Olympiad problems

2012 Bosnia And Herzegovina - Regional Olympiad, 1

Find all possible values of $$\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$$ where $a$, $b$ and $c$ are positive real numbers such that $ab+bc+ca=abc$

1952 Moscow Mathematical Olympiad, 209

Tags: identity , algebra

Prove the identity: a) $(ax + by + cz)^2 + (bx + cy + az)^2 + (cx + ay + bz)^2 =(cx + by + az)^2 + (bx + ay + cz)^2 + (ax + cy + bz)^2$ b) $(ax + by + cz + du)^2+(bx + cy + dz + au)^2 +(cx + dy + az + bu)^2 + (dx + ay + bz + cu)^2 =$ $(dx + cy + bz + au)^2+(cx + by + az + du)^2 +(bx + ay + dz + cu)^2 + (ax + dy + cz + bu)^2$.

2015 Caucasus Mathematical Olympiad, 2

Tags: algebraic identities , identity , algebra

Vasya chose a certain number $x$ and calculated the following: $a_1=1+x^2+x^3, a_2=1+x^3+x^4, a_3=1+x^4+x^5, ..., a_n=1+x^{n+1}+x^{n+2} ,...$ It turned out that $a_2^2 = a_1a_3$. Prove that for all $n\ge 3$, the equality $a_n^2 = a_{n-1}a_{n+1}$ holds.

2017 Romania National Olympiad, 4

Tags: identity , function , algebra , combinatorics

Find the number of functions $ A\stackrel{f}{\longrightarrow } A $ for which there exist two functions $ A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A $ having the properties that $ g\circ h =\text{id.} $ and $ h\circ g=f, $ where $ B $ and $ A $ are two finite sets.

2018 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: real number , identity , algebra

if $a$, $b$ and $c$ are real numbers such that $(a-b)(b-c)(c-a) \neq 0$, prove the equality: $\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca$

India EGMO 2023 TST, 4

Tags: algebra , identity , periodic function , function

Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$ Prove that either $f$ is the identity function or $g$ is periodic. [i]Proposed by Pranjal Srivastava[/i]

2010 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: geometry , identity

It is given acute triangle $ABC$ with orthocenter at point $H$. Prove that $$AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}$$ where $a$, $b$ and $c$ are sides of a triangle, and $h_a$, $h_b$ and $h_c$ altitudes of $ABC$

2017 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , real number , identity

Let $a$, $b$ and $c$ be real numbers such that $abc(a+b)(b+c)(c+a)\neq0$ and $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}$ Prove that $\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017$

2019 District Olympiad, 1

Tags: algebra , inequality , identity , cauchy inequality

Let $n \in \mathbb{N}, n \ge 2$ and the positive real numbers $a_1,a_2,…,a_n$ and $b_1,b_2,…,b_n$ such that $a_1+a_2+…+a_n=b_1+b_2+…+b_n=S.$ $\textbf{a)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k} \ge \frac{S}{2}.$ $\textbf{b)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k}= \sum\limits_{k=1}^n \frac{b_k^2}{a_k+b_k}.$

2023 Bangladesh Mathematical Olympiad, P5

Tags: algebra , identity , manipulation

Let $m$, $n$ and $p$ are real numbers such that $\left(m+n+p\right)\left(\frac 1m + \frac 1n + \frac1p\right) =1$. Find all possible values of $$\frac 1{(m+n+p)^{2023}} -\frac 1{m^{2023}} -\frac 1{n^{2023}} -\frac 1{p^{2023}}.$$

2023 India EGMO TST, P4

Tags: algebra , function , identity , periodic function

Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$ Prove that either $f$ is the identity function or $g$ is periodic. [i]Proposed by Pranjal Srivastava[/i]

1996 Romania National Olympiad, 3

Tags: invariant , algebra , trigonometry , identity

Prove that $ \forall x\in \mathbb{R} $ , $ \cos ^7x+\cos ^7(x+\frac {2\pi}{3})+\cos ^7(x+\frac {4\pi}{3})=\frac {63}{64}\cos 3x $

2022 Indonesia TST, N

Tags: number theory , order , identity , divisibility

Given positive odd integers $m$ and $n$ where the set of all prime factors of $m$ is the same as the set of all prime factors $n$, and $n \vert m$. Let $a$ be an arbitrary integer which is relatively prime to $m$ and $n$. Prove that: \[ o_m(a) = o_n(a) \times \frac{m}{\gcd(m, a^{o_n(a)}-1)} \] where $o_k(a)$ denotes the smallest positive integer such that $a^{o_k(a)} \equiv 1$ (mod $k$) holds for some natural number $k > 1$.

2012 India Regional Mathematical Olympiad, 6

Tags: algebra , identity , algebraic identities

Show that for all real numbers $x,y,z$ such that $x + y + z = 0$ and $xy + yz + zx = -3$, the expression $x^3y + y^3z + z^3x$ is a constant.

2024 Mozambique National Olympiad, P2

Tags: algebra , inequalities , ident , identity

Prove that if $a+b+c=0$ then $a^3+b^3+c^3=3abc$