Olympiad problems

2011 Greece JBMO TST, 1

a) Let $n$ be a positive integer. Prove that $ n\sqrt {x-n^2}\leq \frac {x}{2}$ , for $x\geq n^2$. b) Find real $x,y,z$ such that: $ 2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z$

2011 District Olympiad, 1

Tags: equation , increasing function , easy , district olympiad , algebra

Let $ a,b,c $ be three positive numbers. Show that the equation $$ a^x+b^x=c^x $$ has, at most, one real solution.

2019 LIMIT Category A, Problem 10

Tags: algebra , equation

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

2011 Belarus Team Selection Test, 2

Tags: multiplication , number theory , equation

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]

2015 Bosnia and Herzegovina Junior BMO TST, 1

Tags: equation , diophantine equation , algebra , number theory

Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers

2008 India Regional Mathematical Olympiad, 2

Tags: system of equations , equation , algebra

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

1967 Czech and Slovak Olympiad III A, 1

Tags: complex number , equation , algebra

Find all triplets $(a,b,c)$ of complex numbers such that the equation \[x^4-ax^3-bx+c=0\] has $a,b,c$ as roots.

2006 Cezar Ivănescu, 2

Tags: complex number , conjugate , equation , induction , trigonometry , algebra

[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $ [b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that $$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$ for any nonnegative integer $ n. $

1970 IMO Longlists, 20

Tags: 3d geometry , tetrahedron , volume , geometry , equation

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

2006 Belarusian National Olympiad, 2

Tags: algebra , equation

Find all triples $(x, y,z)$ such that $x, y, z \in (0,1)$ and $$\left(x+\frac{1}{2x}-1\right) \left(y+\frac{1}{2y}-1\right) \left(z+\frac{1}{2z}-1\right) = \left(1-\frac{xy}{z}\right)\left(1-\frac{yz}{x}\right)\left(1-\frac{zx}{y}\right)$$ (D. Bazylev)

1986 Traian Lălescu, 1.1

Tags: equation , algebra , system of equations

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

1994 All-Russian Olympiad, 1

Tags: equation , radical , algebra

Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1$, then $x+y = 0$.

2004 German National Olympiad, 1

Tags: real , system of equations , equation , algebra

Find all real numbers $x,y$ satisfying the following system of equations \begin{align*} x^4 +y^4 & =17(x+y)^2 \\ xy & =2(x+y). \end{align*}

2006 Petru Moroșan-Trident, 1

Tags: equation , floor function , algebra

Let be a natural number $ n\ge 3. $ Solve the equation $ \lfloor x/n \rfloor =\lfloor x-n \rfloor $ in $ \mathbb{R} . $ [i]Constantin Nicolau[/i]

1989 IMO Longlists, 8

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

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

2003 Swedish Mathematical Competition, 3

Tags: floor function , equation , algebra

Find all real solutions $x$ of the equation $$\lfloor x^2-2 \rfloor +2 \lfloor x \rfloor = \lfloor x \rfloor ^2. $$ .

2018 Dutch BxMO TST, 5

Tags: equation , algebra

Let $n$ be a positive integer. Determine all positive real numbers $x$ satisfying $nx^2 +\frac{2^2}{x + 1}+\frac{3^2}{x + 2}+...+\frac{(n + 1)^2}{x + n}= nx + \frac{n(n + 3)}{2}$

1969 IMO Longlists, 37

Tags: trigonometric equation , algebra , equation , trigonometry

$(HUN 4)$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 $y = 0.$

2022 Moldova EGMO TST, 1

Tags: equation

Let $n$ be a positive integer. Solve the equation in $\mathbb{R}$ $$\sqrt[2n+1]{x}+\sqrt[2n+1]{x+1}+\sqrt[2n+1]{x+2}+\dots+\sqrt[2n+1]{x+n}=0.$$

2014 Czech-Polish-Slovak Junior Match, 2

Tags: equation , algebra

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

2016 Macedonia National Olympiad, Problem 1

Tags: equation , number theory

Solve the equation in the set of natural numbers $1+x^z + y^z = LCM(x^z,y^z)$

2004 Nicolae Coculescu, 3

Tags: equation , hermite , floor , fractional , algebra

Prove the identity $ \frac{n-1}{2}=\sum_{k=1}^n \left\{ \frac{m+k-1}{n} \right\} , $ where $ n\ge 2, m $ are natural numbers, and $ \{\} $ denotes the fractional part.

2001 Bosnia and Herzegovina Team Selection Test, 6

Tags: equation , infinitely many solutions , number theory

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 $\mathbb{N}^3$

1980 All Soviet Union Mathematical Olympiad, 301

Tags: equation , floor function , algebra

Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$. ($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)

1983 Spain Mathematical Olympiad, 4

Tags: parameter , equation , algebra

Determine the number of real roots of the equation $$16x^5 - 20x^3 + 5x + m = 0.$$