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Find the sum of the positive factors of 36 that are not divisible by 3. | nvidia |
Let $x$ and $y$ be real numbers such that $3(x^2 + y^2) = xy + 1$. Find the maximum value of $xy$. | nvidia |
A row of five squares rests on a table, with one of the squares hanging over the edge, as shown. The squares are arranged in an increasing arithmetic sequence, with the side length of the smallest square being 2 inches and the side length of the largest square being 10 inches. How many such arrangements are possible if... | nvidia |
Find the smallest three-digit number that is a perfect square, such that the sum of its digits is a prime number, and when the digits are rotated (circular shift) by one position, the resulting number is not a perfect square. | nvidia |
Let $f(x)=2x+1$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-1})$. | nvidia |
Determine the value of $x$ such that
$\log_{0.\overline{9}} \frac{1}{x}=4.$ | nvidia |
Problem:
The sum of the cubes of the digits of a certain positive integer is equal to the square of the sum of its digits. Determine the maximum possible number of digits in such an integer.
Finish response here. | nvidia |
Given a hyperbola $\mathcal{H}$ with the equation $x^2-y^2=1$, a circle $\mathcal{C}$ intersects $\mathcal{H}$ at four distinct points. If three of these points are $(\sqrt{2},1)$, $(1,0)$, and $(\sqrt{5},2)$, find the product of the distances from the center of $\mathcal{H}$ to all four of the intersection points. | nvidia |
The volume of a rectangular box is equal to the surface area of the box. If the length, width, and height of the box are all different positive integers, find the smallest possible length of the longest edge of the box. | nvidia |
The rectangle with the largest area that can be inscribed in a triangle with base length $b$ and height $h$ is a square, and its area is equal to $bh/2$. What is the ratio of the side length of this square to the side length of a square with the same area that is inscribed in the sector of a circle with angle $90^\circ... | nvidia |
Find the number of positive divisors of 60 that are also multiples of 3. | nvidia |
Let $a$ and $b$ be real numbers such that the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ has exactly two solutions $(x,y)$ where $x$ and $y$ are integers. Find the smallest possible product $ab$. | nvidia |
Find the product of the roots of the equation
$$x^3 - x^2 \sqrt{x+1} + x \sqrt[3]{x-1} - 2 = 0,$$
given that all of the roots are real and nonnegative. | nvidia |
Let $a, b, c$, and $d$ be positive real numbers such that
\[\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\ ac&=&bd&=&1000.\end{array}\]
Find the value of $\lceil S \rceil$, where $S=a+b+c+d$. | nvidia |
Find the value of $a$ such that $\log_{2a} (8a^3) = 4$. | nvidia |
Find the maximum value of $2^x - 4^x,$ over all real numbers $x.$ | nvidia |
Let $a$ and $b$ be positive real numbers such that $a + b = 2.$ Find set of all possible values of $\frac{1}{a} + \frac{1}{b}.$ | nvidia |
Let $P$ be a cubic polynomial with leading coefficient equal to $1$ such that $P(x)=0$ has roots $r$, $s$, and $t$. What is the smallest possible value of $r^2+s^2+t^2$ such that $P(x)=0$ has at least two integer roots? | nvidia |
The terms of the sequence $(b_i)$ defined by $b_{n + 2} = b_n + 2b_{n+1}$ for $n \ge 1$ are positive integers. Find the maximum possible value of $\frac{b_1}{b_2}$ such that the sequence does not contain any perfect squares. | nvidia |
How many three-digit positive integers have exactly two digits that are the same? | nvidia |
A $\textit{palindrome}$ is a number that reads the same forward and backward. The number 121 can be expressed as the sum of two palindromes in many ways. What is the minimum positive difference between two such numbers? | nvidia |
To calculate the sum of the cubes of the numbers from 1 to 5, Sarah mentally figures the sum of the cubes of the numbers from 1 to 6 and subtracts a number. What number does she subtract? | nvidia |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | nvidia |
Let $\odot$ be a binary operation defined on the set of positive integers, such that $a\odot b=a^2b$ for all positive integers $a$ and $b$. Find the value of $(2\odot 3)\odot 4$. | nvidia |
Find the value of $x$ such that $\left(\frac{1}{x} \right)^{2} \cdot \left(\frac{1}{3} \right) = \frac{1}{2}$. | nvidia |
Find the sum of all real values of $x$ satisfying
\begin{align*}
\tan{(x^2+y^2)}+\tan{(x^2+z^2)}&=\tan{(y^2+z^2)},\\
\tan{(x^2+2y^2)}+\tan{(x^2+2z^2)}&=\tan{(2y^2+2z^2)},\\
\tan{(x^2+y^2+z^2)}&=7/3,
\end{align*}where $x,$ $y,$ $z$ are real numbers. | nvidia |
Find the largest positive integer $n$ such that there exist distinct positive integers $a$ and $b$, where
$a \equiv 0 \pmod{n}$ and $b \equiv 3 \pmod{n}$, and
$a^2 + b^2$ is not divisible by $n$. | nvidia |
The vertices of a square lie on a circle of radius 2 centered at the origin. One vertex is at point (1, √3). Find the sum of all possible y-coordinates of the remaining vertices. | nvidia |
The lengths of the sides of a triangle with positive area are $\log_{10}12$, $\log_{10}75$, and $\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$. | nvidia |
What is the smallest positive integer $n$ such that $n^2$ has exactly $15$ distinct positive factors? | 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 |
A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively ... | nvidia |
What is the largest integer $n$ such that the number $2^n \times 3^4$ has fewer than 10 digits? | nvidia |
In the triangle $ABC$ with angle $\angle B = 60^\circ$, let $D$ be the midpoint of $\overline{AC}$ and $E$ be the point on $\overline{BD}$ such that $DE=DC$. Find $\tan \angle BAE.$ | nvidia |
Given the equation $ax^2 + bx + c = 0$, determine the value of $a$ such that the sum of the squares of the roots is equal to the product of the roots. | nvidia |
Find the minimum value of n such that the first n positive multiples of 5 contain at least 2014 digits. | nvidia |
The volume of a cube with edge length $x$ inches is twice the volume of a cube with edge length $y$ inches. Find the edge length of the larger cube in the form $c\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ has no perfect cube factors. Compute $c+d$. | nvidia |
Determine the sum of all integer solutions of the equation
$$2x^4+3x^3-16x^2-9x+72=0$$ | nvidia |
Find the sum of all emirps less than 100. Emirps are two-digit prime numbers that when their digits are reversed, form a different prime number. | nvidia |
Let $g(x)$ be a function defined for all positive integers satisfying the conditions $g(x) \geq 0$ for all $x \geq 1$ and
\[g(x + y) = \frac{g(xy) + 1}{g(x) + g(y)}\]for all $x, y \geq 1.$ Determine $g(2010).$ | nvidia |
If the sum of two numbers is $-\sqrt{2}$ and the sum of their squares is 6, find the product of the two numbers. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(x) g(y) + g(xy) = x - y\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(3),$ and let $t$ be the sum of all possible values of $g(3).$ Find $m \times t.$ | nvidia |
Given that $a=2$ and $b=5$, express the value of the following as a common fraction:
$$
\frac
{~\frac{a}{b}~}
{\frac{1}{a+b}}
$$ | nvidia |
Given a set of $n$ positive real numbers $a_1, a_2, \ldots, a_n$, determine the largest constant $k$ such that \[\left(\frac{a_1 + a_2 + \cdots + a_n}{n}\right)^n \leq k(a_1 \cdot a_2 \cdots a_n)^{1/n}\]holds true for any choice of $a_1, a_2, \ldots, a_n$. | nvidia |
Find the least integer $n$ such that $\sqrt{n}+\sqrt{n+1}$ is greater than or equal to 12. | nvidia |
Find all complex numbers $z$ such that the number obtained by reading the digits of $z$ backwards (in the same base) is a multiple of $z$. | nvidia |
Find the maximum value of $3^x - 9^x,$ over all real numbers $x.$ | nvidia |
Let $a$ be a positive real number. Find the sum of the series $a+\frac{1}{a}+\frac{1}{a^2}+\frac{1}{a^3}+\dots$ as a common fraction in terms of $a$, given that the series converges. | nvidia |
Given a regular hexagon with side length $s$, find the largest possible distance between two vertices of a triangle inscribed within the hexagon, such that the triangle has an integer perimeter. | nvidia |
Find the minimum value of
\[\frac{x^2 + 7}{\sqrt{x^2 + 3}}\]over all real numbers $x.$ | nvidia |
Find the sum of the squares of all real solutions of the system of equations
\begin{align*}
x^3 + y^3 + z^3 &= 9,\\
x + y + z &= 3,\\
xyz &= -2.
\end{align*} | nvidia |
In a circle of radius $r$, a regular polygon with $n$ sides is inscribed such that its vertices lie on the circle. Find the sum of the lengths of the perpendicular segments drawn from the center of the circle to each of the sides of the polygon. | nvidia |
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x$, $y$, and $r$ with $|x|>|y|$,
\[(x+y)^r=x^r+rx^{r-1}y^1+\frac{r(r-1)}2x^{r-2}y^2+\frac{r(r-1)(r-2)}{3!}x^{r-3}y^3+\cdots\]What are the first three digits to the right of the decimal point in the decimal representation... | nvidia |
Determine all the positive integers $n$ such that $7^n+1$ is divisible by $8$. | nvidia |
Find the area of the triangle with vertices $(4,2),$ $(1,7),$ and $(-3,1).$ | nvidia |
Determine $k$ such that the function $f(x)=\sqrt {x^2-kx+2}-\sqrt {2x^2+3x+k}$ will have a horizontal asymptote. | nvidia |
Find the number of positive integers less than or equal to 10000 that have no adjacent repeated digits. | nvidia |
The coordinates of the vertices of a rhombus $PQRS$ are all integers, with $P=(12,65)$ and $R=(15,72)$. The rhombus has no horizontal or vertical sides. The sum of the absolute values of all possible slopes for $\overline{PQ}$ is $x/y$, where $x$ and $y$ are relatively prime positive integers. Find $x+y$. | nvidia |
A pair of six-sided dice is loaded in such a way that the numbers 4, 5, and 6 each occur with the same probability, the numbers 2 and 3 each occur with the same probability, and the number 1 occurs with the remaining probability. If the probability of rolling a 6 on the first die is $\frac{1}{6}$, then find the probabi... | nvidia |
Determine the value of $k$ such that $|k+3i|=5$. | nvidia |
Determine the largest positive integer that must divide the sum of the first n terms of any geometric sequence whose terms are positive integers. | nvidia |
Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Find the minimum value of $\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$. | nvidia |
A triangle has vertices at $(0,0)$, $(a,0)$, and $(b,c)$. The line $y=mx$ intersects the hypotenuse of the triangle at a point that divides it into two segments, one of which is twice as long as the other. Given that $a$, $b$, and $c$ are positive integers, and $m$ is a rational number, find the minimum possible value ... | nvidia |
There are 2007 positive numbers with both their sum and the sum of their reciprocals equal to 2008. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$ | nvidia |
Find the smallest positive integer $n$ such that the sum of its divisors is equal to twice its prime factorization. | nvidia |
What is the smallest positive integer $n$ such that the number of digits in the base-$n$ representation of $n^3$ is 4? | nvidia |
Let $\alpha$ and $\beta$ be complex numbers such that $\alpha + \beta$ and $i(\alpha - 2 \beta)$ are both positive real numbers. If $\beta = 3 + 2i,$ compute $\alpha.$
Another problem inspired by this one:
Find all complex numbers $z$ such that $z^2 + iz$ is a real number and the imaginary part of $z$ is equal to th... | 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.
New Problem:
A complex number $z$ is such that $z^2 + z + 1 = 0$ and $|z + 1| = 1$. Find the smallest positive integer $n$ such that $\frac{z^n - 1}{z - 1}$ is a real number.
Thi... | nvidia |
Determine the smallest positive integer $n$ such that the sum of the squares of its digits is equal to $3n$. | nvidia |
The regular polygon $A_1A_2A_3A_4A_5...A_{100}$ is inscribed in a circle of radius $1.$ Point $P$ lies inside the circle so that the distance from $P$ to the center of the circle is $r$, where $0<r<1.$ The rays $\overrightarrow{PA_i}$ divide the circle into $100$ sectors. Determine the maximum number of sectors that ca... | nvidia |
Let $m$ be a positive integer, and let $a_0, a_1, \dots , a_m$ be a sequence of real numbers such that $a_0 = 37$, $a_1 = 72$, $a_m=0$, and $$ a_{k+1} = a_{k-1} - \frac{3}{a_k} $$for $k = 1,
2, \dots, m-1$. Find $m$. | nvidia |
Find the number of positive integers $n \le 1000$ such that $15n$ is a perfect cube. | nvidia |
Determine $x$ such that the sequence
$$1, x, \frac{x^2+1}{2}, \frac{x^3+2x}{3}, \ldots$$
is an arithmetic progression. | nvidia |
Let $a<b<c$ be real numbers such that $\ds\left|x^3+\frac{4}{x}\right|<5$ if and only if $x\in (a,b)\cup(b,c)$. Find the value of $b$. | nvidia |
Find the product of all possible rational values of $a$ such that the cubic equation $x^3 + ax^2 + 2x + 1 = 0$ has an integer root. | nvidia |
A circular cylinder with radius 4 is cut by a plane that passes through the center of the cylinder and intersects the cylinder in an ellipse. The area of one half of the ellipse is equal to the area of the semicircle formed by a perpendicular cross-section of the cylinder. Find the absolute value of the angle between t... | nvidia |
Find the greatest value of $a$ such that $a^2-10a+21 \le 0$. | nvidia |
The product of all positive integers with three distinct digits is denoted by P. What is the sum of the digits of P? | nvidia |
Find the smallest positive integer that is divisible by the cubes of three distinct prime numbers. | nvidia |
The roots of the equation $x^2+kx+5 = 0$ differ by $\sqrt{61}$. Find the greatest possible value of $k$.
Write another problem inspired by this one:
The equation $ax^2 + bx + c = 0$ has two distinct real roots. The sum of the squares of the roots is 13, and the sum of the squares of the coefficients is 50. Find the p... | nvidia |
In the coordinate plane, two lines are given: the line with equation $y = -x + 4$ and the line through points $(1, 1)$ and $(2, 3)$. Find the area of the triangle formed by these two lines and the x-axis. | nvidia |
Determine all real numbers $x$ such that
$\ds\prod_{n=1}^\infty \left(1+\f{x}{2^n}\right)$ is finite. | nvidia |
Find the sum of all positive integer values of $a$ such that the equation $ax^2+12x+16=0$ has a root that is a rational number. | nvidia |
Find the number of distinct possible values of $n$ for which $11^n + 1$ is divisible by $5$, where $n$ is a non-negative integer. | nvidia |
Given that the roots of $x^4+2x^3+3x^2-2x-1=0$ are $p$, $q$, $r$, and $s$, and that the roots of $x^4+ax^3+bx^2+cx+d=0$ are $pq$, $pr$, $qs$, and $rs$. Find $d$. | nvidia |
Let $S$ be the set of complex numbers of the form $x + yi,$ where $x$ and $y$ are real numbers, such that
\[\frac{\sqrt{2}}{2} \le x \le \frac{\sqrt{3}}{2}.\]Find the smallest positive integer $m$ such that for all positive integers $n \ge m,$ there exists a complex number $z \in S$ such that $z^n = 1.$ | nvidia |
Given a positive integer $n$, we define a set $A_n$ as follows:
$$A_n=\left\{\frac{m(n+1)}{2}| m \in \mathbb{Z}, n|m\right\}$$
We call an integer $a \in A_n$ "strict" if there exists an integer $b$ such that $a+b=n$ and $a|b$.
Find the smallest value of $n>5$ such that there exist at least 3 strict elements in $A_n$... | nvidia |
For all composite integers $n$, what is the largest integer that always divides into the difference between $n$ and the cube of $n$? | nvidia |
Find the smallest positive integer that is divisible by exactly three distinct prime numbers. | nvidia |
Find the smallest positive integer $n$ such that the sum of the first $n$ terms of the sequence $\left\{2, 3, 5, 7, 11, \dots \right\}$ is divisible by 25. | nvidia |
What is the sum of the two largest three-digit numbers that are divisible by 7 and the smallest four-digit number that is divisible by 9? | nvidia |
A function $f(x)$ is defined as $f(x) = x^2 + ax + b$, where $a$ and $b$ are constants. Find the sum of all $x$ that satisfy the equation $f(x) = f(f(x))$. | nvidia |
Find the largest prime factor of $n$, where $n$ is the product of all the positive integers from $1$ to $10$. | nvidia |
Let $T$ be the set of positive integers $n$ for which the infinite decimal representation of $\sqrt{n}$ has the property that the sequence of digits repeats in blocks of length 8. How many integers are in $T$ such that $n < 10000$? | nvidia |
The sum of the digits of a two-digit number is equal to the product of its digits. Find the number. | nvidia |
Find the smallest positive integer that is both an integer power of 11 and is not a palindrome. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of
\[x^2y^2z^2.\] | 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 |
Given that $\log_a(b)=c$, find the value of $\log_b(a)$. | nvidia |
In the graph below, five integers are arranged around a circle such that the difference between adjacent numbers is either 2, 3, or 5.
What is the sum of the two integers adjacent to the number 3? | nvidia |
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