Olympiad problems

1959 IMO Shortlist, 2

For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

2013 India IMO Training Camp, 2

Tags: number theory , equation , modular arithmetic

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.

1982 IMO, 1

Tags: diophantine equation , divisibility , equation , number theory

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 $n=2891$.

1988 IMO Shortlist, 22

Tags: quadratic , equation , relatively prime , number theory

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

2014 BMT Spring, 1

Tags: equation , algebra

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

2015 German National Olympiad, 2

Tags: equation , number theory

A positive integer $n$ is called [i]smooth[/i] if there exist integers $a_1,a_2,\dotsc,a_n$ satisfying \[a_1+a_2+\dotsc+a_n=a_1 \cdot a_2 \cdot \dotsc \cdot a_n=n.\] Find all smooth numbers.

1959 IMO, 2

Tags: square root , equation , algebra

For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

2025 District Olympiad, P3

Tags: equation

Determine all positive real numbers $a,b,c,d$ such that $a+b+c+d=80$ and $$a+\frac{b}{1+a}+\frac{c}{1+a+b}+\frac{d}{1+a+b+c}=8.$$

1998 Junior Balkan MO, 3

Tags: equation , induction , algebra , number theory

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

2013 IFYM, Sozopol, 8

Tags: set , equation , number theory

The irrational numbers $\alpha ,\beta ,\gamma ,\delta$ are such that $\forall$ $n\in \mathbb{N}$ : $[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]$. Is it true that the sets $\{ \alpha ,\beta \}$ and $\{ \gamma ,\delta \}$ are equal?

1985 Traian Lălescu, 2.1

Tags: floor , equation , algebra

Solve $ \quad 5\lfloor x^2\rfloor -2\lfloor x\rfloor +2=0. $

1989 IMO Shortlist, 5

Tags: root , diophantine equation , equation , algebra , polynomial

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

1998 Akdeniz University MO, 1

Tags: perfect square , equation , number theory

Prove that, for $k \in {\mathbb Z^+}$ $$k(k+1)(k+2)(k+3)$$ is not a perfect square.

2018 Junior Regional Olympiad - FBH, 1

Tags: equation , buying , splitting money

Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\$$. How much did the ball cost?

1984 IMO Shortlist, 15

Tags: triangle , equation , concurrency , 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 $SD + SE + SF = 2(SA + SB + SC).$

2019 Ramnicean Hope, 1

Tags: equation , algebra , monotone

Solve in the reals the equation $ \sqrt[3]{x^2-3x+4} +\sqrt[3]{-2x+2} +\sqrt[3]{-x^2+5x+2} =2. $ [i]Ovidiu Țâțan[/i]

2023 Moldova EGMO TST, 5

Tags: equation , system of equations

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

2006 Petru Moroșan-Trident, 1

Tags: equation , floor , algebra , monotone

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

1980 IMO, 19

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

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

1986 Traian Lălescu, 2.1

Tags: equation , parametric equation , parameterization , algebra

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.

2017 Junior Regional Olympiad - FBH, 4

Tags: equation , remainder

If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?

1996 Tuymaada Olympiad, 5

Tags: radical , equation , algebra

Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$

2013 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , equation

Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$

2017 Junior Regional Olympiad - FBH, 5

Tags: equation

Fathers childhood lasted for one sixth part of his life, and he married one $8$th after that and he immediately left to army. When one $12$th of his life passed, father returned from the army and $5$ years after he got a son. Son who lived for one half of fathers years, died $4$ years before his father. How many years lived his father, and how many years he had when his son was born?

1961 IMO, 3

Tags: equation , trigonometric equation , algebra , trigonometry

Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.