Olympiad problems

2004 Gheorghe Vranceanu, 4

Prove that $ \left\{ (x,y)\in\mathbb{C}^2 |x^2+y^2=1 \right\} =\{ (1,0)\}\cup \left\{ \left( \frac{z^2-1}{z^2+1} ,\frac{2z}{z^2+1} \right) | z\in\mathbb{C}\setminus \{\pm \sqrt{-1}\} \right\} . $

2012 IMO Shortlist, N4

Tags: number theory , modular arithmetic , equation

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.

1980 IMO Shortlist, 3

Tags: number theory , equation , diophantine equation , algebra

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.

2005 India IMO Training Camp, 2

Tags: number theory , divisor , equation

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

2017 Mathematical Talent Reward Programme, MCQ: P 1

Tags: algebra , equation

The number of real solutions of the equation $\left(\frac{9}{10}\right)^x=-3+x-x^2$ is [list=1] [*] 2 [*] 0 [*] 1 [*] None of these [/list]

2015 German National Olympiad, 1

Tags: algebra , equation , system of equations

Determine all pairs of real numbers $(x,y)$ satisfying \begin{align*} x^3+9x^2y&=10,\\ y^3+xy^2 &=2. \end{align*}

2016 Azerbaijan IMO TST First Round, 3

Tags: equation

Find the solution of the equation $8x(2x^2-1)(8x^4-8x^2+1)=1$ in the interval $[0,1]$?

1968 IMO Shortlist, 6

Tags: algebra , polynomial , real root , equation , root

If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation \[\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n\] has at least $n - 1$ real roots.

2016 Kyiv Mathematical Festival, P1

Tags: algebra , equation

Prove that for every positive integers $a$ and $b$ there exist positive integers $x$ and $y$ such that $\dfrac{x}{y+a}+\dfrac{y}{x+b}=\dfrac{3}{2}.$

1984 IMO Shortlist, 2

Tags: polynomial , number theory , diophantine equation , equation , quadratic

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

2012 Bogdan Stan, 3

Tags: algebra , equation

Find the real numbers $ x,y,z $ that satisfy the following: $ \text{(i)} -2\le x\le y\le z $ $ \text{(ii)} x+y+z=2/3 $ $ \text{(iii)} \frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2} =\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{3}{8} $ [i]Cristinel Mortici[/i]

2017 Germany, Landesrunde - Grade 11/12, 1

Tags: equation , polynomial , algebra

Solve the equation \[ x^5+x^4+x^3+x^2=x+1 \] in $\mathbb{R}$.

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

2024 Euler Olympiad, Round 1, 10

Tags: algebra , equation , euler

Find all $x$ that satisfy the following equation: \[ \sqrt {1 + \frac {20}x } = \sqrt {1 + 24x} + 2 \] [i]Proposed by Andria Gvaramia, Georgia [/i]

1966 IMO Longlists, 25

Tags: trigonometry , trigonometric identities , equation , algebra

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

1996 IMO Shortlist, 4

Tags: algebra , number theory , floor function , equation

Find all positive integers $ a$ and $ b$ for which \[ \left \lfloor \frac{a^2}{b} \right \rfloor \plus{} \left \lfloor \frac{b^2}{a} \right \rfloor \equal{} \left \lfloor \frac{a^2 \plus{} b^2}{ab} \right \rfloor \plus{} ab.\]

2018 Ramnicean Hope, 1

Tags: algebra , equation , monotone

Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $ [i]Mihai Neagu[/i]

2005 Estonia National Olympiad, 4

Tags: algebra , equation , trinomial

Find all pairs of real numbers $(x, y)$ that satisfy the equation $(x + y)^2 = (x + 3) (y - 3)$.

1987 USAMO, 1

Tags: algebra , diophantine equation , equation

Determine all solutions in non-zero integers $a$ and $b$ of the equation \[(a^2+b)(a+b^2) = (a-b)^3.\]

2013 India IMO Training Camp, 2

Tags: number theory , modular arithmetic , equation

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.

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]

1999 Moldova Team Selection Test, 5

Tags: equation

Let $a_1, a_2, \ldots, a_n$ be real numbers, but not all of them null. Show that the equation $$\sqrt{x+a_1}+\sqrt{x+a_2}+\ldots+\sqrt{x+a_n}=n\sqrt{x}$$ has at most one real solution.

2012 EGMO, 5

Tags: modular arithmetic , number theory , prime , equation , gcd

The numbers $p$ and $q$ are prime and satisfy \[\frac{p}{{p + 1}} + \frac{{q + 1}}{q} = \frac{{2n}}{{n + 2}}\] for some positive integer $n$. Find all possible values of $q-p$. [i]Luxembourg (Pierre Haas)[/i]

2015 Hanoi Open Mathematics Competitions, 8

Tags: algebra , equation

Solve the equation $(2015x -2014)^3 = 8(x-1)^3 + (2013x -2012)^3$

1989 IMO Shortlist, 2

Tags: algebra , geometry , area , equation , rectangle

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?