Olympiad problems

2009 Germany Team Selection Test, 1

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]

1966 IMO Longlists, 10

Tags: equation , trigonometry , algebra , trigonometric equation

How many real solutions are there to the equation $x = 1964 \sin x - 189$ ?

1966 German National Olympiad, 5

Tags: trigonometry , equation , trigonometric identities , algebra

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

2015 Ukraine Team Selection Test, 10

Tags: equation , number theory

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]

1988 IMO Longlists, 63

Tags: equation , quadratic , number theory , relatively prime

Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \] [b]OR[/b] Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \]

1980 IMO, 3

Tags: algebra , equation , diophantine equation , number theory

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

1984 IMO, 3

Tags: equation , divisibility , power of 2 , number theory

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

2011 Croatia Team Selection Test, 4

Tags: number theory , relatively prime , diophantine equation , equation

Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

1958 February Putnam, A5

Tags: equation , integration , calculus

Show that the integral equation $$f(x,y) = 1 + \int_{0}^{x} \int_{0}^{y} f(u,v) \, du \, dv$$ has at most one solution continuous for $0\leq x \leq 1, 0\leq y \leq 1.$

1955 Czech and Slovak Olympiad III A, 4

Tags: equation , quadratic polynomial

Given that $a,b,c$ are distinct real numbers, show that the equation \[\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0\] has a real root.

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , real number

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

1966 IMO Longlists, 9

Tags: trigonometry , algebra , trigonometric equation , equation

Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.

1984 Tournament Of Towns, (069) T3

Tags: algebra , number theory , diophantine equation , equation , diophantine

Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions.

2018 Junior Regional Olympiad - FBH, 1

Tags: equation , time

When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?

1966 IMO Shortlist, 31

Tags: function , equation , parameterization , algebra

Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?

2012 Brazil Team Selection Test, 3

Tags: equation , sequence , algebra

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]

1984 IMO Shortlist, 11

Tags: equation , polynomial , diophantine equation , number theory

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

2017 Junior Regional Olympiad - FBH, 4

Tags: equation

Group of $27$ climbers shared among themself $13$ breads. Every man had $2$ breads, every woman half of a bread, and every child $\frac{1}{3}$ of a bread. How many men, women and children where there ?

2009 Romania National Olympiad, 1

Tags: algebra , function , unique solution , equation

[b]a)[/b] Show that two real numbers $ x,y>1 $ chosen so that $ x^y=y^x, $ are equal or there exists a positive real number $ m\neq 1 $ such that $ x=m^{\frac{1}{m-1}} $ and $ y=m^{\frac{m}{m-1}} . $ [b]b)[/b] Solve in $ \left( 1,\infty \right)^2 $ the equation: $ x^y+x^{x^{y-1}}=y^x+y^{y^{x-1}} . $