Olympiad problems

1949-56 Chisinau City MO, 55

Tags: algebra , equation

Find the real roots of the equation $$(5-x)^4+ (x-2)^ 4 = 17$$ and the real roots of a more general equation $$(a - x) ^4+ (x - b)^4 = c$$

1986 Traian Lălescu, 2.1

Tags: algebra , parameterization , parametric equation , equation

Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation $$ 1=2mx(2x-1)(2x-2)(2x-3) $$ are real.

2005 India IMO Training Camp, 2

Tags: equation , divisor , number theory

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 $n$.

2016 Kosovo Team Selection Test, 1

Tags: equation

Solve equation in real numbers $\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1$

2014 Czech-Polish-Slovak Junior Match, 2

Tags: equation , algebra

Solve the equation $a + b + 4 = 4\sqrt{a\sqrt{b}}$ in real numbers

2014 JBMO Shortlist, 1

Tags: algebra , floor function , floor , equation , positive real

Solve in positive real numbers: $n+ \lfloor \sqrt{n} \rfloor+\lfloor \sqrt[3]{n} \rfloor=2014$

1982 IMO Longlists, 31

Tags: number theory , equation , diophantine equation , divisibility

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

1984 IMO Shortlist, 15

Tags: triangle , concurrency , equation , geometry

Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$. (a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$ (b) Prove that $SD + SE + SF = 2(SA + SB + SC).$

1984 IMO Longlists, 25

Tags: number theory , equation , product , perfect square

Prove that the product of five consecutive positive integers cannot be the square of an integer.

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 \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

2015 Finnish National High School Mathematics Comp, 1

Tags: algebra , radical , equation

Solve the equation $\sqrt{1+\sqrt {1+x}}=\sqrt[3]{x}$ for $x \ge 0$.

2006 Estonia National Olympiad, 4

Tags: floor function , equation , algebra

Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

2013 India IMO Training Camp, 2

Tags: modular arithmetic , number theory , 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.

2016 Kyiv Mathematical Festival, P1

Tags: equation , algebra

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

1966 IMO Shortlist, 9

Tags: trigonometry , algebra , equation , trigonometric 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.

1978 Chisinau City MO, 161

Tags: equation , algebra , parameter

For what real values of $a$ the equation $\frac{2^{2x}}{2^{2x}+2^{x+1}+1}+a \frac{2^x}{2^x+1}+(a-1) = 0$ has a single root ?

2004 Nicolae Coculescu, 3

Tags: hermite , algebra , fractional , floor , equation

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.

2022 Moldova EGMO TST, 1

Tags: equation

Let $n$ be a positive integer. Solve the equation in $\mathbb{R}$ $$\sqrt[2n+1]{x}+\sqrt[2n+1]{x+1}+\sqrt[2n+1]{x+2}+\dots+\sqrt[2n+1]{x+n}=0.$$

2011 Hanoi Open Mathematics Competitions, 9

Tags: algebra , equation

Solve the equation $1 + x + x^2 + x^3 + ... + x^{2011} = 0$.

2012 District Olympiad, 1

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

Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.

1997 IMO, 5

Tags: equation , prime factorization , number theory

Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

2000 District Olympiad (Hunedoara), 2

Tags: floor , algebra , equation , iteration , number theory

[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation $$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$ has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $ [b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.

2004 Gheorghe Vranceanu, 4

Tags: diophantine , equation

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

1974 Czech and Slovak Olympiad III A, 3

Tags: digit , sum of digits , permutation , equation , diophantine equation , number theory

Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]

2019 Finnish National High School Mathematics Comp, 1

Tags: algebra , equation

Solve $x(8\sqrt{1-x}+\sqrt{1+x}) \le 11\sqrt{1+x}-16\sqrt{1-x}$ when $0<x\le 1$