Found problems: 15925
2016 CMIMC, 3
Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.
2025 Euler Olympiad, Round 1, 8
Let $S$ be the set of non-negative integer powers of $3$ and $5$, $S = \{1, 3, 5, 3^2, 5^2, \ldots \}$. For every $a$ and $b$ in $S$ satisfying $$ \left| \pi - \frac{a}{b} \right| < 0.1 $$ Find the minimum value of $ab$.
[i]Proposed by Irakli Shalibashvili, Georgia [/i]
2012 Princeton University Math Competition, A3 / B6
Compute $\Sigma_{n=1}^{\infty}\frac{n + 1}{n^2(n + 2)^2}$ .
Your answer in simplest form can be written as $a/b$, where $a, b$ are relatively-prime positive integers. Find $a + b$.
2016 AIME Problems, 6
For polynomial $P(x)=1-\frac{1}{3}x+\frac{1}{6}x^2$, define \[ Q(x) = P(x)P(x^3)P(x^5)P(x^7)P(x^9) = \sum\limits_{i=0}^{50}a_ix^i. \] Then $\sum\limits_{i=0}^{50}|a_i|=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2025 Caucasus Mathematical Olympiad, 6
It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Can it happen that from segments of lengths$$\sqrt{a^2 + bc}, \quad \sqrt{b^2 + ca}, \quad \sqrt{c^2 + ab}$$an obtuse triangle can be formed?
2004 District Olympiad, 2
Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.
2004 Hong kong National Olympiad, 1
Let $a_{1},a_{2},...,a_{n+1}(n>1)$ are positive real numbers such that $a_{2}-a_{1}=a_{3}-a_{2}=...=a_{n+1}-a_{n}$. Prove that $\sum_{k=2}^{n}\frac{1}{a_{k}^{2}}\leq\frac{n-1}{2}.\frac{a_{1}a_{n}+a_{2}a_{n+1}}{a_{1}a_{2}a_{n}a_{n+1}}$
2023 Serbia National Math Olympiad, 5
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following:
[list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list]
(a) Prove that the function $f$ is unique.
(b) Find $f(\frac{\sqrt{5}-1}{2})$.
2022-23 IOQM India, 16
Let $a,b,c$ be reals satisfying\\
$\hspace{2cm} 3ab+2=6b, \hspace{0.5cm} 3bc+2=5c, \hspace{0.5cm} 3ca+2=4a.$\\
\\
Let $\mathbb{Q}$ denote the set of all rational numbers. Given that the product $abc$ can take two values $\frac{r}{s}\in \mathbb{Q}$ and $\frac{t}{u}\in \mathbb{Q}$ , in lowest form, find $r+s+t+u$.
2005 Kazakhstan National Olympiad, 4
Find all polynomials $ P(x)$ with real coefficients such that for every positive integer $ n$ there exists a rational $ r$ with $ P(r)=n$.
2005 Abels Math Contest (Norwegian MO), 4b
Let $a, b$ and $c$ be real numbers such that $ab + bc + ca> a + b + c> 0$. Show then that $a+b+c>3$
2014 Junior Balkan Team Selection Tests - Romania, 1
Let $a, b, c, d$ be positive real numbers so that $abc+bcd+cda+dab = 4$.
Prove that $a^2 + b^2 + c^2 + d^2 \ge 4$
2020 Purple Comet Problems, 1
Find $A$ so that the ratio of $3\frac23$ to $22$ is the same as the ratio of $7\frac56$ to $A$
2016 BMT Spring, 13
The quartic equation $y = x^4 + 2x^3 -20x^2 + 8x+ 64$ contains the points$ (-6, 160)$, $(-3, -113)$ and $(2, 32)$. A cubic $y = ax^3 + bx + c$ also contains these points. Determine the $x$-coordinate of the fourth intersection of the cubic with the quartic.
2024 Chile TST IMO, 4
Let $\alpha$ be a real number. Find all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(f(x+y))=f(x+y) +f(x)f(y)+ \alpha xy$ for all $x,y \in \mathbb{R}$
2005 Vietnam National Olympiad, 3
Let $\{x_n\}$ be a real sequence defined by:
\[x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n\]
For all $n=1,2,3...$ and a is a real number.
Find all $a$ such that $\{x_n\}$ has finite limit when $n\to +\infty$ and find the finite limit in that cases.
2017 Moscow Mathematical Olympiad, 4
3 cyclists rides on track in form circle with length $300$ meters in one direction. Every has constant speed,and speeds are different. Photographer want to make photoshoot with 3 cyclists. It is possible if they will be on the part of track with length $d$ meters. Find minimum $d$ such that it is possible.
2022 Romania Team Selection Test, 3
Let $n\geq 2$ be an integer. Let $a_{ij}, \ i,j=1,2,\ldots,n$ be $n^2$ positive real numbers satisfying the following conditions:
[list=1]
[*]For all $i=1,\ldots,n$ we have $a_{ii}=1$ and,
[*]For all $j=2,\ldots,n$ the numbers $a_{ij}, \ i=1,\ldots, j-1$ form a permutation of $1/a_{ji}, \ i=1,\ldots, j-1.$
[/list]
Given that $S_i=a_{i1}+\cdots+a_{in}$, determine the maximum value of the sum $1/S_1+\cdots+1/S_n.$
1999 Irish Math Olympiad, 2
A function $ f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies:
$ (a)$ $ f(ab)\equal{}f(a)f(b)$ whenever $ a$ and $ b$ are coprime;
$ (b)$ $ f(p\plus{}q)\equal{}f(p)\plus{}f(q)$ for all prime numbers $ p$ and $ q$.
Prove that $ f(2)\equal{}2,f(3)\equal{}3$ and $ f(1999)\equal{}1999.$
2010 Contests, 1
Find the sum of the coefficients of the polynomial $(63x-61)^4$.
2024 Junior Balkan Team Selection Tests - Moldova, 7
Find all the real numbers $x,y,z$ which satisfy the following conditions:
$$
\begin{cases}
3(x^2+y^2+z^2)=1\\
x^2y^2+y^2z^2+z^2x^2=xyz(x+y+z)^3\\
\end{cases}
$$
2012 Iran MO (3rd Round), 5
Let $p$ be an odd prime number and let $a_1,a_2,...,a_n \in \mathbb Q^+$ be rational numbers. Prove that \[\mathbb Q(\sqrt[p]{a_1}+\sqrt[p]{a_2}+...+\sqrt[p]{a_n})=\mathbb Q(\sqrt[p]{a_1},\sqrt[p]{a_2},...,\sqrt[p]{a_n}).\]
2007 Ukraine Team Selection Test, 3
It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.
2014 Contests, 2
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
2011 Philippine MO, 4
Find all (if there is one) functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x\in\mathbb{R}$,
\[f(f(x))+xf(x)=1.\]