Olympiad problems

2017 Junior Regional Olympiad - FBH, 3

Find all real numbers $x$ such that: $$ \sqrt{\frac{x-7}{2015}}+\sqrt{\frac{x-6}{2016}}+\sqrt{\frac{x-5}{2017}}=\sqrt{\frac{x-2015}{7}}+\sqrt{\frac{x-2016}{6}}+\sqrt{\frac{x-2017}{5}}$$

2011 Morocco TST, 1

Tags: modular arithmetic , number theory , equation , lte lemma

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2001 Bosnia and Herzegovina Team Selection Test, 6

Tags: number theory , infinitely many solutions , equation

Prove that there exists infinitely many positive integers $n$ such that equation $(x+y+z)^3=n^2xyz$ has solution $(x,y,z)$ in set $\mathbb{N}^3$

1984 IMO Shortlist, 11

Tags: number theory , equation , polynomial , diophantine equation

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

1984 IMO Shortlist, 2

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

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 $4mn - m- n = p^2.$

1998 Junior Balkan MO, 3

Tags: induction , algebra , number theory , equation

Find all pairs of positive integers $ (x,y)$ such that \[ x^y \equal{} y^{x \minus{} y}. \] [i]Albania[/i]

1996 Tuymaada Olympiad, 7

Tags: algebra , equation , exponential , rational , exponential equation

In the set of all positive real numbers define the operation $a * b = a^b$ . Find all positive rational numbers for which $a * b = b * a$.

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.

1969 IMO Shortlist, 37

Tags: trigonometric equation , algebra , trigonometry , equation

$(HUN 4)$IMO2 If $a_1, a_2, . . . , a_n$ are real constants, and if $y = \cos(a_1 + x) +2\cos(a_2+x)+ \cdots+ n \cos(a_n + x)$ has two zeros $x_1$ and $x_2$ whose difference is not a multiple of $\pi$, prove that $y = 0.$

2011 Bosnia and Herzegovina Junior BMO TST, 1

Tags: positive integer , number theory , equation

Solve equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{5}-\frac{1}{xy}$, where $x$ and $y$ are positive integers.

1988 IMO Longlists, 63

Tags: number theory , relatively prime , quadratic , equation

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 \[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 \]

1955 Czech and Slovak Olympiad III A, 4

Tags: quadratic polynomial , equation

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.

EGMO 2017, 5

Tags: number theory , equation

Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$ a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple. b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple. [i]There are exactly $n$ factors in the product on the left hand side.[/i]

1941 Moscow Mathematical Olympiad, 079

Tags: algebra , equation , absolute value

Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.

1989 IMO Shortlist, 4

Tags: algebra , equation , polynomial , diophantine equation , rational roots

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

1986 Traian Lălescu, 1.1

Tags: equation , system of equations , algebra

Solve: $$ \left\{ \begin{matrix} x+y=\sqrt{4z -1} \\ y+z=\sqrt{4x -1} \\ z+x=\sqrt{4y -1}\end{matrix}\right. . $$

1984 Tournament Of Towns, (069) T3

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

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

1989 IMO Shortlist, 9

Tags: irrational number , number theory , equation , algebra

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

2014 BMT Spring, 1

Tags: algebra , equation

Find all real numbers $x$ such that $4^x-2^{x+2}+3=0$.

2006 Petru Moroșan-Trident, 1

Tags: equation , monotone , floor , algebra

Solve in the reals the equation $ 2^{\lfloor\sqrt[3]{x}\rfloor } =x. $ [i]Nedelcu Ion[/i]

2011 Brazil Team Selection Test, 2

Tags: number theory , equation , multiplication

Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that \[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\] [i]Proposed by Daniel Brown, Canada[/i]