positive integers, such that $y>x$ and $x+y=90$.
Determine the greatest number of digits that $x\cdot y$ can have, depending on $x$ and $y$. | nvidia |
The sum of two positive integers is 80 and their difference is 16. What is the value of the positive difference of the cubes of the integers? | nvidia |
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$. | nvidia |
Find the smallest integer $n$ such that $\lceil x \rceil = n$ for the given $x$, where $x$ satisfies the equation $\sqrt{x^2+4x+3} = 3$. | nvidia |
How many integers $N$ less than $1000$ can be written as the sum of $j$ consecutive positive odd integers from exactly 5 values of $j\ge 1$?
| nvidia |
Find the number of distinct positive integers that can be represented as the sum of two different positive multiples of 9, but not as the sum of two different positive multiples of 12. | nvidia |
In a triangle $ABC$, the lines $AX$, $BY$, and $CZ$ are the angle bisectors. These lines meet the opposite sides in $X$, $Y$, and $Z$, respectively.
Prove that the area of triangle $XYZ$ is equal to $\frac{1}{4}$ the area of triangle $ABC$. | nvidia |
Let $S$ be a set of points on the unit circle in the complex plane. Find the minimum number of points that $S$ must contain so that for every partition of $S$ into two subsets, at least one of the subsets contains three points $a$, $b$, and $c$ such that $a$, $b$, and $c$ are the vertices of an equilateral triangle. | nvidia |
Let $S$ be the set of all positive integers $n$ such that $\sqrt{n}$ is not an integer. For each element $n$ in $S$, let $f(n)$ be the smallest positive integer $k$ such that $k^2 > n$. Find the sum of all values of $f(n)$ for $n$ in $S$ and $1 \leq n \leq 20$. | nvidia |
Determine the number of ways to arrange the letters of the word ABCDEFGH such that no letter is in its original position. | nvidia |
Determine the smallest positive integer $n$ such that $5^n\equiv n^5\pmod 3$. | nvidia |
If $0.8= \frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers. | nvidia |
In a regular 15-gon, a certain diagonal divides the polygon into two smaller polygons. Determine the difference between the sum of the interior angles of the larger polygon and the sum of the interior angles of the smaller polygon. | nvidia |
A positive integer is 35 less than its cube root multiplied by its square root. Determine the maximum possible sum of such integers. | nvidia |
A square with side length $x$ is inscribed in an isosceles triangle whose base angle measures $75^\circ$. Given that the length of the segment connecting the midpoints of the sides containing the base angle is $\frac{15\sqrt{3}}{2}$, find the value of $x$. | nvidia |
Let a and b be two positive integers such that 7a+19b and 13a-2b are divisible by 25. Determine the remainder when 11a+12b is divided by 25. | nvidia |
In a certain country, license plates consist of a sequence of four letters followed by a sequence of two digits. The probability that a randomly chosen license plate will contain at least one pair of consecutive letters that are the same is $\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find ... | nvidia |
Find the positive difference between the sum of the squares of the first 10 positive integers and the sum of the cubes of the first 5 positive integers. | nvidia |
How many three-digit positive integers have at least one digit that is a prime number? | nvidia |
What is the largest positive integer $n$ such that there are exactly two different positive integers $k$ such that $k^2+n^2=25^2$? | nvidia |
Find the set of values for $x$ for which the function $\frac{x^3-2x^2+4x+7}{x^2+2x-3}$ takes only positive values. | nvidia |
Find the smallest positive value of $x$ that satisfies the inequality $|x-1| < |2-x|$. | nvidia |
Find the largest integer $n < 1000$ such that its square root has exactly three digits when expressed in base $n$. | nvidia |
Interior numbers begin in the third row of Pascal's Triangle. The sum of the interior numbers in the fourth row is 6. The sum of the interior numbers of the fifth row is 14. What is the sum of the interior numbers of the seventh row? | nvidia |
Find the sum of all values of $x$ such that $x^3 = 64$. | nvidia |
A number is expressed in the form of a recurring decimal as $0.\overline{ab}$, where $a$ and $b$ are digits. Prove that if $a + b$ is a multiple of 3, then the number can be expressed as a fraction with a denominator of the form $3^k$, where $k$ is a positive integer. | nvidia |
Consider a sequence of consecutive positive integers that are all nonprime and are followed by a prime number. Let's define a "prime gap" as the difference between the prime number and the largest nonprime integer in the sequence. For example, if the sequence is 24, 25, 26, 27, 28, followed by the prime number 29, then... | nvidia |
Compute the value of $a$ such that $\sqrt{a} = \sqrt[3]{128}$. | nvidia |
Find the smallest positive integer such that when divided by each prime number less than 10, it leaves a remainder of 2. | nvidia |
What is the sum of all values of $x$ such that $x^3=4x^2-3x+5$? | nvidia |
Find the product of all values of $x$ for which the expression $\frac{x-2}{x^2+3x-10}$ is undefined. | nvidia |
How many positive integers $n$ less than 200 have a corresponding integer $m$ divisible by 5 such that the roots of $x^2-nx+m=0$ are consecutive negative integers? | nvidia |
Determine the largest positive integer $n$ such that $n^3 - 1$ is divisible by 42 and $\sqrt{n}$ is between 3 and 4. | nvidia |
Find all real values of $x$ that satisfy $\frac{x^2-2x^3+x^4}{x+x^2+x^3} \le \frac{1}{2}.$ (Give your answer in interval notation.) | nvidia |
Find the sum of all positive integers less than 100 that are cubes of numbers whose squares are also perfect squares. | nvidia |
Determine the number of integer values of $n$ between $1$ and $299$ (inclusive) for which the decimal representation of $\sqrt{n}$ terminates. | nvidia |
Suppose that there exist two polynomials f(x) and g(x) of degree 4 with nonzero complex coefficients a, b, c, d, e and f such that the roots of f(x) are exactly the reciprocals of the roots of g(x). If the product of the roots of f(x) is 2, determine the possible values of the coefficient a. | nvidia |
Find the largest positive real number $k,$ such that for all positive real numbers $x$ and $y,$
\[\frac{2}{\frac{1}{x} + \frac{1}{y}} + k \left| \frac{1}{x} - \frac{1}{y} \right| \le \sqrt{xy}.\] | nvidia |
Let $S$ be the sum of the first $200$ positive even integers. Find the largest integer $m$ such that $S$ is divisible by $5^m$. | nvidia |
In a quadrilateral $ABCD$, the measures of two adjacent angles $A$ and $B$ are in the ratio $3:4$. The measures of the other two angles $C$ and $D$ are in the ratio $2:3$. If the sum of the measures of angles $A$ and $C$ is $150^\circ$, find the measure of the largest angle. | nvidia |
A quadratic equation $ax^2+bx+c=0$ has two distinct real solutions if and only if the discriminant $b^2-4ac$ is positive. Consider the quadratic equation $x^2+px+(2p+1)=0$, where $p$ is a parameter. For what values of $p$ does this equation have two distinct real solutions? | nvidia |
Find the smallest value of $n$ for which the sum $\sum_{k=2}^n \frac{1}{\log_k 2002}$ is greater than 2002. | nvidia |
What is the smallest positive integer that has exactly 12 positive divisors? | nvidia |
Find the smallest positive integer $n$ such that $n^6 + 3^4 + 7^3 - 2^{10}$ is divisible by 5. | nvidia |
Determine the value of $k$ such that the product of the roots of the equation $x^3 - kx^2 + kx - 1 = 0$ is equal to the sum of the squares of the roots. | nvidia |
Determine the value of $y$ that satisfies $\sqrt[3]{y\sqrt{y^2}}=2$. | nvidia |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | nvidia |
A five-digit number is formed by using the digits 1, 2, 3, 4, and 5 without repetition. What is the sum of all the five-digit numbers that can be formed such that the sum of the digits in each number is a multiple of 3? | nvidia |
What is the largest possible value of $n$ for which the $n$ consecutive positive integers $2n+1, 2n+2, \ldots, 3n$ all have exactly one positive divisor other than 1 and themselves? | nvidia |
Find the product of all values of $x$ for which the expression $\frac{x-2}{x^2+3x-10}$ is undefined. | nvidia |
Consider a function $g : \mathbb{R} \to \mathbb{R}$ that satisfies the following conditions:
$g(0) = 0$ and
$g(x^2 + y^2) = (x + y) (g(x) - g(y))$
for all real numbers $x$ and $y$.
Let $m$ be the number of possible values of $g(1)$, and let $t$ be the sum of all possible values of $g(1)$.
Find $m \times t$. | nvidia |
Let $f(n)$ be the sum of the proper positive integer divisors of $n$. For how many values of $n$ is $f(n)$ a prime number, given that $3 \le n \le 100$? | nvidia |
Find two positive integers whose product and difference both end with the same digit. What is the sum of these two integers? | nvidia |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | nvidia |
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are... | nvidia |
Compute the sum of the roots of the equation \[x\sqrt{x} - 6x + 7\sqrt{x} - 1 = 0,\]given that all of the roots are real and nonnegative. | nvidia |
A square grid is divided into rectangles by drawing horizontal and vertical lines through some of the grid points. Each rectangle contains at least one grid point. Prove that there is a rectangle that contains an odd number of grid points. | nvidia |
Let $r$ and $s$ be the roots of the equation
$$x^2+20x+7=0.$$ Determine all the values of $k$ for which the equation
$$x^2-(r+s)x+(rs+k)=0$$ has only one solution. | nvidia |
The function $f(x)$ is defined as $f(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers. Given that $f(1) = 1$, $f(2) = 8$, and $f(f(x)) = x^3$ for all real numbers $x$, find the unique value of $x$ such that $f(x) = 0$. | nvidia |
Find the smallest positive integer $n$ such that $n!$ is divisible by $1001!$ and $n$ is a multiple of $77$. | nvidia |
Let $f(n)$ denote the number of sequences of length $n$ consisting of $0$s and/or $1$s, such that no three consecutive $1$s appear. Find the smallest positive integer $k$, such that $f(k)$ is divisible by $1000$. | nvidia |
What is the minimum number of consecutive positive integers that can be multiplied together before the product exceeds 400? | nvidia |
In a triangle, three circles are inscribed such that each circle is tangent to two sides of the triangle and the other two circles. The triangle's sides are 9, 12, and 15 units long. A line passing through the triangle's centroid is such that the total perimeter of the parts of the three circles on one side of the line... | nvidia |
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?
$\textbf{(A)}\ 560 \qquad \textbf{(B)}\ 564 \qquad \textbf{(C)}\ 568 \qquad \textbf{(D)}\ 1498 \qquad ... | nvidia |
Let $T$ be the set of real numbers of the form $x + \sqrt{y},$ where $x$ and $y$ are rational numbers, such that
\[\frac{1}{2} \le x \le 1.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a real number $t \in T$ such that $t^n = 2.$ | nvidia |
Determine the sum of the values of $x$ that satisfy the equation $x^2 = |x^2-3x-18|$. | nvidia |
Determine all real values of $r$ for which the equation $r^{13} + 1 = 0$ has exactly one solution. | nvidia |
A positive integer is such that when its digits are reversed, the resulting number is equal to the original number plus 297. What is the original number? | nvidia |
The function $f(x)$ satisfies
$f(x+y)=f(x)f(y)$ for all real numbers $x$ and $y$, where $f(x)$ is an increasing function. Find all possible values of $f(1)$. | nvidia |
Find the greatest integer value of $x$ such that $\frac{x^2 - 3x + 7}{x+2}$ is an integer. | nvidia |
Find the sum of the first 8 even positive integers. | nvidia |
Find the sum of all positive integers $m$ for which $2^m - 1$ is a divisor of a perfect cube greater than $1$. | nvidia |
Suppose that $a$ and $b$ are positive integers such that $a-b=6$ and $\text{gcd}\left(\frac{a^3+b^3}{a+b}, ab\right) = 9$. Find the smallest possible value of $b$. | nvidia |
Find all real solutions to the equation $x^3 + 2x^2 - 4x - 8 = 0$, given that the sum of the squares of the roots is 24. | nvidia |
A $3\times 3\times 3$ cube is formed by gluing together 27 smaller cubes. Determine the maximum number of small cubes that can be painted red such that no two red cubes are adjacent. | nvidia |
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