Olympiad problems

2008 Brazil Team Selection Test, 2

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

2013 District Olympiad, 1

Tags: diophantine , algebra

Prove that the equation $$\frac{1}{\sqrt{x} +\sqrt{1006}}+\frac{1}{\sqrt{2012 -x} +\sqrt{1006}}=\frac{2}{\sqrt{x} +\sqrt{2012 -x}}$$ has $2013$ integer solutions.

2011 Hanoi Open Mathematics Competitions, 4

Tags: inequalities , polynomial , algebra

Prove that $1 + x + x^2 + x^3 + ...+ x^{2011} \ge 0$ for every $x \ge - 1$ .

2004 Nicolae Păun, 1

Tags: decomposition , injectivity , algebra , function

Prove that any function that maps the integers to themselves is a sum of any finite number of injective functions that map the integers to themselves. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

2005 Alexandru Myller, 4

Tags: algebra , superior algebra , polynomial , function

Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$. [i]Marian Andronache[/i]

1996 Romania National Olympiad, 4

Tags: algebra , geometric inequality , geometric inequalities , 3d geometry , inequalities , geometry

a) Let $AB CD$ be a regular tetrahedron. On the sides $AB$, $AC$ and $AD$, the points $M$, $N$ and $P$, are considered. Determine the volume of the tetrahedron $AMNP$ in terms of $x, y, z$, where $x=AM$, $y=AN$, $z=AP$. b) Show that for any real numbers $x, y, z, t, u, v \in (0, 1)$ : $$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$

2018 Peru Cono Sur TST, 2

Tags: algebra

Let $ x $ be a positive real number such that the numbers $ x^{-1} $, $ x $, and $ x^{2018} $ have the same fractional part: $$ \{x^{-1}\} = \{x\} = \{x^{2018}\}. $$ Prove that $ x = 1 $. [b]Note:[/b] If $ x $ is a real number, its fractional part is $ \{x\} = x - \lfloor x \rfloor $, where $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $ x $.

2018 Thailand TSTST, 1

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients satisfying: $P(2017) = 2016$ and $$(P(x)+1)^2=P(x^2+1).$$

1993 All-Russian Olympiad, 3

Tags: domain , functional equation , function , algebra

Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.

2004 China Team Selection Test, 1

Tags: quadratic , function , algebra

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

2019 PUMaC Algebra B, 1

Tags: number theory , algebra , relatively prime

Let $a,b$ be positive integers such that $a+b=10$. Let $\tfrac{p}{q}$ be the difference between the maximum and minimum possible values of $\tfrac{1}{a}+\tfrac{1}{b}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.

2012 China National Olympiad, 3

Tags: algebra , ceiling function

Prove for any $M>2$, there exists an increasing sequence of positive integers $a_1<a_2<\ldots $ satisfying: 1) $a_i>M^i$ for any $i$; 2) There exists a positive integer $m$ and $b_1,b_2,\ldots ,b_m\in\left\{ -1,1\right\}$, satisfying $n=a_1b_1+a_2b_2+\ldots +a_mb_m$ if and only if $n\in\mathbb{Z}/ \{0\}$.

Dumbest FE I ever created, 5.

Tags: algebra , algebruh , rutthee ahh function , function

Find all non decreasing function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ and $m,n \in \mathbb{N}_0$ such that $m+n \neq 0$ there exist $m',n' \in \mathbb{N}_0$ such that $m'+n'=m+n+1$ and $$f(f^m(x)+f^n(y))=f^{m'}(x)+f^{n'}(y)$$ . Note : $f^0(x)=x$ and $f^{n}(x)=f(f^{n-1}(x))$ for all $n \in \mathbb{N}$ . [hide=original]Find all non decreasing functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ $$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(x))+y$$ .[/hide]

2015 India PRMO, 2

Tags: polynomial , quadratic , algebra

$2.$ The equations $x^2-4x+k=0$ and $x^2+kx-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$

1999 IMO Shortlist, 4

Tags: coloring , ramsey theory , partition , algebra , combinatorics

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.

2017 Mathematical Talent Reward Programme, MCQ: P 3

Tags: polynomial , algebra

Let $p(x)=x^4-4x^3+2x^2+ax+b$. Suppose that for every root $\lambda$ of $p$, $\frac{1}{\lambda}$ is also a root of $p$. Then $a+b=$ [list=1] [*] -3 [*] -6 [*] -4 [*] -8 [/list]

2024 Singapore Senior Math Olympiad, Q5

Tags: algebra , sequence , bounded

Let $a_1,a_2,\dots$ be a sequence of positive numbers satisfying, for any positive integers $k,l,m,n$ such that $k+n=m+l$, $$\frac{a_k+a_n}{1+a_ka_n}=\frac{a_m+a_l}{1+a_ma_l}.$$Show that there exist positive numbers $b,c$ so that $b\le a_n\le c$ for any positive integer $n$.

1981 IMO Shortlist, 16

Tags: recurrence relation , limit , sequence , algebra , function

A sequence of real numbers $u_1, u_2, u_3, \dots$ is determined by $u_1$ and the following recurrence relation for $n \geq 1$: \[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\] Describe, with proof, the behavior of $u_n$ as $n \to \infty.$

2017 Bosnia Herzegovina Team Selection Test, 5

Tags: algebra

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

2019 Belarus Team Selection Test, 8.2

Tags: functional equation , function , algebra

Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions: 1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$; 2. $f$ takes all integer values. [i](I. Voronovich)[/i]

2024 Malaysian IMO Training Camp, 2

Tags: algebra

Let $k$ be a positive integer. Find all collection of integers $(a_1, a_2,\cdots, a_k)$ such that there exist a non-linear polynomial $P$ with integer coefficients, so that for all positive integers $n$ there exist a positive integer $m$ satisfying: $$P(n+a_1)+P(n+a_2)+...+P(n+a_k)=P(m)$$ [i]Proposed by Ivan Chan Kai Chin[/i]

2008 Bulgarian Autumn Math Competition, Problem 12.1

Tags: algebra , interval , parameterization , inequalities

Determine the values of the real parameter $a$, such that the solutions of the system of inequalities $\begin{cases} \log_{\frac{1}{3}}{(3^{x}-6a)}+\frac{2}{\log_{a}{3}}<x-3\\ \log_{\frac{1}{3}}{(3^{x}-18)}>x-5\\ \end{cases}$ form an interval of length $\frac{1}{3}$.

2013 Hanoi Open Mathematics Competitions, 12

Tags: quadratic , root , inequalities , algebra

If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$, prove that the equation $f(x) = 2x^2 - 1$ has two real roots.

1975 Polish MO Finals, 6

Tags: algebra

On the interval $[0,1]$ are given functions $S(x) = 1 - x$ and $T(x) = x/2$. Does there exist a function of the form $f = g_1\circ g_2\circ ... \circ g_n$, where $n \in N$ and each $g_k$ is either $S(x)$ or $T(x)$, such that $$f\left(\frac12\right)=\frac{1975}{2^{1975}} \, ?$$

2025 Macedonian Balkan MO TST, 3

Tags: algebra , functional equation

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy \[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\] for every $x, y \in \mathbb{R}$, and $f(kx) > kf(x)$ for every $x \in \mathbb{R}$ and $k \in \mathbb{R}$, such that $k > 1$.