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Find the greatest possible value of $x+y$ given that $x$ and $y$ are positive integers such that $\frac{1}{x} - \frac{1}{y} = \frac{1}{9}$. | nvidia |
Find the smallest positive integer $n$ such that the graph of the equation $|z^n - 1| = |z - 1|^n$ in the complex plane forms a circle. | nvidia |
Find the smallest positive integer that has exactly three different prime factors, all of which are greater than 5. | nvidia |
Given that the roots of the equation $x^2-5x-2=0$ are $\alpha$ and $\beta$, find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$. | nvidia |
Find the positive integer $m$ such that
\[\sin \left( \frac{\pi}{m} \right) + \cos \left (\frac{\pi}{m} \right) = \frac{\sqrt{m+1}}{2}.\] | nvidia |
Determine the number of rational numbers of the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers between 1 and 1000 (inclusive), whose decimal representation terminates and the sum of the digits in the numerator is equal to the sum of the digits in the denominator. | nvidia |
Find the sum of all positive integers whose square root is greater than 8 but less than 10. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Find the minimum value of $a^2 + b^2 + c^2$. | nvidia |
Find the side length of the largest square that can be inscribed in a regular hexagon with sides of length 1. The square must be aligned with the hexagon's sides and vertices. Express your answer in the form a*sqrt(b)/c, where a, b, and c are positive integers, and b is not divisible by the square of any prime. | nvidia |
Let $x$ and $y$ be relatively prime positive integers such that $\dfrac xy=\dfrac{2^1}{3^1}+\dfrac{2^2}{3^2}+\dfrac{2^3}{3^3}+\cdots$, where the numerators are powers of $2$, and the denominators are powers of $3$, with exponents increasing by $1$ for each subsequent term. Compute $x+y$. | nvidia |
Find the value of $a$ such that the vectors $\begin{pmatrix} 2 \\ a \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ 2 \\ a \end{pmatrix}$ are orthogonal, given that the dot product of the vectors is equal to the sum of their magnitudes. | nvidia |
Let $a,b,c$ be positive real numbers such that $a^2b^3c^4=1$, $b^5c^6a^7=1$ and $c^8a^9b^{10}=1$. Find $a+b+c$. | nvidia |
Given that $f(x)$ is a third-degree polynomial and its graph has a point of inflection at $(1,-2)$, determine $a$ in the expression $f(x)=ax^3+bx^2+cx+d$ where $a$, $b$, $c$, and $d$ are integers. | nvidia |
The operation $\oplus$ is defined as $a \oplus b = a \sqrt{b-a}$, where $a$ and $b$ are positive real numbers. Find the value of $x$ such that $x \oplus (x+1) = 3 \oplus 4$. | nvidia |
Two circles with centers at $(2,4)$ and $(-3,1)$ intersect at two points. Find the equation of the line passing through these two points of intersection. | nvidia |
Consider a function $g(x)$ defined on the set of real numbers, such that it satisfies the equation
$$[g(x)]^2-3xg(x)-7x^2-2=0.$$
Given that $g(1)=5$, determine the value of the function at $x=3$, i.e., find $g(3)$. | nvidia |
Find the number of integers $n$ such that $n^2$ can be written as a three-digit number in base $3$ and the sum of its digits is equal to $6$. | nvidia |
Let $P$ be a point on the parabola $y=2x^2+1$ and let $S$ be a point on the line $y=k$. Let $O$ be the origin, and let $Q$ and $R$ be points on the positive $x$-axis such that $QR=2$ and $OQ=OS$. Determine the value of $k$ such that $OP$ is perpendicular to $SR$. | 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 |
Let $M$ be the number of ordered pairs of nonempty sequences $A$ and $B$ that have the following properties:
$A$ and $B$ are sequences of distinct elements from the set $\{1,2,3,4,5,6,7,8,9,10,11,12\}$,
$A$ and $B$ have the same length, which is not an element of either sequence,
The sum of the elements of $A$ is eq... | nvidia |
Let $g$ be a function defined on the positive integers, such that
\[g(x) + g(y) = g(x + y) + g(\gcd(x, y))\]for all positive integers $x$ and $y.$ Given $g(5) = 7$ and $g(10) = 9,$ find $g(15).$ | nvidia |
Find the length of the segment of the number line whose endpoints satisfy $|x-\sqrt[3]{9}|=2$. | nvidia |
Five friends want to ride on a Ferris wheel, but the ride operator has a rule that no two people of the same height can ride together. The heights of the friends are 155, 160, 165, 170, and 175 cm. How many different ways can the friends be grouped into two cars, each with at least one person, such that no two people o... | nvidia |
A spherical ball with radius $r$ is divided into two parts by a plane passing through its center. The part containing a smaller portion of the sphere's surface area is then further divided into two parts by another plane passing through the center of the sphere, such that the volume of the smaller part is one-eighth of... | nvidia |
Let $p$ be a prime number and let $a$ be an integer such that $p \nmid a$. Prove that the equation $ax \equiv 1 \pmod{p^2}$ has a solution if and only if $a \equiv 1 \pmod{p}$. If this is the case, prove that the solution is unique modulo $p^2$. | nvidia |
How many different pairs of positive integers can be formed such that their product equals 200? | nvidia |
In triangle $ABC,$ the feet of the angle bisectors meet at point $I,$ and the feet of the medians meet at point $G.$ The distance between $I$ and $G$ is $\sqrt{10}.$ Find the area of triangle $ABC.$ | nvidia |
Let $f(x)=3x^4-x^3-23x^2-25x+22$, and let the graph of $f(x)$ intersect the graph of $y=k$ at four points, where $k$ is a real number. If the distance between the two $x$-coordinates of the points on the left is twice the distance between the two $x$-coordinates of the points on the right, find the sum of the four $x$... | nvidia |
What is the largest positive integer $m$ for which there is a unique integer $p$ such that $\frac{3}{5} < \frac{m}{m + p} < \frac{2}{3}$? | nvidia |
If $x$ and $y$ are positive real numbers such that $2x^2 + 5xy = x^3 + x^2 y^2,$ what is the value of $x$? | nvidia |
Let $ABCDE$ be a convex pentagon with $AB \parallel CE, BC \parallel AD, AC \parallel DE, \angle ABC=135^\circ, AB=2, BC=7,$ and $DE = 21.$ Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | nvidia |
Find the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. | nvidia |
In the given equation
$z = \displaystyle\frac{x}{2x-3}$,
determine the value of $x$ such that the difference between $z$ and its reciprocal equals $-2$. | nvidia |
Find the sum of the digits in the base 6 representation of the number obtained by reversing the digits of $2143_{10}$. | nvidia |
Circle $\omega$ has radius 8 and is centered at $O$. Point $A$ lies outside $\omega$ such that $OA=15$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$. Comput... | nvidia |
Find the smallest positive integer $n$ such that $n \equiv 3 \pmod{5}$, $n \equiv 2 \pmod{7}$, and $n \equiv 11 \pmod{13}$. | nvidia |
Find the number of real values of $x$ for which $\left|\frac{x^2-4}{x+5}\right|>3$. | nvidia |
Find the smallest positive integer x for which the inequality $x^3 + x > 2x^2$ holds. | nvidia |
The roots of the equation $2x^2+kx+8=0$ have a sum that is $\sqrt{17}$ times their product. Find the smallest possible value of $k$. | nvidia |
The real function $g$ has the property that, whenever $x,$ $y,$ $m$ are positive integers such that $x \cdot y = 3^m,$ the equation
\[g(x) + g(y) = m^3\]holds. What is $g(729)$? | nvidia |
The sum of the squares of three consecutive positive integers is equal to 50. Determine the smallest of these integers. | nvidia |
Find all $x$ such that $x^2+5x<6$. Express your answer in interval notation. | nvidia |
Let $x$ be the number of positive multiples of $9$ that are less than $45$. Let $y$ be the number of positive integers that are less than $45$, and a multiple of $3$ and a multiple of $3$. Compute $(x - y)^2$. | nvidia |
Compute the number of ways to choose 3 elements from the set {1, 2,..., 15} such that the sum of the chosen elements is a multiple of 3. | nvidia |
Let $a, b, c, d, e, f$ be distinct real numbers, such that there exists a quadratic polynomial $P(x)$ with $a, b, c, d, e, f$ as its roots. The expression $(a+c+e)^2+(b+d+f)^2$ attains its maximum value. Find the maximum value of the expression $(ab+cd+ef)+(bc+de+fa)+(ad+be+cf)$. | nvidia |
Count the number of three-digit positive integers, all of whose digits are different and are 1, 2 or 3. | nvidia |
The square of a positive integer $n$ can be represented as the sum of $n$ consecutive positive integers starting from 1. Determine the smallest possible value of $n$. | nvidia |
Let $g(x)$ be a function defined for all positive integers $x$, such that $g(1) = 0$ and $g(xy) = xg(y) + yg(x)$ for all positive integers $x$ and $y$. Find $g(1986)$. | nvidia |
How many positive integers less than 1000 can be expressed as the sum of distinct powers of 3? | nvidia |
A fair coin is flipped eight times. The probability that the number of heads obtained is a power of two is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | nvidia |
Determine the maximum number of real roots to a polynomial of the form $x^2 + ax + b = 0$, where $a$ and $b$ are positive integers such that $a$ is odd and $b$ is even. | nvidia |
The vertices of a regular hexagon lie on the parabola $y^2 = x$, and the center of the hexagon is at the focus of the parabola. Find the area of the hexagon. | nvidia |
Given that $a$ and $b$ are positive integers and that $a+b=24$, what is the value of $ab$ if $2ab + 10a = 3b + 222$? | nvidia |
Let $S$ be the set of all nonzero real numbers. Define a binary operation $\otimes$ on $S$ that satisfies the following properties:
(1) For any $a$, $b$, and $c$ in $S$, $(ab) \otimes c = a(b \otimes c)$.
(2) For any $a$ and $b$ in $S$, if $a \otimes b = 1$, then $b \otimes a = 1$.
(3) For any $a$ in $S$, there exi... | nvidia |
Determine the sum of all real solutions to $x^2 > x+3 > 1$. | nvidia |
A group of friends are playing a game where they take turns saying a positive integer. The rule is that the next person's number must be a divisor of the previous number. If someone says a number that is not a divisor of the previous number, they lose. The game starts with the number 120. What is the maximum number of ... | nvidia |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? | nvidia |
A domino is a $1$ by $2$ rectangular tile with two numbers written on it, one on each half. A set of dominoes is said to be "balanced" if the sum of the numbers on each half of each domino is the same. For example, the set $\{(1,2), (2,3), (3,1)\}$ is balanced because the sum of the numbers on each half of each domino ... | nvidia |
A triangle has a base of length $x$ and a height of length $y$. The area of the triangle is given by the equation $A = \frac{1}{2}xy$. If the length of the hypotenuse of the triangle is 10 units, find the value of $x$ such that the area of the triangle is 21 square units. | nvidia |
Find the minimum value of
\[\frac{(x+1)^2+(y+1)^2+(z+1)^2}{\sqrt{x^2+y^2+z^2}}\]
over all real numbers $x, y,$ and $z.$ | nvidia |
Find all non-negative real numbers $b$ such that the numbers $b^2,$ $b,$ and $2-b$ form a geometric sequence, in that order. | nvidia |
A rhombus $ABCD$ is inscribed in a circle with center $O$. The diagonals $AC$ and $BD$ intersect at $P$, and the length of the diagonal $AC$ is 12 cm. The point $E$ is on the circumference such that the line $PE$ bisects the angle $\angle BPD$. Let $F$ be the foot of the perpendicular from $P$ to $OE$. If $OF = 2$, fin... | nvidia |
Find $x$ such that $\log_{12}3x=2$. | nvidia |
Find the area of the region enclosed by the graph of $|x-60|+|y|=\left|\frac{x}{4}\right|.$
New Problem:
A pentagon is formed by joining the midpoints of the sides of a square with sides of length $a$. Find the area of the region inside the pentagon that is not inside the square. | nvidia |
The equation
$$3x^2-7x-8=0$$
has two roots that can be written as $x_1$ and $x_2$.
It is known that $x_1-x_2$ equals $\ds\f{\sqrt m}{n}$.
Determine the expression for the product of the two roots, that is to say, $x_1\cdot x_2$, in a form of fraction $\f mn$, so that $n$ is an integer and $m$ is an integer not divisi... | nvidia |
When three positive integers are divided by $24$, the remainders are $10,$ $4,$ and $12,$ respectively.
When the sum of the three integers is divided by $24$, what is the remainder? | nvidia |
A real number $b$ is chosen randomly and uniformly from the interval $[-15, 20]$. Find the probability that the roots of the polynomial
\[x^3 - 3bx^2 + (3b + 1)x - 1\]are all real. | nvidia |
Find the number of 4-digit numbers that do not contain any repeated digits and whose digits are in strictly increasing order. | nvidia |
Let $A$ and $B$ be real numbers such that $\frac{A}{x-5}+B(x+1)=\frac{-3x^2+12x+22}{x-5}$. What is $A+B$? | nvidia |
Let $a$ and $b$ be positive integers. Determine the smallest value of $ab$ such that the equation $x^2+ax+2b=0$ has rational roots and $a+b$ is a perfect square. | nvidia |
A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial? | nvidia |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of five consecutive positive integers all of which are nonprime? | nvidia |
Find the smallest integer $n$ such that the decimal representation of $\sqrt{2}$ starts with $n$ digits, all of which are zeros, after the decimal point.
Note: This problem requires a different approach than the original one, as it involves the decimal representation of an irrational number, whereas the original probl... | nvidia |
In the multiplication problem
$$
\begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & \triangle & 3 & 5_6\\
& & & \times & \triangle & 4_6\\
\cline{2-6}
& & & 2 & 5 & 4_6\\
& + & \triangle & \triangle & \triangle & \triangle_6\\
\cline{2-6}
& & 3 & \triangle & 0 & 4_6\\
\end{array}
$$
the digit represented by $\triangle$ is ... | nvidia |
Find all real solutions to $x^4+(2-x)^4=34$. Enter all the solutions, separated by commas. | nvidia |
Find the smallest positive integer $m$ such that $216m \equiv 729m \pmod{15}$. | nvidia |
Find the smallest possible positive integer value of $x$ if $\displaystyle\frac{x^2}{x} < 10$. | nvidia |
Find $t$ such that $(t,5)$ lies on the line through $(0,3)$ and $(-8,0)$. | nvidia |
A deck of cards contains 52 cards, numbered from 1 to 52. What is the smallest number of consecutive cards that can be removed from the deck so that the sum of the remaining cards is divisible by 13? | nvidia |
Given the equation of an ellipse in standard form, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ where $a$ and $b$ are positive real numbers, find the equation of the circle that passes through the foci of the ellipse and has its center at the origin. | nvidia |
Find the area of a quadrilateral with points $A(-2,3)$, $B(4,5)$, $C(7,3)$, and $D(1,-2)$ on a Cartesian coordinate plane, given that the diagonals of the quadrilateral are perpendicular to each other. | nvidia |
Find the range of values of $a$ such that the expression $\frac{\ln(2x-a)}{\sqrt{x+1}}$ has a domain that includes all real numbers greater than or equal to 1. | nvidia |
The sum of the squares of two positive numbers is 65, and the difference of their squares is 33. Determine the sum of these two numbers. | nvidia |
For what value of $x$ is the expression $\frac{2x^3+3}{x^2-20x+100}$ not defined?
Determine $k$ such that the expression $\frac{x^2+kx-3}{x^2+2x-6}$ is defined for all real values of $x$. | nvidia |
The parabola shown has its vertex on the y-axis and its focus on the parabola itself.
Let $b$ be the y-coordinate of the focus and let $a$ be the y-coordinate of the point of the parabola with x-coordinate $2$.
Determine the product $ab$. | nvidia |
Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number. | nvidia |
If $a$, $b$, and $c$ are positive integers such that $\left(\frac{a}{b}\right)^c=1$, what is the smallest possible value of $a+b+c$? | nvidia |
Find the number of positive integers $m$ that satisfy $80 < m^3 < 400$. | nvidia |
Find all values of $x$ with $0 \le x < 2 \pi$ that satisfy $\sin x + \cos x = \sqrt{2}.$ Enter all the solutions, separated by commas. | nvidia |
Let $f(x)$ be a function defined on the set of all integers, such that $f(x) = 3x$ if $x$ is even, and $f(x) = x^2 + 5$ if $x$ is odd. Determine the number of integers $x$ for which $f(f(x)) = f(x)$ holds. | nvidia |
Let $T$ be the sum of all integers $c$ for which the polynomial $x^3+cx^2+2010c^2$ can be expressed as the product of three linear factors with integer coefficients. Find $T$. | nvidia |
Find the number of positive integers less than or equal to $100,000$ that are simultaneously $17$-safe, $19$-safe, and $23$-safe. | nvidia |
Find the smallest positive integer x for which the inequality $x^3 + x > 2x^2$ holds. | nvidia |
Let n be a positive integer. How many distinct numbers can be obtained as the product of two integers whose sum is n? | nvidia |
Determine the largest power of 3 that divides the product of all positive integers less than or equal to $3^{10}$. | nvidia |
In a set of 5 points on a circle, each pair of points is connected by a line segment. What is the probability that at least one of the line segments has a length that is an integer multiple of the radius of the circle? | nvidia |
How many non-congruent trapezoids have side lengths that are distinct positive integers in geometric progression? | nvidia |
Find the number of positive integers $n$ such that the roots of the equation $x^2-nx+n=0$ are integers and their sum of squares is a perfect square. | nvidia |
A rectangle has side lengths which are positive integers and the area of the rectangle is a perfect square. What is the smallest possible perimeter of the rectangle? | nvidia |
Find the value of
$$\sqrt[3]{30^3+40^3+50^3}$$
assuming that $x$, $y$, and $z$ are the roots of the equation $t^3-80t^2+2000t-50000=0$. | nvidia |
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