Olympiad problems

2007 Balkan MO, 2

Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.

2024 South Africa National Olympiad, 4

Tags: functional equation , functional equation in n , algebra

Find all functions $f$ from integers to integers such that \[ f(m+n) + f(m-n) - 2f(m) = 6mn^2\] for all integers $m$ and $n$.

1967 IMO Shortlist, 1

Tags: number theory , perfect cube , decimal representation , algebra

Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

2024 TASIMO, 5

Tags: algebraic number theory , algebra , functional equation

Find all functions $f: \mathbb{Z^+} \to \mathbb{Z^+}$ such that for all integers $a, b, c$ we have $$ af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac). $$ [i]Note. The set $\mathbb{Z^+}$ refers to the set of positive integers.[/i] [i]Proposed by Mojtaba Zare, Iran[/i]

2019 Switzerland Team Selection Test, 4

Tags: prime number , number theory , polynomial , algebra

Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties: $(a)$ $P(x)>x$ for all positive integers $x$. $(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.

2008 Germany Team Selection Test, 3

Tags: inequalities , polynomial , algebra

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

2014 Cuba MO, 5

Tags: system , system of equations , algebra

Determine all real solutions to the system of equations: $$x^2 - y = z^2$$ $$y^2 - z = x^2$$ $$z^2 - x = y^2$$

1999 Portugal MO, 5

Tags: sum , number theory , algebra

Each of the numbers $a_1,...,a_n$ is equal to $1$ or $-1$. If $a_1a_2 + a_2a_3 + ··· + a_{n-1}a_n + a_na_1 = 0$, proves that $n$ is divisible by $4$.

2021 BMT, 4

Tags: algebra

Compute the sum of all real solutions to $4^x - 2021 \cdot 2^x + 1024 = 0$.

2021 HMNT, 5

Tags: algebra

Let $n$ be the answer to this problem. The polynomial $x^n+ax^2+bx+c$ has real coefficients and exactly $k$ real roots. Find the sum of the possible values of $k$.

2005 Austrian-Polish Competition, 2

Tags: polynomial , ratio , algebra

Determine all polynomials $P$ with integer coefficients satisfying \[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]

1981 Austrian-Polish Competition, 5

Tags: algebra , polynomial , rational

Let $P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4$ be a polynomial with rational coefficients. Show that if $P(x)$ has exactly one real root $\xi$, then $\xi$ is a rational number.

2004 USAMTS Problems, 3

Tags: algebra , polynomial

Find, with proof, a polynomial $f(x,y,z)$ in three variables, with integer coefficients, such that for all $a,b,c$ the sign of $f(a,b,c)$ (that is, positive, negative, or zero) is the same as the sign of $a+b\sqrt[3]{2}+c\sqrt[3]{4}$.

2016 Switzerland - Final Round, 3

Tags: algebra , number theory

Find all primes $p, q$ and natural numbers $n$ such that: $p(p+1)+q(q+1)=n(n+1)$

2010 Korea Junior Math Olympiad, 5

Tags: trigonometric inequality , inequalities , trigonometry , algebra

If reals $x, y, z $ satises $tan x + tan y + tan z = 2$ and $0 < x, y,z < \frac{\pi}{2}.$ Prove that $$sin^2 x + sin^2 y + sin^2 z < 1.$$

2013 Saudi Arabia GMO TST, 2

Tags: algebra , inequalities

For positive real numbers $a, b$ and $c$, prove that $$\frac{a^3}{a^2 + ab + b^2} +\frac{b^3}{b^2 + bc + c^2} +\frac{c^3}{ c^2 + ca + a^2} \ge\frac{ a + b + c}{3}$$

2013 Brazil Team Selection Test, 3

Tags: arithmetic sequence , algebra , floor function

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[ X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...). \] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

1983 IMO Shortlist, 20

Tags: system of equations , algebra

Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

1980 IMO, 2

Tags: recurrence relation , sequence , inequality , algebra

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

2015 Saudi Arabia GMO TST, 2

Tags: algebra , increasing , sequence

Find the number of strictly increasing sequences of nonnegative integers with the first term $0$ and the last term $15$, and among any two consecutive terms, exactly one of them is even. Lê Anh Vinh

2009 Kazakhstan National Olympiad, 1

Tags: binomial coefficient , algebra

Let $S_n$ be number of ordered sets of natural numbers $(a_1;a_2;....;a_n)$ for which $\frac{1}{a_1}+\frac{1}{a_2}+....+\frac{1}{a_n}=1$. Determine 1)$S_{10} mod(2)$. 2)$S_7 mod(2)$. (1) is first problem in 10 grade, (2)- third in 9 grade.

2009 Putnam, B2

Tags: algebra , integration , calculus , function , polynomial , limit

A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$

1990 Romania Team Selection Test, 2

Tags: binomial , sum , combinatorics , algebra

Prove the following equality for all positive integers $m,n$: $$\sum_{k=0}^{n} {m+k \choose k} 2^{n-k} +\sum_{k=0}^m {n+k \choose k}2^{m-k}= 2^{m+n+1}$$

Russian TST 2019, P2

Tags: rational number , set , algebra

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

2019 VJIMC, 4

Tags: inequalities , algebra , real analysis

Determine the largest constant $K\geq 0$ such that $$\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\frac{c^c(a^2+b^2)}{(c^c-1)^2}\geq K\left (\frac{a+b+c}{abc-1}\right)^2$$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$. [i]Proposed by Orif Ibrogimov (Czech Technical University of Prague).[/i]