Olympiad problems

2007 Italy TST, 3

Tags: function , algebra , polynomial , algebra proposed , functional equation

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

2005 Germany Team Selection Test, 1

Tags: inequalities , algebra proposed , algebra

Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.

2003 Iran MO (3rd Round), 17

Tags: algebra , binomial theorem , algebra proposed

A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/,$ \sqrt{\mbox{}}$) How can you find $ 3^{\sqrt{2}}$ with accuracy of 6 digits.

1998 Spain Mathematical Olympiad, 2

Tags: function , induction , algebra proposed , algebra

Find all strictly increasing functions $f:\mathbb{N}\rightarrow\mathbb{N}$ that satisfy \[f(n+f(n))=2f(n)\quad\text{for all}\ n\in\mathbb{N} \]

1997 Vietnam National Olympiad, 3

Tags: function , number theory , relatively prime , algebra proposed , algebra

Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying: (i) $ f(1) \equal{} 1$ (ii) $ f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997$ for every natural numbers n.

2024 Dutch IMO TST, 3

Tags: inequalities , inequalities proposed , algebra proposed , algebra

Let $a,b,c$ be real numbers such that $0 \le a \le b \le c$ and $a+b+c=1$. Show that \[ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.\]

2010 District Olympiad, 4

Tags: search , function , algebra proposed , algebra

Consider the sequence $ a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1$, where $ z\in \mathbb{C}^*$ is given. i) Prove that if $ a_1>2$, then: \[ a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^*\] ii) Prove that if there is a $ k\in \mathbb{N}^*$ such that $ a_k\le 2$, then $ a_1\le 2$.

2012 ELMO Shortlist, 3

Tags: algebra , polynomial , function , algebra proposed

Prove that any polynomial of the form $1+a_nx^n + a_{n+1}x^{n+1} + \cdots + a_kx^k$ ($k\ge n$) has at least $n-2$ non-real roots (counting multiplicity), where the $a_i$ ($n\le i\le k$) are real and $a_k\ne 0$. [i]David Yang.[/i]

2014 Middle European Mathematical Olympiad, 2

Tags: inequalities , function , algebra proposed , algebra

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ xf(xy) + xyf(x) \ge f(x^2)f(y) + x^2y \] holds for all $x,y \in \mathbb{R}$.

1996 Baltic Way, 13

Tags: function , algebra proposed , algebra

Consider the functions $f$ defined on the set of integers such that \[f(x)=f(x^2+x+1)\] for all integer $x$. Find $(a)$ all even functions, $(b)$ all odd functions of this kind.

2006 Pre-Preparation Course Examination, 1

Tags: algebra , polynomial , algebra proposed

Find out wich of the following polynomials are irreducible. a) $t^4+1$ over $\mathbb{R}$; b) $t^4+1$ over $\mathbb{Q}$; c) $t^3-7t^2+3t+3$ over $\mathbb{Q}$; d) $t^4+7$ over $\mathbb{Z}_{17}$; e) $t^3-5$ over $\mathbb{Z}_{11}$; f) $t^6+7$ over $\mathbb{Q}(i)$.

2008 Stars Of Mathematics, 1

Tags: floor function , function , algebra proposed , algebra

Prove that for any positive integer $m$, the equation \[ \frac{n}{m}\equal{}\lfloor\sqrt[3]{n^2}\rfloor\plus{}\lfloor\sqrt{n}\rfloor\plus{}1\] has (at least) a positive integer solution $n_{m}$. [i]Cezar Lupu & Dan Schwarz[/i]

2010 Mediterranean Mathematics Olympiad, 1

Tags: algebra , system of equations , algebra proposed

Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[ x^{2}-yz-zu-yu=a\] \[ y^{2}-zu-ux-xz=b\] \[ z^{2}-ux-xy-yu=c\] \[ u^{2}-xy-yz-zx=d\]

2012 Macedonia National Olympiad, 3

Tags: function , floor function , inequalities , induction , ceiling function , algebra , algebra proposed

Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions: $f(x+y) < f(x) + f(y)$ $f(f(x)) = \lfloor {x} \rfloor + 2$

2012 ELMO Shortlist, 5

Tags: algebra , polynomial , pigeonhole principle , number theory , relatively prime , algebra proposed

Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.) [i]David Yang.[/i]

2010 All-Russian Olympiad Regional Round, 9.1

Tags: quadratics , algebra , polynomial , algebra proposed

Three quadratic polynomials $f_1(x) = x^2+2a_1x+b_1$, $f_2(x) = x^2+2a_2x+b_2$, $f_3(x) = x^2 + 2a_3x + b_3$ are such that $a_1a_2a_3 = b_1b_2b_3 > 1$. Prove that at least one polynomial has two distinct roots.

2013 Korea - Final Round, 2

Tags: function , induction , algebra proposed , algebra

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions. (a) $ f(x) \ge 0 $ for all $ x \in \mathbb{R} $. (b) For $ a, b, c, d \in \mathbb{R} $ with $ ab + bc + cd = 0 $, equality $ f(a-b) + f(c-d) = f(a) + f(b+c) + f(d) $ holds.

2014 District Olympiad, 1

Tags: conics , ellipse , complex numbers , algebra proposed , algebra

Solve for $z\in \mathbb{C}$ the equation : \[ |z-|z+1||=|z+|z-1|| \]

1990 IMO Longlists, 73

Tags: function , induction , algebra proposed , algebra

Let $\mathbb Q$ be the set of all rational numbers and $\mathbb R$ be the set of real numbers. Function $f: \mathbb Q \to \mathbb R$ satisfies the following conditions: (i) $f(0) = 0$, and for any nonzero $a \in Q, f(a) > 0.$ (ii) $f(x + y) = f(x)f(y) \qquad \forall x,y \in \mathbb Q.$ (iii) $f(x + y) \leq \max\{f(x), f(y)\} \qquad \forall x,y \in \mathbb Q , x,y \neq 0.$ Let $x$ be an integer and $f(x) \neq 1$. Prove that $f(1 + x + x^2+ \cdots + x^n) = 1$ for any positive integer $n.$

2009 USA Team Selection Test, 7

Tags: algebra , polynomial , trigonometry , system of equations , algebra proposed

Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations \[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\] [i]Razvan Gelca.[/i]

2024 Middle European Mathematical Olympiad, 1

Tags: Sequences , algebra , algebra proposed , inequalities , inequalities proposed

Consider two infinite sequences $a_0,a_1,a_2,\dots$ and $b_0,b_1,b_2,\dots$ of real numbers such that $a_0=0$, $b_0=0$ and \[a_{k+1}=b_k, \quad b_{k+1}=\frac{a_kb_k+a_k+1}{b_k+1}\] for each integer $k \ge 0$. Prove that $a_{2024}+b_{2024} \ge 88$.

2014 Baltic Way, 1

Tags: trigonometry , modular arithmetic , algebra proposed , algebra

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]

2001 Austrian-Polish Competition, 2

Tags: limit , algebra , system of equations , pigeonhole principle , algebra proposed

Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.

2010 Bulgaria National Olympiad, 2

Tags: function , floor function , induction , algebra proposed , algebra

Let $f: \mathbb N \to \mathbb N$ be a function such that $f(1)=1$ and \[f(n)=n - f(f(n-1)), \quad \forall n \geq 2.\] Prove that $f(n+f(n))=n $ for each positive integer $n.$

2010 ELMO Shortlist, 7

Tags: algebra proposed , algebra

Find the smallest real number $M$ with the following property: Given nine nonnegative real numbers with sum $1$, it is possible to arrange them in the cells of a $3 \times 3$ square so that the product of each row or column is at most $M$. [i]Evan O' Dorney.[/i]