Olympiad problems

2010 IMO Shortlist, 2

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]

1989 IMO Longlists, 8

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

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 \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

1966 IMO Longlists, 12

Tags: number theory , decimal representation , equation

Find digits $x, y, z$ such that the equality \[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\] holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.

1977 IMO Shortlist, 11

Tags: algebra , recurrence relation , sequence , equation , floor function

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

2019 Purple Comet Problems, 15

Tags: number theory , equation

Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.

2007 Nicolae Coculescu, 2

Tags: newton s binom , diophantine , equation

[b]a)[/b] Prove that there exists two infinite sequences $ \left( a_n \right)_{n\ge 1} ,\left( b_n \right)_{n\ge 1} $ of nonnegative integers such that $ a_n>b_n $ and $ (2+\sqrt 3)^n =a_n (2+\sqrt 3) -b_n , $ for any natural numbers $ n. $ [b]b)[/b] Prove that the equation $ x^2-4xy+y^2=1 $ has infinitely many solutions in $ \mathbb{N}^2. $ [i]Florian Dumitrel[/i]

2006 Petru Moroșan-Trident, 2

Tags: system of equations , algebra , equation , monotone

Solve in the positive real numbers the following system. $$ \left\{\begin{matrix} x^y=2^3\\y^z=3^4\\z^x=2^4 \end{matrix}\right. $$ [i]Aurel Ene[/i]

2013 Taiwan TST Round 1, 5

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.

2005 Grigore Moisil Urziceni, 1

Tags: system of equations , algebra , equation

Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system: $$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$

2011 IMO Shortlist, 2

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]

2004 Thailand Mathematical Olympiad, 5

Tags: algebra , sum , equation , radical

Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$

1998 Croatia National Olympiad, Problem 1

Tags: complex number , algebra , equation

Solve the equation $2z^3-(5+6i)z^2+9iz+1-3i=0$ over $\mathbb C$ given that one of the solutions is real.

2015 Brazil Team Selection Test, 2

Tags: number theory , equation

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

2002 Baltic Way, 1

Tags: equation , symmetric , algebra

Solve the system of simultaneous equations \[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\] in real numbers.

2022 Cyprus JBMO TST, 1

Tags: algebra , floor function , equation

Determine all real numbers $x\in\mathbb{R}$ for which \[ \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor=x-2022. \] The notation $\lfloor z \rfloor$, for $z\in\mathbb{R}$, denotes the largest integer which is less than or equal to $z$. For example: \[\lfloor 3.98 \rfloor =3 \quad \text{and} \quad \lfloor 0.14 \rfloor =0.\]

2017 Junior Regional Olympiad - FBH, 3

Tags: equation , square root

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

1968 IMO, 2

Tags: number theory , decimal representation , digit , equation

Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

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

2004 Nicolae Coculescu, 2

Tags: quadratic , algebra , trigonometry , logarithm , equation , monotone

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: equation , real number , algebra

Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$

1971 IMO Longlists, 31

Tags: equation , root , polynomial , algebra

Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

1966 IMO Shortlist, 34

Tags: number theory , equation

Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$

1984 IMO Longlists, 32

Tags: geometry , triangle , concurrency , equation

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 $SD + SE + SF = 2(SA + SB + SC).$

2017 Saudi Arabia JBMO TST, 2

Tags: algebra , equation

Find all pairs of positive integers $(p; q) $such that both the equations $x^2- px + q = 0 $ and $ x^2 -qx + p = 0 $ have integral solutions.

1993 Moldova Team Selection Test, 4

Tags: equation

Solve in positive integers the following equation $$\left [\sqrt{1}\right]+\left [\sqrt{2}\right]+\left [\sqrt{3}\right]+\ldots+\left [\sqrt{x^2-2}\right]+\left [\sqrt{x^2-1}\right]=125,$$ where $[a]$ is the integer part of the real number $a$.