Olympiad problems

2014 Bosnia And Herzegovina - Regional Olympiad, 1

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

2018 VJIMC, 1

Tags: equations , real numbers , powers

Find all real solutions of the equation \[17^x+2^x=11^x+2^{3x}.\]

2018 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , identity , real numbers

if $a$, $b$ and $c$ are real numbers such that $(a-b)(b-c)(c-a) \neq 0$, prove the equality: $\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca$

Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the mass of any material (the mass can be any real number between $0$ and $20.2$ kg) with an error of at most $10$ g.

1989 Bundeswettbewerb Mathematik, 2

Tags: Inequality , algebra , real numbers

Find all pairs $(a,b)$ of real numbers such that $$|\sqrt{1-x^2 }-ax-b| \leq \frac{\sqrt{2} -1}{2}$$ holds for all $x\in [0,1]$.

2007 German National Olympiad, 5

Tags: arithmetic sequence , Sets , finite , algebra , real numbers

Determine all finite sets $M$ of real numbers such that $M$ contains at least $2$ numbers and any two elements of $M$ belong to an arithmetic progression of elements of $M$ with three terms.

2018 Bundeswettbewerb Mathematik, 2

Tags: algebra , equation , real numbers , floor function , function

Find all real numbers $x$ satisfying the equation \[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]

2017 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: identity , real numbers , algebra

Let $a$, $b$ and $c$ be real numbers such that $abc(a+b)(b+c)(c+a)\neq0$ and $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}$ Prove that $\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017$

2003 Bosnia and Herzegovina Team Selection Test, 6

Tags: algebra , real numbers

Let $a$, $b$ and $c$ be real numbers such that $\mid a \mid >2$ and $a^2+b^2+c^2=abc+4$. Prove that numbers $x$ and $y$ exist such that $a=x+\frac{1}{x}$, $b=y+\frac{1}{y}$ and $c=xy+\frac{1}{xy}$.

2005 Spain Mathematical Olympiad, 2

Tags: inequalities , real numbers , minimum

Let $r,s,u,v$ be real numbers. Prove that: $$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$

2020 Dürer Math Competition (First Round), P3

Tags: algebra , real numbers , counting , Extremal combinatorics

At least how many non-zero real numbers do we have to select such that every one of them can be written as a sum of $2019$ other selected numbers and a) the selected numbers are not necessarily different? b) the selected numbers are pairwise different?

2024 Abelkonkurransen Finale, 2b

Tags: algebra , functional equation , algebra proposed , real numbers , involution

Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying \[xf(f(x)+y)=f(xy)+x^2\] for all $x,y \in \mathbb{R}$.

1954 Putnam, A6

Tags: Putnam , Sequence , real numbers

Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that $$u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots$$ Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$.