Olympiad problems

2008 India Regional Mathematical Olympiad, 2

Solve the system of equation $$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$

2011 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , equation

For positive integers $a$ and $b$ holds $a^3+4a=b^2$. Prove that $a=2t^2$ for some positive integer $t$

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: \[\lfloor 3.98 \rfloor =3 \quad \text{and} \quad \lfloor 0.14 \rfloor =0.\]

1939 Moscow Mathematical Olympiad, 046

Tags: radical , parameter , equation , algebra

Solve the equation $\sqrt{a-\sqrt{a+ x}} = x$ for $x$.

1985 All Soviet Union Mathematical Olympiad, 414

Tags: algebra , equation , recurrence relation

Solve the equation ("$2$" encounters $1985$ times): $$\dfrac{x}{2+ \dfrac{x}{2+\dfrac{x}{2+... \dfrac{x}{2+\sqrt {1+x}}}}}=1$$

2004 Thailand Mathematical Olympiad, 4

Tags: algebra , radical , equation

Find all real solutions $x$ to the equation $$x =\sqrt{x -\frac{1}{x}} +\sqrt{1 -\frac{1}{x}}$$

1968 IMO Shortlist, 6

Tags: real root , algebra , polynomial , equation , root

If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation \[\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n\] has at least $n - 1$ real roots.

2020 Canadian Mathematical Olympiad Qualification, 8

Tags: radical , rational , equation , algebra

Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$

2009 Brazil Team Selection Test, 1

Tags: algebra , number theory , equation

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

2007 Nicolae Coculescu, 2

Tags: equation , trigonometry , algebra

Solve in the real numbers the equation $ \cos \left( \pi\log_3 (x+6) \right)\cdot \cos \left( \pi \log_3 (x-2) \right) =1. $

1992 IMO Longlists, 14

Tags: equation , divisibility , modular arithmetic , number theory

Integers $a_1, a_2, . . . , a_n$ satisfy $|a_k| = 1$ and \[ \sum_{k=1}^{n} a_ka_{k+1}a_{k+2}a_{k+3} = 2,\] where $a_{n+j} = a_j$. Prove that $n \neq 1992.$

2012 Brazil Team Selection Test, 3

Tags: equation , sequence , algebra

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]

2019 LIMIT Category A, Problem 10

Tags: algebra , equation

Number of solutions of the equation $3^x+4^x=8^x$ in reals is $\textbf{(A)}~0$ $\textbf{(B)}~1$ $\textbf{(C)}~2$ $\textbf{(D)}~\infty$

2017 District Olympiad, 2

Tags: equation , diophantine , parameterization

Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $ [b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $ [b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $

2016 India National Olympiad, P5

Tags: geometry , inradius , trigonometry , equation

Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$, prove that \[ \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}. \]

2016 Kosovo Team Selection Test, 1

Tags: equation

Solve equation in real numbers $\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{…+\sqrt{4^nx+3}}}}}-\sqrt{x}=1$

2000 Bosnia and Herzegovina Team Selection Test, 1

Tags: algebra , root , equation

Find real roots $x_1$, $x_2$ of equation $x^5-55x+21=0$, if we know $x_1\cdot x_2=1$

2019 EGMO, 1

Tags: algebra , equation

Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and $$a^2b + c = b^2c + a = c^2a + b.$$

2018 Ramnicean Hope, 1

Tags: equation , monotone , algebra

Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $ [i]Mihai Neagu[/i]

2012 District Olympiad, 2

Tags: function , equation , algebra

[b]a)[/b] Solve in $ \mathbb{R} $ the equation $ 2^x=x+1. $ [b]b)[/b] If a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has the property that $$ (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} , $$ then $ f(0)+f(1)=1. $

2020 Abels Math Contest (Norwegian MO) Final, 3

Tags: algebra , solutions , real solutions , equation

Show that the equation $x^2 \cdot (x - 1)^2 \cdot (x - 2)^2 \cdot ... \cdot (x - 1008)^2 \cdot (x- 1009)^2 = c$ has $2020$ real solutions, provided $0 < c <\frac{(1009 \cdot1007 \cdot ... \cdot 3\cdot 1)^4}{2^{2020}}$ .

2003 Gheorghe Vranceanu, 1

Tags: floor function , algebra , equation

Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $

2017 China Team Selection Test, 1

Tags: combinatorics , equation , generating functions

Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

1984 IMO Shortlist, 15

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

2023 District Olympiad, P1

Tags: algebra , equation

Determine all real numbers $x{}$ satisfying $2^{x-1}+2^{1/\sqrt{x}}=3$.