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The sum of the digits of a certain integer is equal to the number of its digits. If the digits of the number are increased by 1, the resulting number will be a multiple of 3. How many digits does the original number have? | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f((x + y)^2) = f(x)^2 + 2xf(y) - y^2\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $f(0),$ and let $t$ be the sum of all possible values of $f(0).$ Find $m \times t.$ | nvidia |
Find the area of the triangle with vertices at points $3-2i$, $4+2i$, and $6+3i$ in the complex plane. | nvidia |
Find the number of positive integers x such that the fraction $\frac{2x+5}{x-3}$ is greater than or equal to 3. | nvidia |
Determine the value of \(b\) such that the expression \(\frac{2^{2b} \cdot 3^{3b}}{6^b} = 27\) holds true. | nvidia |
Find the number of integers in the range from 1 to 1000 that are not divisible by any of the prime numbers 2, 3, 5, or 7. | nvidia |
Let $g$ be a non-constant polynomial such that
\[g(x - 2) + g(x) + g(x + 2) = \frac{[g(x)]^2}{2015x^2}\]
for all nonzero real numbers $x.$ Find the sum of all possible values of $g(2)$. | nvidia |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 3.$ Find the minimum value of
\[\frac{4}{x} + \frac{9}{y} + \frac{16}{z}.\] | nvidia |
If $c,$ $d,$ $u,$ and $v$ are real numbers such that $cu+dv=5,$ $cu^2+dv^2=11,$ $cu^3+dv^3=25,$ and $cu^4+dv^4=61,$ find $cu^5+dv^5.$ | nvidia |
Compute the value of the infinite series
\[
\sum_{n=1}^{\infty} \frac{n^3+2n^2+5n+5}{3^n \cdot \left(n^3+1\right)}
\] | nvidia |
What is the largest number of non-overlapping dominoes that can be placed on a 10x10 grid such that each domino covers exactly two adjacent cells, and no domino extends beyond the grid? | nvidia |
Consider the function
\[f(x) = 3x^2 + \frac{8}{x}\]
for positive real numbers $x$.
Determine the value of $x$ for which the function is smallest.
Which positive integer $n$ is closest to this value of $x$? | nvidia |
Find all real values of $a$ for which the system of equations
\begin{align*}
x^{\log_a y} &= y^{\log_a x}, \\
\sqrt{\log_a x \cdot \log_a y} &= a^a
\end{align*}
has a solution $(x,y)$ in real numbers. | nvidia |
Find the number of positive integers $n$ less than $10^6$ such that $n$ and $n^2$ are both divisible by 24. | nvidia |
Given that the absolute value of the difference between two integers is at most 5, what is the greatest integer value for the larger integer such that their product is less than or equal to 50? | nvidia |
Find the largest positive integer that is not divisible by any of the first four prime numbers. | nvidia |
Prove that for any complex number $z$, $$|1+z^n|=|1+z^{-n}|,$$
where $n$ is an integer. | nvidia |
How many of the positive integers less than or equal to 1000 can be expressed as the sum of two perfect cubes, where the two cubes may be positive, negative, or zero? | nvidia |
Find the area of the region $B$ in the complex plane that consists of all points $z$ such that the arguments of $z$, $2z$, and $3z$ are in the ratio $1:2:3$. | nvidia |
There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$ | nvidia |
Find the smallest positive integer whose square has exactly 5 positive integer factors, not counting 1 and itself. | nvidia |
Let $$n! = \prod_{i=1}^ni$$
be the factorial of the integer $n$, where
$n>0$. For every positive integer $n$, determine the largest integer $m$ such that $m^n!$ is a divisor of $$\prod_{i=1}^n (n+i)!$$ | nvidia |
Consider a pyramid with a square base, where the vertices of the base are given by the points (0,0,0), (2,0,0), (2,2,0), and (0,2,0). The apex of the pyramid is given by the point (1,1,h), where h is a parameter. Determine the values of h for which the pyramid has a symmetry plane that contains the base edge from (0,0,... | nvidia |
The function $g(x)$ satisfies
\[g(x+y)=g(x)g(y)\]
for all real numbers $x$ and $y$, and $g(2)=3.$ Find $g(3).$ | nvidia |
Problem:
In the system of equations
\begin{align*}
x + ty &= 1, \\
tx + y &= 2,
\end{align*}determine the value of $t$ such that the system has no solutions in $x$ and $y$. | nvidia |
Let $a$ and $d$ be real numbers. If the polynomial $x^2+ax+d$ has exactly one real root and $a=2d-3$, find the value of the sum of all possible values of $d$. | nvidia |
A set of $n$ people participate in an online video meeting. Each person has a different favorite color. Determine the number of ways to arrange the $n$ people in a row such that no two people with the same favorite color are seated next to each other. | nvidia |
A hyperbola in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(5,3)$ and the other focus is at $(5,e).$ Compute $e$ such that the asymptotes of the hyperbola intersect at a point with coordinates $(2,2)$. | nvidia |
The roots $x_1$, $x_2$ and $x_3$ of the equation
$$x^3+px^2+qx+r=0$$
are such that
$x_1+x_2\sqrt 2+x_3\sqrt 3=1$.
Determine $p$, $q$ and $r$. | 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 number of ordered triples $(x, y, z)$ of positive integers such that the sum of the reciprocals of the three numbers is equal to the square of the smallest number. | nvidia |
A sequence of $2n$ distinct numbers is given. The sequence is split into two subsequences: one contains the terms at even positions, and the other contains the terms at odd positions. The subsequences are then merged in a special way: the smallest number of the first subsequence is placed first, then the smallest numbe... | nvidia |
My co-worker Erich is very odd. He only likes numbers that are divisible by 5. How many different last digits are possible in numbers that Erich likes? | nvidia |
Let $f(n)$ be a function that, given an integer $n$, returns an integer $k$, where $k$ is the smallest possible integer such that $k!$ is divisible by $n$. Given that $n$ is a multiple of 24, what is the smallest value of $n$ such that $f(n) > 20$? | nvidia |
Find the maximum value of the expression $ab+bc+ca$, where $a$, $b$, and $c$ are positive integers whose sum is 25. | nvidia |
A circle is inscribed inside a right triangle. The radius of the circle is 6 cm, and the lengths of the two legs of the triangle are in the ratio 3:4. Find the area of the triangle. | nvidia |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f((x - y)^2) = f(x)^2 - 2xf(y) + y^2\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(1),$ and let $s$ be the sum of all possible values of $f(1).$ Find $n \times s.$ | nvidia |
Let $T_i$ be the set of all positive integers $n$ such that $10^i \leq n < 10^{i+1}$ and the sum of the digits of $n$ is divisible by $3$. For example, $T_2$ is the set ${102, 105, 108, \ldots, 999}$. How many of the sets $T_1, T_2, T_3, \ldots, T_{10}$ contain a number that is a power of $2$? | nvidia |
How many 4-digit positive integers exist that satisfy the following conditions: (A) Each of the first two digits must be 1, 4, or 5, and (B) the last two digits cannot be the same digit, and (C) each of the last two digits must be 5, 7, or 8? | nvidia |
In a city, the number of engineers is triple the number of architects. There are 90 architects, and none of the architects are engineers. The total number of men is three times the number of women. If all the engineers are men and all the architects are women, how many people are there in total? | nvidia |
Let $Q(x)$ be a polynomial with real coefficients such that its set of complex roots is $\{-1, 2i, -2i, 3, 4\}$. Suppose that $Q(x)$ can be factored into the product of two monic polynomials $R(x)$ and $S(x)$, each having at least two distinct roots. Find the least possible degree of $R(x)$ such that the constant term ... | nvidia |
Let $d$ be a positive number such that when $109$ is divided by $d$, the remainder is $4.$ Compute the sum of all possible two-digit values of $d$. | nvidia |
Find the sum of all integer divisors of $210$ that are also multiples of $3$ and divide $100$. | nvidia |
A polynomial $P(x)$ of degree $100$ is defined such that the product of its roots is equal to $1$ and $P(1) = 1$. Furthermore, it is known that the product of its coefficients is equal to $100!$. Determine the constant term of $P(x)$. | nvidia |
The equation $x^2-mx-24=0$ has only integer solutions for certain positive integers $m$. What is the sum of all such values of $m$? | 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 smallest five-digit positive integer with all different digits that is divisible by the sum of its digits. | nvidia |
Find the number of positive integers less than 1000 that are divisible by both 13 and 7 but not by 3. | nvidia |
Find the value of $a$ such that the graphs of $y=ax^3+2x^2-1$ and $y=2x-3$ intersect at exactly two points. | nvidia |
Consider the polynomial function:
$$f(x) = x^4 + mx^3 + nx^2 + px + q$$
which has four distinct x-intercepts, one of which is at (1,0). Which of the following coefficients must be non-zero?
$\textbf{(A)}\ m \qquad \textbf{(B)}\ n \qquad \textbf{(C)}\ p \qquad \textbf{(D)}\ q$ | nvidia |
Given that $a$ and $b$ are positive integers, find the smallest value of $n$ such that
$$\prod_{k=1}^{n} \left(1 + \frac{a}{b^k}\right) \geq 3.$$ | nvidia |
Find all real numbers $c$ for which there exists a nonzero polynomial $P(x)$ of degree less than 3 such that
\[P(2x) + P(x) = c P(x).\]
Enter all the solutions, separated by commas. | nvidia |
Find the product of all positive integral values of $m$ for which $\frac{m^2+2m-3}{m+1}$ is an odd integer. | nvidia |
There are 2 complex numbers $x+yi$, $z+wi$. If $x=2$, $z=-x$, and the sum of the numbers is $3i$, find $y+w$. | nvidia |
Find the area of the triangle with vertices $(0,7,10),$ $(-1,6,6),$ and $(-4,9,6).$ | nvidia |
Find the sum of all values of $y$ for which $\sqrt{(y-5)^2} = 9$. | nvidia |
Find the number of ways to arrange the letters of the word "referee" so that the vowels (E and E) are always together and the consonants (R, F, and R) are always together. | nvidia |
Let $\zeta$ be a function from the set of integers to itself, such that for all integers $m$, $n$
\[\zeta(\zeta(m)+\zeta(n))=\zeta(m)+\zeta(n+1)+1.\]
Let $t$ be the smallest possible value of $\zeta(10)$ and let $r$ be the largest possible value of $\zeta(10)$. Find $t\times r$. | 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 minimum possible value of the sum
\[\frac{a}{2b} + \frac{b}{4c} + \frac{c}{8a},\]where $a,$ $b,$ and $c$ are positive real numbers. | nvidia |
Find the value of $a$ such that the sum of the first six terms in the geometric sequence $a,2a,4a,...$ is equal to $48$. | nvidia |
On the faces of a rectangular solid (that is, a "box") are written six positive integers, one per face. Each edge of the solid is labeled with the sum of the two numbers on the faces adjacent to the edge. If the sum of the numbers on the edges is equal to $F$, then what is the sum of the numbers written on the faces of... | nvidia |
Find all complex numbers $z$ such that
$$\cos(z)+i\sin(z)=\cos(2z).$$ | nvidia |
Let $a_1, a_2, \dots, a_n$ be a sequence of positive real numbers such that $\prod_{i=1}^n a_i = 1$. Find the maximum value of
\[\sum_{i=1}^n \frac{a_i^2}{1 + a_i}.\] | nvidia |
Find the smallest integer $n$ such that the product $12n$ is greater than 350 and is also a perfect square. | nvidia |
Consider a set of positive integers $x$ such that the sum of any two elements is divisible by 7, and the sum of any three elements is divisible by 11. Determine the smallest number of elements in such a set.
Note: I've created a problem that requires a different approach to solve, focusing on divisibility and modular ... | nvidia |
Find all real solutions to $x^3+(x+1)^3+(x+2)^3=(x+3)^3$. Enter all the solutions, separated by commas. | nvidia |
Consider the sequence of points on the coordinate plane defined by $(2n, n^2)$ for positive integers $n$. Determine the smallest value of $n$ for which the slope of the line passing through $(2n, n^2)$ and $(2(n+1), (n+1)^2)$ exceeds 20. | nvidia |
Determine the maximum value of $x^3 + ax^2 + bx + c$ when the sum of the coefficients is 12, the sum of the roots is -2, and the sum of the reciprocals of the roots is 0.5. | nvidia |
Let $p(x)$ be a polynomial with integer coefficients such that $\ds\f{p(n)}{p(n+1)}$ is a terminating decimal for every positive integer $n$. What is the largest possible degree of $p(x)$? | nvidia |
For how many three-digit positive integers is the sum of the digits equal to $5?$ | nvidia |
Let $a$ and $b$ be positive integers. Consider the product of two numbers, $3^a$ and $2^b$, that ends with the digits...16. Find the smallest possible value of $a+b$. | 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 $a$ be the maximum possible value of $f(1),$ and let $b$ be the minimum possible value of $f(1).$ Find $a - b.$ | nvidia |
The product of two positive integers is 360 and their quotient is $\frac{3}{4}$. What is the value of the smaller of the two integers? | nvidia |
The blue curve is the graph of the function
\[f(x) = a\cos^2 x + b\sin^2 x.\]
Using the fact that the graph passes through the points $(0,2)$ and $(\pi/2,6)$, and that the graph has a maximum at $x=\pi/4$, find the value of $f(\pi/3)$. | nvidia |
Let $a$ and $b$ be two positive integers such that their product is 240. Determine the sum of all possible values of the sum of $a$ and $b$. | nvidia |
Let $g : \mathbb{R} \to \mathbb{R}$ be a function such that
\[g(g(x) - y) = g(y - x^2) + 2g(x) y\]for all real numbers $x$ and $y.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the product of all possible values of $g(2).$ Find $m \times t.$ | nvidia |
Let $ABCD$ be a convex quadrilateral and let $E$ be the intersection of its diagonals. If $P$, $Q$, $R$, and $S$ are points on segments $AB$, $BC$, $CD$, and $DA$ such that $\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DS}{SA}$, show that the area of quadrilateral $PQRS$ is equal to half the area of quadrilateral $A... | nvidia |
Let $a$ and $b$ be positive integers. Determine the number of ordered pairs $(a,b)$ such that $|a + b\sqrt[3]{2}| < 1$. | nvidia |
Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be... | nvidia |
Let $T$ be the sum of all integers $c$ for which the polynomial $x^3+cx^2+2009c$ can be expressed as the product of three integer-coefficient linear factors. Compute $|T|$. | nvidia |
Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Find the set of all possible values of $\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx}$. | nvidia |
Let $A$ be a set of six non-zero digits such that for any two different digits $a$ and $b$ in $A$, the two-digit number formed by $a$ and $b$ is divisible by the sum of $a$ and $b$. What is the smallest possible sum of the digits in $A$? | nvidia |
Let $\mathbf{B}$ be a $3 \times 3$ matrix, with real entries, such that $\mathbf{B}^2 = \mathbf{I}.$ Find the number of different possible determinants that $\mathbf{B}$ can have. If you think the answer is infinite, then enter "infinite". | nvidia |
What is the probability that the sum of the numbers shown on two fair eight-sided dice will be 7? | nvidia |
Find the number of positive integer solutions to the system of equations:
$$\frac{x}{y} = a, \quad \frac{x+b}{y+b} = c$$
where $a$, $b$, and $c$ are fixed positive integers, and $x$ and $y$ are positive integers such that $y < x \leq 1000$. | nvidia |
Find the largest real number $c$ such that $\left|\left(1 + i\right)^n\right| \geq c^n$ for all positive integers $n$. | nvidia |
Find the positive integer $n$ such that the equation $x^2 - nx + 2 = 0$ has solutions with a product of 20. | nvidia |
Find the largest value of $n$ such that $5x^2+nx+48$ can be factored as the product of two linear factors with integer coefficients. | nvidia |
Find the largest possible value of $n$ for which $2^n$ can be represented as the difference of two consecutive perfect cubes. | nvidia |
How many nonnegative integers can be written in the form\[a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,\]where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$?
$\textbf{(A) } 512 \qquad \textbf{(B) } 729 \qquad \textbf{(C) } 1094 \qquad \textbf{(D) } 3281 \qquad \textbf{(E)... | nvidia |
The sequence $a_1, a_2, \ldots$ is arithmetic with $a_1=a$ and common difference $d$, where $a$ and $d$ are positive integers. Given that $\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_{12}} = \frac{2006}{3}$, find the number of possible ordered pairs $(a,d)$. | nvidia |
Determine the area of the region enclosed by the graph of the equation $r^2 = 3 \cos 2\theta$, where $r$ and $\theta$ are polar coordinates, that lies inside the circle $x^2+y^2=4$. | nvidia |
A set of five points is given in the plane, and it is known that no three of the points are collinear. A "divided triangle" is a triangle formed by three of the points, where the side lengths are each divided into two segments by a vertex of the triangle. If the sum of the lengths of the longer segments for all possibl... | nvidia |
How many positive integers less than $700$ can be expressed as the difference of two positive perfect squares? | nvidia |
Let $f(x)$ be a non-constant polynomial with real coefficients, such that
\[\frac{f(x^2)}{x^2+1}\le \frac{3x^2+3x+5}{x^2+1}\] for all real numbers $x$. Find the sum of the coefficients of $f(x)$. | nvidia |
Let $a$, $b$, and $c$ be positive integers greater than $1$ such that $a^2=b^3=c^4$. Determine the least possible value of $a+b+c$. | nvidia |
A polynomial of degree 6 has 3 distinct real roots, and it satisfies the condition that the product of its derivative and its primitive is always a multiple of the square of the polynomial. Determine the sum of the reciprocals of the real roots of this polynomial. | nvidia |
Determine the value of $b$ such that the quadratic expression $bx^2 + 16x + 16$ can be expressed as the square of a binomial. | nvidia |
For what real values of $x$ does the expression $\frac{x^2+1}{x+\sqrt{x^2+1}}$ take on only positive values? | nvidia |
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