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In a triangle with sides of length $x$, $2x$, and $x+4$, the length of the longest side is equal to the sum of the lengths of the two shorter sides. Determine the value of $x$. | nvidia |
Let $a$, $b$, $c$, and $d$ be real numbers such that
$|ad-bc|=1$ and
$\ds\frac{a}{\sqrt{a^2+b^2}}=\ds\frac{c}{\sqrt{c^2+d^2}}$.
Find the smallest possible value of $\sqrt{a^2+b^2+c^2+d^2}$. | nvidia |
The point $(5,3)$ is on the graph of $y=g(x)$. What is the product of the coordinates of the point that must be on the graph of $3y=2g(3x)-1$? | nvidia |
What is the smallest positive integer $n$ that can be expressed as both the sum of 12 consecutive positive integers and the sum of 20 consecutive positive integers? | nvidia |
Find the sum of the roots of the equation
$$z^3-5z^2+2z-1=0,$$
given that one of the roots is $z=2.$ | nvidia |
Find the sum of the solutions of: $\sqrt{x^2-2x+5}=2(\sqrt{x^2-2x+5}-1)$ | nvidia |
Given that \[\sqrt{17} < 5\], determine the smallest positive integer \(n\) such that \[\sqrt{n} > 5\]. | nvidia |
For how many positive integers $n$ less than or equal to 1000 is $$(\cos t+i\sin t)^n=\cos nt+i\sin nt$$true for all real $t$? | nvidia |
The product of positive integers x, y, and z equals 504. What is the maximum possible value of the sum x+y+z? | nvidia |
Find the sum of all values of $y$ that satisfy the equation $2^y + 2^{y-1} = 6$. | nvidia |
Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$. | nvidia |
Given that $0<x<1$, find the value of $x$ such that $e^{\frac{1}{\log_{10}x}} + \log_{10} x = 0$.
If no such $x$ exists, show that the equation has no solution. | nvidia |
Determine the number of positive integers $a$ less than $24$ such that the congruence $ax\equiv 3\pmod{24}$ has a solution in $x$ and the greatest common divisor of $a$ and $24$ is $6$. | nvidia |
Find the smallest integer $m$, greater than $2$, such that $m^{-1}\pmod{1350}$ is defined. | nvidia |
Find the greatest $a$ such that $\frac{7\sqrt{(2a)^2+(1)^2}-4a^2-1}{\sqrt{1+4a^2}+3}=2$. | nvidia |
Let $a$ and $b$ be two positive integers such that $a > b$.
The product of all integers from $a$ to $b$ inclusive is divisible by $7^7$.
Determine the smallest possible value of $a+b$. | nvidia |
Find the sum of the squares of the roots of the equation $x^3-6x^2+11x-6=0$. | nvidia |
Solve for the sum of all possible values of $x$ when $2^{x^2+x+1}=4^{x^2-3x+2}$. | nvidia |
Find all values of $x$ such that $\lceil x \rceil - x = 0.75$ and $x$ is a non-negative real number. | nvidia |
Find the maximum area of a triangle that can be formed by three complex numbers $z_1$, $z_2$, and $z_3$ such that $z_1+z_2+z_3=0$ and $|z_1|+|z_2|+|z_3|=10$, where $|z|$ denotes the magnitude of the complex number $z$. | nvidia |
Find all values of $x$ that satisfy the inequality $|\sin x| \leq \frac{1}{2}$ for $x$ in the interval $[-2\pi, 3\pi]$. | nvidia |
Let's consider the set $A$ of all real numbers $x$ such that $11\le x\le12$.
Find the number of elements of the set $B=\{\lfloor \arctan x \rfloor: x\in A\}$. | nvidia |
Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find ... | nvidia |
The sum of two positive integers $a$ and $b$ is 1001. What is the largest possible value of $\gcd(a,b)$? | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers? | nvidia |
A cylindrical can is placed on a flat surface and a cylindrical hole is cut through its center. The hole has a diameter that is equal to one-third of the diameter of the can. The can is then painted red on its outside surface. How many of the square centimeters of the can's surface are painted red on at least two sides... | nvidia |
Given that a circle passes through the points $(1,2)$, $(4,3)$, and $(6,5)$, find the slope of the line connecting the center of the circle to the point $(0,0)$. | nvidia |
Let $a$ and $b$ be positive real numbers such that the imaginary part of $(a + bi)^n$ is equal to the real part of $(a - bi)^n$ for some positive integer $n$. Compute the minimum possible value of $\frac{b}{a}$. | nvidia |
The product of two positive integers is 18. The positive difference of these two integers is 3. What is the sum of the two integers? | nvidia |
Find all the complex numbers $z$ such that $z^3=az^2+\overline{z}$, where $\overline{z}$ denotes the complex conjugate of $z$, and $a$ is a real number. | nvidia |
What is the smallest positive integer $n$ such that $5n \equiv 2187 \pmod{27}?$ | nvidia |
Consider the congruence $ax+b\equiv 0\pmod{m}$ where a, b, m are positive integers such that a and m are coprime. Find the least natural number n for which the given congruence has exactly n incongruent solutions modulo mn. | nvidia |
Find a cubic polynomial with real coefficients such that two of its roots are 2 and 2 + 3i. | nvidia |
Let $x$ be a real number such that $2x^3+7x^2-7x+1=\frac{1}{\cos x}$. Find the mean of the set $\left\{\frac{1}{x}, x, \sin x, \cos x\right\}$. | nvidia |
Let $f$ be a function from the set of positive real numbers to the set of all real numbers such that
(i) $f(1) = 2,$
(ii) $f(xy) = f(x)f(y)$ for all positive real numbers $x$ and $y$, and
(iii) $f(x) + f(1/x) = 3$ for all positive real numbers $x.$
Determine the number of possible functions $f(x).$ | nvidia |
How many four-digit positive integers exist, all of whose digits are 1's, 3's, and/or 7's, and such that no two consecutive digits are the same? | nvidia |
Find the coordinates of the point that divides the line segment joining the points $(2,3)$ and $(6,5)$ into three equal parts. | nvidia |
Find the sum of the squares of three consecutive integers such that the product of the smallest integer and the largest integer is 48. | nvidia |
Let $a$ and $b$ be complex numbers such that $|a|=|b|=1$. Determine all positive integers $n$ such that $(a+b)^n=a^n+b^n$ for all possible choices of $a$ and $b$. | nvidia |
Find the sum of the roots of the equation $(x-2)^2(x-3)(x-5)=0$, where each root is weighted by its multiplicity. | nvidia |
In the sequence $a_n=3\cdot2^n-2n$, determine $n$ such that the sum of the first $n$ terms equals the sum of the first $2n$ terms of the sequence $b_n=n\cdot2^n$. | nvidia |
The number $s$ is the sum of all the two-digit positive integers whose squares end with the digits 01. What is the sum of the digits of $s$? | nvidia |
Find the largest constant $C$ such that
$$\frac{x^2 + y^2 + z^2}{3} \ge C \sqrt{\frac{x^2 + y^2 + z^2}{2} + xyz}$$
for all real numbers $x, y,$ and $z.$ | nvidia |
Find the largest constant $C$ so that
\[x^2 + y^2 + 1 \ge C(x + y)\]for all real numbers $x$ and $y.$ | nvidia |
Let $A$ and $B$ be two points on a circle such that the arc length between them is $5\pi/3$ units. Suppose that $A$ and $B$ are connected by two chords, $CD$ and $EF$, such that the arc lengths $AC$, $DE$, $FB$, and $BA$ are all equal. If the length of chord $CD$ is $4$ units, what is the length of chord $EF$? | nvidia |
Determine the value of $x$ such that the decimal representation of $\frac{x}{999}$ has a block of three digits that repeats infinitely, and the sum of the digits in this block is 15. | nvidia |
Find the number of pairs of positive integers $(a,b)$ that satisfy $a^2-b^2=99$. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(x) f(y) - f(xy) = x + y\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | nvidia |
Given that $D$ is a regular dodecahedron, that $I$ is the icosahedron whose vertices are the centers of the faces of $D,$ and that the ratio of the volume of $D$ to that of $I$ is $\frac pq,$ where $p$ and $q$ are relatively prime integers, find $p+q.$ | nvidia |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes? | nvidia |
Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $1 \le n \le 25$, is $f(n)$ prime? | nvidia |
Find the value of $x$ such that
$$\left(\frac{2}{3} \right)^{x} \cdot \left(\frac{5}{7} \right) = \left(\frac{5}{7} \right)^{x} \cdot \left(\frac{2}{3} \right).$$ | nvidia |
Find the value of $x$ such that the sum of the first three terms of the arithmetic sequence $x, x+2, x+4, \dots$ is equal to the sum of the first four terms of the geometric sequence $2, 6, 18, \dots$. | nvidia |
For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i^2 = -1.$ Find $b.$ | 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 edges of a tetrahedron are labeled with positive real numbers in such a way that for every edge, the product of the three numbers assigned to the edges meeting at its endpoints is constant. Determine the smallest possible constant for which this is possible. | nvidia |
The sum of the squares of two consecutive positive integers is $85$. What is the greater of the two integers? | nvidia |
Problem:
Determine the number of positive integer divisors of $3^7\cdot 11^5$ that are less than or equal to $10^6$. | nvidia |
The sum of the digits of a two-digit number is equal to the product of its digits. Determine the sum of all such numbers. | nvidia |
What is the largest integer $n$ such that $n!$ is divisible by $2^{20}$? | nvidia |
Let $x$, $y$, and $z$ be distinct real numbers such that $x^2+y^2+z^2=1$. Compute \[ \dfrac {x^3+y^3+z^3}{xyz}. \] | nvidia |
Find the sum of all integers from 25 through 35, inclusive. | nvidia |
Find the sum of all values of $z$ that satisfy the equation $\frac{4}{2z^2+5z-2}+\frac{5}{z^2+6z+9}=1$. | nvidia |
Given that the quadratic inequality $-x^2+bx-5<0$ holds only when $x\in (-\infty, 1)\cup(5,\infty)$, find the value of $b$.
Write another problem:
The equation $ax^2 + bx + c = 0$ has two distinct real roots. The sum of the roots is twice the product of the roots. If the product of the roots is 3, find the sum of the... | nvidia |
How many positive integers less than $500$ can be written as the sum of two positive perfect cubes?
Here is another problem:
The number $n$ can be expressed as the sum of two perfect cubes, and also as the sum of two perfect squares. What is the minimum value of $n$ that satisfies both conditions? | nvidia |
Find the smallest positive integer $m$ such that $m^3 + m^2$ is divisible by some but not all integer values of $j$ when $2 \leq j \leq m$. | nvidia |
Let $\mathcal{T}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 2 adjacent place numbers are not both 0's. For example, $10101$ works, but $10011$ does not. Find the number of elements in $\mathcal{T}_{10}$ that start and end with 1. | nvidia |
Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. | nvidia |
Let $S$ be the sum of the first $100$ positive odd integers. Find the largest integer $k$ such that $S^2$ is divisible by $5^k$. | nvidia |
Given that $$(x+y+z)(xy+xz+yz)=10$$and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=3$$for real numbers $x$, $y$, and $z$, what is the value of $xyz$? | nvidia |
Increasing the radius of a cylinder by $6$ units increased the volume by $y$ cubic units. Increasing the height of the cylinder by $6$ units also increases the volume by $y$ cubic units. If the original height is $2$, then the original radius is:
$\text{(A) } 2 \qquad \text{(B) } 4 \qquad \text{(C) } 6 \qquad \text{(D)... | 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 |
Find the smallest positive integer $k$ such that \[k^3 \equiv 27 \pmod{8}.\] | nvidia |
There are four complex numbers $z$ such that
\[z \overline{z}^3 + \overline{z} z^3 = 350,\]and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by the four complex numbers as vertices. | nvidia |
Suppose that $g(x)$ is a function such that
\[g(xy) - y = yg(x) - g(y)\]for all real numbers $x$ and $y.$ If $g(2) = 3$ then compute $g(1002).$ | nvidia |
Let $a\otimes b=a^2+3ab+2b^2$ for all real numbers $a$ and $b$. Find $x$ such that $2\otimes x = 3\otimes (x-1)$. | nvidia |
The sum of the numbers $5x^2-10x+1$ and $3x^2+4x-2$ is divisible by the number $x^2-5x+1$. Determine the remainder. | nvidia |
A list of five positive integers has all of the following properties:
$\bullet$ The only integer in the list that occurs more than once is $8,$
$\bullet$ its median is $9,$ and
$\bullet$ its average (mean) is $10.$
What is the largest possible integer that could appear in the list? | nvidia |
Find the smallest positive integer that has exactly 5 distinct prime factors. | nvidia |
Compute the number of ways to arrange 5 red balls and 11 blue balls in a row such that no two red balls are adjacent. | nvidia |
Let $a$ and $b$ be two positive real numbers such that the expression $\frac{\sqrt[3]{a}+\sqrt[3]{b}}{\sqrt[3]{a\cdot b}}$ is as large as possible. Find the product $a\cdot b$ if $\frac{a}{b}=\frac{27}{64}$. | nvidia |
Let $x$ and $y$ be positive integers such that $xy = 360.$ Find the minimum value of $|x - y|.$ | nvidia |
Among all pairs of real numbers $(a, b)$ such that $\cos \cos a = \cos \cos b$ with $-15 \pi \le a, b \le 15 \pi$, Maya randomly selected a pair $(A, B)$. Compute the probability that $A = B$. | nvidia |
The digits of a four-digit number form a geometric progression. If the sum of the digits is 18, find the smallest possible value of the number. | nvidia |
Consider two real numbers $x$ and $y$ such that $-3 < x - y < 7$. If $\frac{x+y}{y}$ is an integer, determine its value. | nvidia |
Find the largest integer $n$ such that the increasing sequence $\left\lfloor \frac{2^{k_1}+2^{k_2}+\ldots+2^{k_n}}{n}\right\rfloor$, where $k_i\geq 1$ are distinct integers for all $i$, contains all positive integers. | nvidia |
Find the value of $a$ such that the geometric sequence
$a,\frac{2}{3}+\frac{1}{a}, \frac{1}{3}(1+\frac{2}{a}), ...$ has a finite sum. | nvidia |
How many non-negative real values of $x$ satisfy the equation $\sqrt[3]{144-x}=y$, where $y$ is a positive integer? | nvidia |
The new problem is:
Let $S$ be the set of all positive integers $n$ such that $n^2$ can be written as the sum of $n$ consecutive positive integers. Find the probability that a randomly chosen positive integer is in $S$. | nvidia |
Determine the smallest positive integer $k$ such that the sum of the proper divisors of $k$ is equal to $k-1$. | nvidia |
The first three digits of any integer divided by 9 always leaves the same remainder as the sum of the digits of that integer divided by 9. What are the possible remainders when the first three digits of any number are divided by 9? | nvidia |
Find the sum of all the even factors of 240 that are not divisible by 4. | nvidia |
Determine the number of positive integers $a$ less than $24$ such that the equation $ax+7y=1$ has a solution in integers $x$ and $y$. | nvidia |
Find the sum of the squares of the solutions of $x^2-13x+4=0$. | nvidia |
A circle with 23 points on its circumference is given. Each pair of points is connected by a chord. Determine the number of pairs of chords that intersect each other inside the circle. | nvidia |
Find the sum of all values of $k$ for which the quadratic $x^2 + 2(k-1)x + 3k^2 - 2k + 1$ has a real root that is greater than or equal to 2. | nvidia |
A set of non-negative integers has the property that each number in the set is the square of a distinct integer. What is the maximum number of elements in such a set if no two numbers in the set are congruent modulo 10? | nvidia |
How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$? | nvidia |
If the point $(3,6)$ is on the graph of $y=g(x)$, and $h(x)=(g(x))^2$ for all $x$, then there is one point that must be on the graph of $y=h(x)$. What is the sum of the coordinates of that point? | nvidia |
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