Found problems: 15925
2023 India IMO Training Camp, 3
Prove that for all integers $k>2$, there exists $k$ distinct positive integers $a_1, \dots, a_k$ such that $$\sum_{1 \le i<j \le k} \frac{1}{a_ia_j} =1.$$
[i]Proposed by Anant Mudgal[/i]
2011 Estonia Team Selection Test, 4
Let $a,b,c$ be positive real numbers such that $2a^2 +b^2=9c^2$.Prove that $\displaystyle \frac{2c}{a}+\frac cb \ge\sqrt 3$.
2012 IMO, 4
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds:
\[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]
(Here $\mathbb{Z}$ denotes the set of integers.)
[i]Proposed by Liam Baker, South Africa[/i]
2024 Serbia Team Selection Test, 3
Let $S$ be the set of all convex cyclic heptagons in the plane. Define a function $f:S \rightarrow \mathbb{R}^+$, such that for any convex cyclic heptagon $ABCDEFG,$ $$f(ABCDEFG)=\frac{AC \cdot BD \cdot CE \cdot DF \cdot EG \cdot FA \cdot GB} {AB \cdot BC \cdot CD \cdot DE \cdot EF \cdot FG \cdot GA}. $$
a) Show that for any $M \in S$, $f(M) \geq f(\prod)$, where $\prod$ is a regular heptagon.
b) If $f(M)=f(\prod)$, is it true that $M$ is a regular heptagon?
2005 MOP Homework, 4
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x^3)-f(y^3)=(x^2+xy+y^2)(f(x)-f(y))$.
2002 District Olympiad, 4
Let $ n\ge 2 $ be a natural number. Prove the following propositions:
[b]a)[/b] $ a_1,a_2,\ldots ,a_n\in\mathbb{R}\wedge a_1+\cdots +a_n=a_1^2+\cdots +a_n^2\implies a_1+\cdots +a_n\le a_n. $
[b]b)[/b] $ x\in [1,n]\implies\exists b_1,b_2,\ldots ,b_n\in\mathbb{R}_{\ge 0}\quad x=b_1+\cdots +b_n=b_1^2 +\cdots +b_n^2 . $
2007 Hanoi Open Mathematics Competitions, 5
Let be given an open interval $( \alpha; \beta)$ with $\alpha - \beta = \frac{1}{27}$. Determine the maximum number of irreducible fractions $\frac{a}{b}$
in $( \alpha; \beta)$ with $1 \leq b \leq 2007$?
2016 Balkan MO Shortlist, A5
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.
Find the minimum value and the maximum value of the product $abcd$.
1988 Swedish Mathematical Competition, 4
A polynomial $P(x)$ of degree $3$ has three distinct real roots.
Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$.
2024 India IMOTC, 13
Find all functions $f:\mathbb R \to \mathbb R$ such that
\[
xf(xf(y)+yf(x))= x^2f(y)+yf(x)^2,
\]
for all real numbers $x,y$.
[i]Proposed by B.J. Venkatachala[/i]
2020 Canadian Mathematical Olympiad Qualification, 8
Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$
1962 All-Soviet Union Olympiad, 13
Given are $a_0,a_1, ... , a_n$, satisfying $a_0=a_n = 0$, and $a_{k-1} - 2a_k+a_{k+1}\ge 0$ for $k=0, 1, ... , n-1$. Prove that all the numbers are negative or zero.
2016 IFYM, Sozopol, 1
Find all functions $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ with the following property: $a,b,$ and $c$ are lengths of sides of a triangle, if and only if $f(a),f(b),$ and $f(c)$ are lengths of sides of a triangle.
1967 IMO Shortlist, 5
Solve the system of equations:
$
\begin{matrix}
x^2 + x - 1 = y \\
y^2 + y - 1 = z \\
z^2 + z - 1 = x.
\end{matrix}
$
2015 India Regional MathematicaI Olympiad, 2
Let $P_1(x) = x^2 + a_1x + b_1$ and $P_2(x) = x^2 + a_2x + b_2$ be two quadratic polynomials with integer coeffcients. Suppose $a_1 \ne a_2$ and there exist integers $m \ne n$ such that $P_1(m) = P_2(n), P_2(m) = P_1(n)$. Prove that $a_1 - a_2$ is even.
2010 ELMO Shortlist, 4
Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$.
[i]Evan O' Dorney.[/i]
2018 German National Olympiad, 1
Find all real numbers $x,y,z$ satisfying the following system of equations:
\begin{align*}
xy+z&=-30\\
yz+x &= 30\\
zx+y &=-18
\end{align*}
2023 BMT, 4
Let f$(x)$ be a continuous function over the real numbers such that for every integer $n$, $f(n) = n^2$ and $f(x) $ is linear over the interval $[n, n + 1]$. There exists a unique two-variable polynomial $g$ such that $g(x, \lfloor x \rfloor) = f(x)$ for all $x$. Compute $g(20, 23)$. (Here, $\lfloor x \rfloor$ is defined as the greatest integer less than or equal to $x$. For example, $\lfloor 2\rfloor = 2$ and $\lfloor -3.5 \rfloor = -4$.)
2023 Mongolian Mathematical Olympiad, 1
Let $u, v$ be arbitrary positive real numbers. Prove that \[\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.\]
2013 CHMMC (Fall), 3
Let $p_n$ be the product of the $n$th roots of $1$. For integral $x > 4$, let $f(x) = p_1 - p_2 + p_3 - p_4 + ... + (-1)^{x+1}p_x$. What is $f(2010)$?
2019 New Zealand MO, 3
Let $a, b$ and $c$ be positive real numbers such that $a + b + c = 3$. Prove that $$a^a + b^b + c^c \ge 3$$
2022 Stanford Mathematics Tournament, 1
If $x$, $y$, and $z$ are real numbers such that $x^2+2y^2+3z^2=96$, what is the maximum possible value of $x+2y+3z$?
2011 Uzbekistan National Olympiad, 2
Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.
2024 Serbia National Math Olympiad, 3
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ be reals. Show that for any positive integer $1 \leq m \leq n$, there exist two distinct reals $\alpha, \beta$, $\alpha^2+\beta^2>0$, such that $p_m=\min\{p_1, p_2, \ldots, p_n\}$, where $$p_j=\sum_{i=1}^n|\alpha(a_i-a_j)+\beta(b_i-b_j)|$$ for $1\leq j \leq n$.
2019 PUMaC Algebra A, 3
Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.