Olympiad problems

2013 India IMO Training Camp, 2

An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers. [b]a)[/b] Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$. [b]b)[/b] Decide whether $a=2$ is friendly.

2004 Nicolae Coculescu, 3

Tags: equation , hermite , fractional , floor , algebra

Prove the identity $ \frac{n-1}{2}=\sum_{k=1}^n \left\{ \frac{m+k-1}{n} \right\} , $ where $ n\ge 2, m $ are natural numbers, and $ \{\} $ denotes the fractional part.

2015 Germany Team Selection Test, 1

Tags: number theory , equation

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2009 Germany Team Selection Test, 1

Tags: algebra , number theory , equation

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

2010 JBMO Shortlist, 3

Tags: algebra , equation

Find all pairs $(x,y)$ of real numbers such that $ |x|+ |y|=1340$ and $x^{3}+y^{3}+2010xy= 670^{3}$ .

1966 IMO Shortlist, 25

Tags: trigonometry , algebra , trigonometric identities , equation

Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]

2009 District Olympiad, 3

Tags: equation , algebra , logarithm

Let $ A $ be the set of real solutions of the equation $ 3^x=x+2, $ and let be the set $ B $ of real solutions of the equation $ \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . $ Prove the validity of the following subpoints: [b]a)[/b] $ A\subset B. $ [b]b)[/b] $ B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} . $

1958 Czech and Slovak Olympiad III A, 1

Tags: equation

Find all real solutions of equation $x + \sqrt{2p - x^2} = 8$ with real parameter $p$.

1989 IMO Shortlist, 5

Tags: algebra , polynomial , root , equation , diophantine equation

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

MathLinks Contest 6th, 2.1

Tags: algebra , equation

Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.

2018 IFYM, Sozopol, 1

Tags: number theory , equation

Find the number of solutions to the equation: $6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $ With {x} we denote the fractional part of the number x.

2011 Laurențiu Duican, 1

Tags: equation , integer part , fractional part , floor , algebra

Solve in the real numbers the equation $ 2^{1+x} =2^{[x]} +2^{\{x\}} , $ where $ [],\{\} $ deonotes the ineger and fractional part, respectively. [i]Aurel Bârsan[/i]

2012 India IMO Training Camp, 1

Tags: algebra , equation , sequence

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

1980 IMO Shortlist, 14

Tags: algebra , sequence , equation

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

2018 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: parameterization , algebra , equation

Find all values of real parameter $a$ for which equation $2{\sin}^4(x)+{\cos}^4(x)=a$ has real solutions

1989 IMO Longlists, 83

Tags: number theory , diophantine equation , quadratic , equation

Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

1960 Putnam, B1

Tags: integer , equation , number theory

Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$

1966 IMO Shortlist, 12

Tags: number theory , decimal representation , equation

Find digits $x, y, z$ such that the equality \[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\] holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.

2016 Macedonia National Olympiad, Problem 1

Tags: equation , number theory

Solve the equation in the set of natural numbers $1+x^z + y^z = LCM(x^z,y^z)$

2002 All-Russian Olympiad, 1

Tags: algebra , polynomial , quadratic , ring theory , equation , functional equation

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

1985 Traian Lălescu, 2.1

Tags: equation , algebra , floor

Solve $ \quad 5\lfloor x^2\rfloor -2\lfloor x\rfloor +2=0. $

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , real number

Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$

2018 Hanoi Open Mathematics Competitions, 4

Tags: equation , algebra

Find the number of distinct real roots of the following equation $x^2 +\frac{9x^2}{(x + 3)^2} = 40$. A. $0$ B. $1$ C. $2$ D. $3$ E. $4$

2004 Gheorghe Vranceanu, 4

Tags: equation , diophantine

Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $

1967 IMO Longlists, 44

Tags: algebra , polynomial , summation , equation , diophantine equation

Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$