Olympiad problems

2020 Middle European Mathematical Olympiad, 4#

Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that $$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$

2004 Nicolae Coculescu, 2

Tags: floor function , equation , algebra

Let be a natural number $ n\ge 2. $ Find the real numbers $ a $ that satisfy the equation $$ \lfloor nx \rfloor =\sum_{k=1}^{n} \lfloor x+(k-1)a \rfloor , $$ for any real numbers $ x. $ [i]Marius Perianu[/i]

2015 Hanoi Open Mathematics Competitions, 13

Tags: algebra , rational , equation

Give rational numbers $x, y$ such that $(x^2 + y^2 - 2) (x + y)^2 + (xy + 1)^2 = 0 $ Prove that $\sqrt{1 + xy}$ is a rational number.

2013 Poland - Second Round, 4

Tags: algebra , equation

Solve equation $(x^4 + 3y^2)\sqrt{|x + 2| + |y|}=4|xy^2|$ in real numbers $x$, $y$.

2021 Brazil National Olympiad, 5

Tags: number theory , equation , square , triplets

Find all triples of non-negative integers $(a, b, c)$ such that \[a^{2}+b^{2}+c^{2} = a b c+1.\]

2004 IMO Shortlist, 2

Tags: function , number theory , greatest common divisor , equation

The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\] a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$. b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution. c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.

1984 IMO Shortlist, 10

Tags: number theory , equation , product , perfect square

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

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 ?

1991 IMO Shortlist, 17

Tags: modular arithmetic , algebra , number theory , equation

Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$

1989 IMO Shortlist, 3

Tags: calculus , algebra , rectangle , equation , area

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. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

2023 Moldova EGMO TST, 5

Tags: equation , system of equations

Find all pairs of real numbers $(x, y)$, that satisfy the system of equations: $$\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.$$

2016 Kosovo National Mathematical Olympiad, 2

Tags: equation

Find all real numbers $x$ which satisfied $|2x+1|+|x-1|=2-x$ .

1983 IMO Longlists, 63

Tags: trigonometry , number theory , equation , trigonometric identities , trigonometric equation

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

1967 IMO Longlists, 48

Tags: algebra , equation , root , exponential equation

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

1962 IMO Shortlist, 4

Tags: trigonometry , algebra , equation , trigonometric equation

Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$

1966 IMO Longlists, 48

Tags: algebra , equation , number theory , root , parametric equation

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

1965 Czech and Slovak Olympiad III A, 3

Tags: algebra , parameter , radical , equation , real root

Find all real roots $x$ of the equation $$\sqrt{x^2-2x-1}+\sqrt{x^2+2x-1}=p,$$ where $p$ is a real parameter.

2001 IMO Shortlist, 5

Tags: inequalities , algebra , recurrence relation , equation , integer sequence

Find all positive integers $a_1, a_2, \ldots, a_n$ such that \[ \frac{99}{100} = \frac{a_0}{a_1} + \frac{a_1}{a_2} + \cdots + \frac{a_{n-1}}{a_n}, \] where $a_0 = 1$ and $(a_{k+1}-1)a_{k-1} \geq a_k^2(a_k - 1)$ for $k = 1,2,\ldots,n-1$.

1966 IMO Longlists, 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.

2006 Cezar Ivănescu, 2

Tags: complex number , induction , trigonometry , conjugate , algebra , equation

[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $ [b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that $$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$ for any nonnegative integer $ n. $

2010 JBMO Shortlist, 3

Tags: algebra , equation

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

2018 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: algebra , equation , floor function , frac function

Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number

2014 Korea National Olympiad, 4

Tags: function , algebra , ratio , equation

Prove that there exists a function $f : \mathbb{N} \rightarrow \mathbb{N}$ that satisfies the following (1) $\{f(n) : n\in\mathbb{N}\}$ is a finite set; and (2) For nonzero integers $x_1, x_2, \ldots, x_{1000}$ that satisfy $f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)$, then $x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0$.

1978 USAMO, 3

Tags: induction , algebra , equation , additive number theory

An integer $n$ will be called [i]good[/i] if we can write \[n=a_1+a_2+\cdots+a_k,\] where $a_1,a_2, \ldots, a_k$ are positive integers (not necessarily distinct) satisfying \[\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1.\] Given the information that the integers 33 through 73 are good, prove that every integer $\ge 33$ is good.

2022 Thailand Online MO, 1

Tags: algebra , equation

Determine, with proof, all triples of real numbers $(x,y,z)$ satisfying the equations $$x^3+y+z=x+y^3+z=x+y+z^3=-xyz.$$