Olympiad problems

2012 Centers of Excellency of Suceava, 4

Solve in the reals the following system. $$ \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right. $$ [i]Gheorghe Marchitan[/i]

2013 Bogdan Stan, 4

Tags: equation , logarithm , algebra

Solve in the real numbers the equation $ 3^{\sqrt[3]{x-1}} \left( 1-\log_3^3 x \right) =1. $ [i]Ion Gușatu[/i]

1966 IMO Shortlist, 29

Tags: number theory , summation , equation

A given natural number $N$ is being decomposed in a sum of some consecutive integers. [b]a.)[/b] Find all such decompositions for $N=500.$ [b]b.)[/b] How many such decompositions does the number $N=2^{\alpha }3^{\beta }5^{\gamma }$ (where $\alpha ,$ $\beta $ and $\gamma $ are natural numbers) have? Which of these decompositions contain natural summands only? [b]c.)[/b] Determine the number of such decompositions (= decompositions in a sum of consecutive integers; these integers are not necessarily natural) for an arbitrary natural $N.$ [b]Note by Darij:[/b] The $0$ is not considered as a natural number.

2006 Mathematics for Its Sake, 1

Tags: equation , fractional part , algebra , floor

Solve in the set of real numbers the equation $$ 16\{ x \}^2-8x=-1, $$ where $ \{\} $ denotes the fractional part.

2014 Czech-Polish-Slovak Junior Match, 2

Tags: equation , algebra

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

2024 Euler Olympiad, Round 1, 10

Tags: equation , euler , algebra

Find all $x$ that satisfy the following equation: \[ \sqrt {1 + \frac {20}x } = \sqrt {1 + 24x} + 2 \] [i]Proposed by Andria Gvaramia, Georgia [/i]

2021 Turkey Team Selection Test, 6

Tags: algebra , equation , n-variable equation , construction

For which positive integers $n$, one can find real numbers $x_1,x_2,\cdots ,x_n$ such that $$\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}$$ and $i\leq x_i\leq 2i$ for all $i=1,2,\cdots ,n$ ?

2025 District Olympiad, P3

Tags: equation

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

1974 All Soviet Union Mathematical Olympiad, 194

Tags: algebra , equation

Find all the real $a,b,c$ such that the equality $$|ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x|+|y|+|z|$$ is valid for all the real $x,y,z$.

2023 China Northern MO, 3

Tags: equation , floor function , algebra

Find all solutions of the equation $$sin\pi \sqrt x+cos\pi \sqrt x=(-1)^{\lfloor \sqrt x \rfloor }$$

2015 District Olympiad, 3

Tags: equation , complex number , algebra

Solve in $ \mathbb{C} $ the following equation: $ |z|+|z-5i|=|z-2i|+|z-3i|. $

2017 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: equation , algebra , root

If $a$ is real number such that $x_1$ and $x_2$, $x_1\neq x_2$ , are real numbers and roots of equation $x_2-x+a=0$. Prove that $\mid {x_1}^2-{x_2}^2 \mid =1$ iff $\mid {x_1}^3-{x_2}^3 \mid =1$

2010 Laurențiu Panaitopol, Tulcea, 1

Tags: equation , trigonometric equation , floor function , floor , monotone functions , algebra

Solve in the real numbers the equation $ \arcsin x=\lfloor 2x \rfloor . $ [i]Petre Guțescu[/i]

1971 Bulgaria National Olympiad, Problem 2

Tags: algebra , equation

Prove that the equation $$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$ has no real solutions.

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 $0<x\le 1$

1966 IMO Shortlist, 31

Tags: function , parameterization , equation , algebra

Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation?

2017 Mathematical Talent Reward Programme, MCQ: P 1

Tags: equation , algebra

The number of real solutions of the equation $\left(\frac{9}{10}\right)^x=-3+x-x^2$ is [list=1] [*] 2 [*] 0 [*] 1 [*] None of these [/list]

1967 IMO Longlists, 48

Tags: algebra , equation , root , exponential equation

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

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: number theory , equation

Solve the equation: $$ \frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}=3$$ where $x$, $y$ and $z$ are integers

2012 EGMO, 5

Tags: number theory , modular arithmetic , equation , gcd , prime

The numbers $p$ and $q$ are prime and satisfy \[\frac{p}{{p + 1}} + \frac{{q + 1}}{q} = \frac{{2n}}{{n + 2}}\] for some positive integer $n$. Find all possible values of $q-p$. [i]Luxembourg (Pierre Haas)[/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.

1997 Croatia National Olympiad, Problem 1

Tags: algebra , equation

Let $n$ be a natural number. Solve the equation $$||\cdots|||x-1|-2|-3|-\ldots-(n-1)|-n|=0.$$

2019 Romania National Olympiad, 3

Tags: algebra , diophantine , equation

Prove that the number of solutions in $ \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right) $ of the parametric equation $$ \sqrt{x^2+y+n}+\sqrt{y^2+x+n} = z, $$ is greater than zero and finite, for nay natural number $ n. $

2004 Gheorghe Vranceanu, 4

Tags: equation , complex number , algebra

Prove that $ \left\{ (x,y)\in\mathbb{C}^2 |x^2+y^2=1 \right\} =\{ (1,0)\}\cup \left\{ \left( \frac{z^2-1}{z^2+1} ,\frac{2z}{z^2+1} \right) | z\in\mathbb{C}\setminus \{\pm \sqrt{-1}\} \right\} . $

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 $k = 1,2,\ldots,n-1$.