Olympiad problems

2017 Greece National Olympiad, 4

Tags: polynomial , algebra

Let $u$ be the positive root of the equation $x^2+x-4=0$. The polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$. 1) Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$. 2) Find the minimum possible value of $a_0+a_1+...+a_n$.

2011 Morocco National Olympiad, 2

Tags: algebra , quadratic

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

2006 Indonesia MO, 1

Tags: algebra

Find all pairs $ (x,y)$ of real numbers which satisfy $ x^3\minus{}y^3\equal{}4(x\minus{}y)$ and $ x^3\plus{}y^3\equal{}2(x\plus{}y)$.

1989 IMO Longlists, 53

Tags: algebra , pigeonhole principle , floor function

Let $ \alpha$ be the positive root of the equation $ x^2 \minus{} 1989x \minus{} 1 \equal{} 0.$ Prove that there exist infinitely many natural numbers $ n$ that satisfy the equation: \[ \lfloor \alpha n \plus{} 1989 \alpha \lfloor \alpha n \rfloor \rfloor \equal{} 1989n \plus{} \left( 1989^2 \plus{} 1 \right) \lfloor \alpha n \rfloor.\]

2014 Kosovo National Mathematical Olympiad, 2

Tags: algebra

Solve $|x-1|-2|x+5|>3+x$.

2008 Indonesia TST, 2

Tags: functional , algebra , functional equation

Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.

2022 Switzerland Team Selection Test, 12

Tags: functional equation , algebra

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that \[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\] for all $x,y>0.$

2008 Princeton University Math Competition, A3

Tags: algebra

A sequence $\{a_i\}$ is defined by $a_1 = c$ for some $c > 0$ and $a_{n+1} = a_n + \frac{n}{a_n}$. Prove that $\frac{a_n}{n}$ converges and find its limit.

2009 Greece JBMO TST, 4

Tags: algebra , minimum , sum , system of equations

Find positive real numbers $x,y,z$ that are solutions of the system $x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.

2006 Switzerland Team Selection Test, 3

Tags: limit , domain , function , algebra

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

2009 Finnish National High School Mathematics Competition, 2

Tags: modular arithmetic , polynomial , function , algebra

A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?

1996 IMO Shortlist, 6

Tags: polynomial , algebra , function

Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that \[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\] for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$ Note: This is variant A6' of the three variants given for this problem.

2016 Balkan MO Shortlist, A6

Tags: algebra , function , functional equation

Prove that there is no function from positive real numbers to itself, $f : (0,+\infty)\to(0,+\infty)$ such that: $f(f(x) + y) = f(x) + 3x + yf(y)$ ,for every $x,y \in (0,+\infty)$ by Greece, Athanasios Kontogeorgis (aka socrates)

2004 Thailand Mathematical Olympiad, 5

Tags: equation , algebra , sum , radical

Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =|2nx - n - 2n^2|$

VI Soros Olympiad 1999 - 2000 (Russia), 10.1

Tags: inequalities , algebra

For real numbers $x,y, \in [1,2]$, prove the inequality $3(x + y)\ge 2xy + 4$

2006 Estonia Math Open Junior Contests, 6

Tags: 3d geometry , quadratic , geometry , algebra

Find all real numbers with the following property: the difference of its cube and its square is equal to the square of the difference of its square and the number itself.

2020 Iranian Our MO, 6

Tags: algebra , functional equation , polynomial

Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$ [i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]

2021 Bulgaria National Olympiad, 3

Tags: algebra , functional equation

Find all $f:R^+ \rightarrow R^+$ such that $f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+$ @below: [url]https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems[/url] [quote]Feel free to start individual threads for the problems as usual[/quote]

2019 MOAA, 2

Tags: team , algebra , geometry

The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?

2017 OMMock - Mexico National Olympiad Mock Exam, 5

Tags: function , functional equation , algebra

Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation $$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$ for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$. [i]Proposed by Maximiliano Sánchez[/i]

2013 Saudi Arabia GMO TST, 1

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ which satisfy $f \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x$ and $f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right) $ for all $x, y \in R$, with $y \ne 0$

2010 Contests, 1

Tags: limit , algebra , polynomial

suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

1994 Tournament Of Towns, (428) 5

Tags: algebra , sequence , periodic

The periods of two periodic sequences are $7$ and $13$. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.) (AY Belov)

2023 Quang Nam Province Math Contest (Grade 11), Problem 3

Tags: polynomial , algebra

Given a polynomial $P(x)$ with real coefficents satisfying:$$P(x).P(x+1)=P(x^2+x+1),\forall x\in \mathbb{R}.$$ Prove that: $\deg(P)$ is an even number and find $P(x).$

1998 Romania National Olympiad, 2

Tags: algebra , complex number , inequalities

Let $a \ge1$ be a real number and $z$ be a complex number such that $| z + a | \le a$ and $|z^2+ a | \le a$. Show that $| z | \le a$.