Olympiad problems

2012 German National Olympiad, 4

Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$, then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$

2017 Vietnamese Southern Summer School contest, Problem 1

Tags: convergence , sequence , real number , analysis

Given a real number $a$ and a sequence $(x_n)_{n=1}^\infty$ defined by: $$\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right.$$ for all positive integers $n$. 1. For $a=0$, prove that $(x_n)$ converges. 2. Determine the largest possible value of $a$ such that $(x_n)$ converges.

2008 India Regional Mathematical Olympiad, 3

Tags: inequality , positive integer , real number , inequalities

Prove that for every positive integer $n$ and a non-negative real number $a$, the following inequality holds: $$n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).$$

2020 Dürer Math Competition (First Round), P3

Tags: counting , extremal combinatorics , real number , algebra

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?

2018 VJIMC, 1

Tags: equation , power , real number

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

2024 Abelkonkurransen Finale, 2b

Tags: involution , real number , functional equation , algebra

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

2005 Spain Mathematical Olympiad, 2

Tags: minimum , real number , inequalities

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

2022 JBMO TST - Turkey, 6

Tags: real number , inequality

Let $c$ be a real number. If the inequality $$f(c)\cdot f(-c)\ge f(a)$$ holds for all $f(x)=x^2-2ax+b$ where $a$ and $b$ are arbitrary real numbers, find all possible values of $c$.

1954 Putnam, A6

Tags: real number , sequence

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 $k$.

2019 Polish MO Finals, 5

Tags: real number , algebra , inequalities

The sequence $a_1, a_2, \ldots, a_n$ of positive real numbers satisfies the following conditions: \begin{align*} \sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1 \end{align*} for all $i\in \lbrace 1, 2, \ldots, n \rbrace$, where $a_0$ is an integer. Prove that \begin{align*} n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i} \end{align*}

2018 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: real number , identity , algebra

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

2016 Polish MO Finals, 5

Tags: algebra , real number , sum of squares

There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions: $1$. $a< \frac{p}{q} < \frac{r}{s} < b$. $2.$ $p^2+q^2=r^2+s^2$.

KoMaL A Problems 2017/2018, A. 712

Tags: sequence , real number , algebra

We say that a strictly increasing positive real sequence $a_1,a_2,\cdots $ is an [i]elf sequence[/i] if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\cdots$. Furthermore, we say that $a_n$ is a [i]hat[/i] if $a_{n-i}+a_{n+i}<2a_n$ for $\displaystyle 1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?

1999 Czech and Slovak Match, 3

Tags: real number , congruent triangles , combinatorics

Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$.

1989 Bundeswettbewerb Mathematik, 2

Tags: real number , inequality , algebra

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 $x\in [0,1]$.

2023 Brazil National Olympiad, 4

Tags: real number , algebra , inequalities , system of equations

Let $x, y, z$ be three real distinct numbers such that $$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.

2015 Germany Team Selection Test, 1

Tags: real number , integer , root , algebra , polynomial

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

1994 Bundeswettbewerb Mathematik, 1

Tags: algebra , pigeonhole , decimal representation , real number , digit

Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.

2007 German National Olympiad, 5

Tags: arithmetic sequence , algebra , real number , set , finite

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 number , function , floor function

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

2014 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , real number

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

2024 Brazil National Olympiad, 5

Tags: functional equation , real number , algebra

Let $ \mathbb{R} $ be the set of real numbers. Determine all functions $ f: \mathbb{R} \to \mathbb{R} $ such that, for any real numbers $ x $ and $ y $, \[ f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1. \]

2016 Indonesia MO, 6

Tags: real number

For a quadrilateral $ABCD$, we call a square $amazing$ if all of its sides(extended if necessary) pass through distinct vertices of $ABCD$(no side passing through 2 vertices). Prove that for an arbitrary $ABCD$ such that its diagonals are not perpendicular, there exist at least 6 $amazing$ squares

2006 Spain Mathematical Olympiad, 1

Tags: algebra , functional equation , real number

Find all the functions $f:(0,+\infty) \to R $ that satisfy the equation $$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$ for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$

2023 OMpD, 1

Tags: functional equation , function , algebra , real number

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f(x)(x+f(f(y))) = f(x^2)+xf(y)$$