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Determine all real numbers $\alpha > 1$ for which there exist infinitely many positive integers $n$ such that $$[\alpha^n]=\alpha^{[\alpha^{n-1}]}.$$ | nvidia |
Find the smallest positive integer that has exactly four different prime factors, and whose total number of divisors is equal to the square of the number of its prime factors.
(Note: This problem requires a different approach, as it involves the relationship between the number of divisors and the number of prime facto... | nvidia |
Let $f(z)$ be a function on the complex plane defined by $f(z)=z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots$. Find the sum of all integer values of $z$ such that $\left|z\right|<20$ and $f(z) = \frac{2z-1}{z+1}$. | nvidia |
How many positive four-digit integers less than 800 have at least two consecutive digits that are the same? | nvidia |
Find the sum of the first seven prime numbers that have a units digit of 7. | nvidia |
Prove that $\dbinom{n}{k}=\dbinom{n}{n-k}$ for any positive integer $n$ and $k$, $0\leq k \leq n$. | nvidia |
In the city of Umberville, there are three streets: Pine, Oak, and Maple. Each street has three houses, and each house can be painted with one of three colors: red, blue, or green. The city has a peculiar rule that no two adjacent houses on the same street can have the same color. How many different ways can the houses... | nvidia |
If the quadratic $x^2+4mx+m$ has exactly one real root, find the positive value of $m$. | nvidia |
Find $x$ such that $\log_x 4=\log_{27} 3$. | nvidia |
Find the sum of all positive integers n, such that 2n+1 and 3n+2 are both prime numbers. | nvidia |
How many of the integers between 100 and 999, inclusive, have three distinct digits such that one of them is the sum of the other two? | nvidia |
A fair 12-sided die is rolled twice. What number should be removed from the set of possible sums of the two rolls, so that the probability that the sum of the two rolls is even is maximized? | nvidia |
In the triangle $ABC$ the length of the side $AC$ is equal to the length of the segment $BD$, where $D$ is a point on the side $BC$. The angle $\angle ADB$ is equal to $60^\circ$. If the length of the side $BC$ is equal to 24, find the length of the side $AB$. | nvidia |
Let $g(x)$ be a function defined on all real numbers satisfying $x^2g(y) + y^2g(x) = xy(g(x) + g(y))$ for all real numbers $x$ and $y$. If $g(10) = 15$, determine $g(20)$. | nvidia |
Find the side length of the largest square that can be inscribed in the region bounded by the curves $y=\frac{1}{x}$ and $y=-\frac{1}{x}$ and the lines $x=1$ and $x=-1$. | nvidia |
Given the equation $\dfrac{r^2-3r-17}{r+4}=2r+7$, find the value of $r$ for which the equation holds true.
Alternatively, consider the equation $\dfrac{x^2-2x-15}{x-3}=x+5$. Find the sum of the solutions of this equation. | nvidia |
A positive integer $n$ has exactly 8 factors, and the number 18 is one of them. Find the number $n$ such that the sum of its proper divisors (excluding $n$ itself) is minimal. | nvidia |
Find the sum of all the positive integers less than or equal to 100 that have exactly three distinct factors. | nvidia |
Given the sum of the cubes of the first 20 positive integers, determine the sum of the first 10 positive integers squared. | nvidia |
Find the point on the surface of the sphere $x^2 + y^2 + z^2 = 9$ that is farthest from the point $(2,2,2)$. | nvidia |
Find the sum of all the divisors of a number, given that its prime factorization is $2^4 \times 3^2 \times 5$. | 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 |
Let $S$ be the set of all positive integers that are the sum of the squares of two positive integers.
Let $A$ be a set of positive integers such that for all $a\in A$ and for all positive integers $k$, $ka\in S$ if and only if $k\in S$.
Find the smallest possible number of elements in $A$. | nvidia |
What is the smallest positive integer that can be expressed as the sum of three distinct positive perfect cubes, where each cube has a different prime factorization? | nvidia |
If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? | nvidia |
Find a positive integer that is divisible by 18 and has a square root between 32 and 32.5. | nvidia |
What is the sum of all two-digit positive integers whose squares end with the digits 01? | nvidia |
Let $f(x)$ and $g(x)$ be two polynomials such that $f(x)=x^4-x^2+1$ and $g(x)=x^4+x^2+1$. Determine the product of all possible values of $x$ satisfying $\ds\dfrac{f(x)}{g(x)}=-\ds\dfrac{1}{2}$. | nvidia |
Consider the equation $2^{x^2-6x+9}=4^{x-3}$. Find the product of all possible values of $x$ that satisfy this equation. | nvidia |
When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.
When the sum of the three integers is divided by $12$, what is the remainder? | nvidia |
A regular hexagon is inscribed in a circle. The area of the hexagon is 18 square units. Find the area of the region enclosed by the circle that is not covered by the hexagon. | nvidia |
At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$ | nvidia |
Determine the value of $k$ such that the line $y=3x-2$ is tangent to the circle $(x-k)^2+y^2=3$. | nvidia |
Find the number of non-decreasing sequences of four numbers from the set {1, 2, 3, 4, 5, 6}, where each number can be used any number of times. | 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 |
The product of two consecutive positive integers is 7020. What is the difference of the two integers? | nvidia |
Let $x$ and $y$ be two real numbers such that $x+y = 6$ and $\frac{1}{2}x^2 + \frac{1}{3}y^2 = 12$. Determine the value of $x$ that minimizes the expression $x^2 - 3y^2$. | nvidia |
A number in the form of $a^b$ where $a$ and $b$ are positive integers and $b > 1$ is called a perfect power. Find the perfect power that is closest to the cube root of 1999, and determine the sum of its base and exponent. | nvidia |
Find the number of positive perfect cube integers that are factors of the product $(2^{12})(3^{18})(5^{24})$. | nvidia |
The boundaries of a square with sides of length $2001$ units are drawn on a piece of graph paper with a grid of $1$ unit by $1$ unit squares. In the interior of the $2001$ by $2001$ square a small square of $10$ units by $10$ units is drawn that can be moved along the lines of the grid but can't intersect the boundary ... | nvidia |
The vertices of a parallelogram $ABCD$ are $A(2,1)$, $B(4,3)$, $C(6,5)$ and $D(x,y)$. Given that the diagonals of the parallelogram bisect each other, find the value of $xy$. | nvidia |
Determine the smallest non-negative integer $n$ such that $3^n \equiv 17 \pmod{23}$. | nvidia |
Consider the ellipse with equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. If the foci of this ellipse are at $(c,0)$ and $(-c,0)$, and the area of the triangle formed by the origin and these two foci is $\frac{5\pi ab}{16}$, find the length of the major axis of the ellipse. | nvidia |
Problem:
The three roots of the equation $ax^3+bx^2+cx+d=0$ are $1,$ $2,$ and $3.$
Compute $\frac{c}{d}.$
Another problem inspired by this one:
Let $\alpha$ be a root of the equation \[3x^3-x^2-7x+3=0.\]If $1<\alpha<2$ and $\beta$ is an integer such that $\frac{1}{\alpha}+\frac{1}{\beta}=1$, find $\beta$. | nvidia |
What are the last two digits in the sum of factorials of the first 100 positive integers? | nvidia |
Let $g$ be a function defined on the positive real numbers, such that
$$g(x+y)=g(x)g(y)$$
for all positive real numbers $x$ and $y.$ Given that $g(2)=3$ and $g(4)=9$, find the value of $g(5)$. | nvidia |
Determine the number of distinct ordered pairs of positive integers $(m,n)$ such that the sum of the reciprocals of $m$ and $n$ equals the reciprocal of the product of $m$ and $n$. | nvidia |
Let $r$ and $s$ be the roots of the equation $x^2+ax+6a=0$. Determine the number of possible values of $a$ for which $r^3+s^3$ is an integer. | nvidia |
All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the ... | nvidia |
Given that the matrix
$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$
has real entries such that
$$a_{13}+a_{31}=0$$
$$a_{23}+a_{32}=0$$
$$a_{33}=0$$
and
$$\det(A)=1$$
find the value of
$$a_{11}a_{22}-a_{12}a_{21}$$ | nvidia |
What is the smallest positive integer $n$ such that $435n \equiv 917n \pmod{16}?$ | nvidia |
Find the smallest positive integer greater than 2000 that is a multiple of 7 and has the sum of its digits equal to 18. | nvidia |
Find the units digit of the sum of all the even positive integers between 20 and 120. | nvidia |
Find the equation of the ellipse with foci at $(3,0)$ and $(-3,0)$ that passes through the point $(2,3)$. | nvidia |
In the graph of $\frac{x^2-4x+3}{x^3-2x^2-x+2}$, let $p$ be the number of holes in the graph, $q$ be the number of vertical asymptotes, $r$ be the number of horizontal asymptotes, and $s$ be the number of oblique asymptotes. Find $p+2q+3r+4s$. | nvidia |
Let $n$ be the product of all positive integers less than $6!$ which are relatively prime to $6!$. Find the remainder when $n$ is divided by $6!$. | nvidia |
Find the smallest positive integer $n$ such that $\sqrt{n}$ and $\sqrt[3]{n}$ are both integers, and $\sqrt{n}+\sqrt[3]{n}$ is a perfect square. | nvidia |
Count the number of positive integers $n$ such that $n!$ is a perfect square, where $n!$ is the product of all positive integers up to $n$. | nvidia |
Find the value of $a$ such that the circle $x^2+y^2=a$ passes through the centroid of the triangle with vertices $(1,2)$, $(3,4)$, and $(5,6)$. | nvidia |
Let $x, y, z$ be positive integers such that $x^2+y^2+z^2=3xyz$. Find the maximum value of $x+y+z$. | nvidia |
Let $f(x) = \frac{x^3 - x}{2}$.
Find the sum of all integers $a$ for which there are distinct integers $x$ and $y$ such that $f(x) + f(y) = a.$ | nvidia |
Determine the value of $a$ so that the minimum value of
\[5x^2 - 8axy + (4a^2 + 1) y^2 - 2x - 8y + 9\]
over all real numbers $x$ and $y$ that satisfy $x \ge 0$ and $y \ge 0$ is equal to $1$. | nvidia |
Find the maximum value of $n$ such that there exist positive integers $a$ and $b$ satisfying \[\frac{a}{b}<\sqrt{2}<\frac{a+n}{b+n}.\] | nvidia |
A right pyramid with a square base has a volume of 54 cubic units. The height of the pyramid is twice the length of the side of the base. Find the area of the base of the pyramid. | nvidia |
Solve for the sum of all possible values of $x$ when $3^{x^2+4x+4}=9^{x+2}$. | nvidia |
Find all real numbers $x$ such that the equation
$$\left| \frac{e^{ix} + e^{-ix}}{2} \right| = \left| \frac{e^{ix} - e^{-ix}}{2i} \right|$$
holds true. | nvidia |
Let $P(x)=1-x+x^2-x^3+\cdots+x^{2008}$, where $x$ is a complex number. Find the sum of all values of $x$ such that $P(x)P(-x)=3$. | nvidia |
Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 11? | nvidia |
Let $w$ and $z$ be two complex numbers such that $|w|=|z|=1$ and $|w-z|=1$. Determine the number of possible values of $\ds\left|\frac{w+z}{w-z}\right|$.
Please note that this is just one possible problem inspired by the given one. There may be many other problems that can be created depending on the approach and the ... | nvidia |
The sum of the lengths of all the edges of a rectangular prism is 120 cm. Find the number of cubic centimeters in the volume of the prism. | nvidia |
In the equation
$$\frac{x^2+1}{x^2-4} = \frac{x+1}{x-2}$$
determine $x$ such that the difference between the two sides of the equation is minimized. | nvidia |
Let p(x) be a polynomial of degree 4 with integer coefficients.
Assume that there exists a positive integer k such that p(3k)=5 for any integer k.
What is the smallest possible value of p(0)? | nvidia |
A sequence of 25 consecutive positive integers has a mean of $n$ and a standard deviation of $\sqrt{n/2}$. Find the largest possible value of $n$. | nvidia |
What is the maximum number of consecutive positive integers that can be added together before the sum exceeds 400? | nvidia |
Let $P(x)$ be a polynomial of degree 6 such that
\[P(x) = P(2016-x).\]
Find the maximum number of real roots that $P(x) - 1$ can have. | nvidia |
Find all real numbers $x$ such that
\[\frac{8^x + 27^x}{12^x + 18^x} = \frac{7}{6}.\]Enter all the solutions, separated by commas. | nvidia |
The positive integers $a$, $b$, $c$, $d$, and $e$ satisfy
$$\sqrt[5]{a^4b^3c^2d^5e^1}=5\sqrt[5]{e^3c^2b^4a^5d^1}$$
Find the smallest possible value of $a+b+c+d+e$. | nvidia |
Determine $p$ if the equation
$$2y^3-3py^2+p^2y-1=0$$
has three roots $y_1$, $y_2$, and $y_3$ such that
$$\frac{1}{y_1}+\frac{1}{y_2}+\frac{1}{y_3}=0.$$ | nvidia |
Find the sum of the smallest positive integer greater than or equal to $\sqrt{n}$ for each $n$ that is not a perfect square, where $1\leq n \leq 100$. In other words, find $\sum\limits_{n=1}^{100} \lceil \sqrt{n} \rceil$ excluding the terms where $n$ is a perfect square. | nvidia |
Compute the number of ways to arrange the digits of the number 123456789 to form a 9-digit number that is divisible by 9. | nvidia |
Find the sum of the cubes of all positive integers from 1 to 10, inclusive, and divide it by the sum of their squares. Determine the resulting fraction. | nvidia |
Determine the value of x such that 3^(2x+1) = 81^(x-1). | nvidia |
Find all real numbers $a$ such that the matrix $\begin{pmatrix} 1 & a & 0 \\ 0 & 1 & a \\ a & 0 & 1 \end{pmatrix}$ has a non-trivial solution to the equation $\begin{pmatrix} 1 & a & 0 \\ 0 & 1 & a \\ a & 0 & 1 \end{pmatrix} \mathbf{x} = \mathbf{0}$. | nvidia |
Let $\sigma(n)$ denote the sum of the positive divisors of $n$, including 1 and $n$. Define $T(n)$ by $T(n) = \sigma(1) + \sigma(2) + \cdots + \sigma(n)$. Find the smallest positive integer $n$ such that $T(n)$ is divisible by $n$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$ Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\] | nvidia |
The sum of the squares of the sides of a parallelogram is equal to 100, and the sum of the squares of the diagonals is equal to 150. Find the largest possible value of the length of one of the diagonals. | nvidia |
Find the least positive integer $x$ that satisfies $x+4609 \equiv 2104 \pmod{12}$. | nvidia |
Let's generate a problem similar to the one provided by the user.
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?
Another problem inspire... | nvidia |
Find the number of integers that are both greater than 10 and less than 50, and are also multiples of 3 or 5. | nvidia |
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 4.$ Find the minimum value of $\frac{1}{a} + \frac{1}{b^2} + \frac{1}{c^3}.$ | nvidia |
What is the smallest positive integer $n$ such that $531n \equiv 1067n \pmod{24}?$ | nvidia |
Determine the value of $x$ such that the expression $3x^2-2x-15$ is divisible by $x^2+2x+2$. | nvidia |
Find the value of $x$ such that $\sin(x+30^\circ) = \sin(2x-45^\circ)$. | nvidia |
What is the sum of all integer values of $n$ such that $\frac{30}{3n - 2}$ is an integer? | nvidia |
Find the smallest value of $b$ such that $\left|b-\frac{3}{b}\right|\geq 2$. | nvidia |
Find all real numbers $a$ such that the roots of the polynomial
$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. | nvidia |
The sum of the squares of two consecutive integers is 157. What is the sum of the cubes of these integers? | nvidia |
Determine the minimum number of points in space that can determine a set of 12 planes such that no two planes intersect in more than one point. | nvidia |
Let $x$ and $y$ be two positive numbers such that $\frac{x}{y}=5$ and $y=\frac{10}{\frac{1}{x}+\frac{1}{5}}$.
Determine $x$. | nvidia |
There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$. | nvidia |
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