Olympiad problems

2017 German National Olympiad, 1

Tags: quadratics , system of equations , equation , quadratic equation , algebra , algebra proposed , Germany

Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$: \begin{align*} x^2+py+q&=0,\\ y^2+px+q&=0. \end{align*} Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.

2005 Greece National Olympiad, 1

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

Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.

2024 Dutch IMO TST, 2

Tags: function , functional equation , algebra proposed , algebra , Functional inequality

Find all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}$ with \[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\] for all $x,y,z \in \mathbb{R}_{\ge 0}$.

2007 Moldova National Olympiad, 12.1

Tags: algebra , polynomial , algebra proposed

For $a\in C^{*}$ find all $n\in N$ such that $X^{2}(X^{2}-aX+a^{2})^{2}$ divides $(X^{2}+a^{2})^{n}-X^{2n}-a^{2n}$

2003 Serbia Team Selection Test, 1

Tags: algebra , polynomial , algebra proposed

If $ p(x)$ is a polynomial, denote by $ p^n(x)$ the polynomial $ p(p(...(p(x))..)$, where $ p$ is iterated $ n$ times. Prove that the polynomial $ p^{2003}(x)\minus{}2p^{2002}(x)\plus{}p^{2001}(x)$ is divisible by $ p(x)\minus{}x$

1995 Baltic Way, 9

Tags: induction , logarithms , algebra proposed , algebra

Prove that \[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]

1970 IMO Longlists, 54

Tags: algebra , polynomial , algebra proposed

Let $P,Q,R$ be polynomials and let $S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)$ be a polynomial of degree $n$ whose roots $x_1,\ldots, x_n$ are distinct. Construct with the aid of the polynomials $P,Q,R$ a polynomial $T$ of degree $n$ that has the roots $x_1^3 , x_2^3 , \ldots, x_n^3.$

2011 Baltic Way, 5

Tags: function , algebra proposed , algebra

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that \[f(f(x))=x^2-x+1\] for all real numbers $x$. Determine $f(0)$.

2007 All-Russian Olympiad Regional Round, 11.2

Tags: quadratics , algebra , polynomial , algebra proposed , Polynomials

Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|\plus{}|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}\equal{} g^{2}$ for some $ g\in\mathbb{R}[x]$.

2013 India IMO Training Camp, 3

Tags: geometry , geometric transformation , algebra , polynomial , algebra proposed

For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.

2003 Romania Team Selection Test, 1

Tags: algebra proposed , algebra

Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$ \[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \] Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.

2013 Iran MO (2nd Round), 3

Tags: function , inequalities , algebra proposed , algebra , Recurrence

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers for which \[ a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. \] Prove that there exists a positive integer $m$ such that $a_m=4$ and $a_{m+1} \in\{3,4\}$. [b]Note.[/b] $[x]$ is the greatest integer not exceeding $x$.

2004 Tuymaada Olympiad, 1

Tags: algebra , polynomial , calculus , integration , algebra proposed

Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$ [i]Proposed by A. Golovanov[/i]

2006 Iran MO (3rd Round), 2

Tags: analytic geometry , algebra , polynomial , algebra proposed

A liquid is moving in an infinite pipe. For each molecule if it is at point with coordinate $x$ then after $t$ seconds it will be at a point of $p(t,x)$. Prove that if $p(t,x)$ is a polynomial of $t,x$ then speed of all molecules are equal and constant.

2012 Romanian Masters In Mathematics, 3

Tags: function , algebra proposed , algebra , combinatorics , combinatorics proposed , Additive combinatorics

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

2012 Iran MO (3rd Round), 3

Tags: algebra , polynomial , algebra proposed

Suppose $p$ is a prime number and $a,b,c \in \mathbb Q^+$ are rational numbers; [b]a)[/b] Prove that $\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})$. [b]b)[/b] If $\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})$, prove that for a nonnegative integer $k$ we have $\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q$. [b]c)[/b] If $\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q$, then prove that numbers $\sqrt[p]{a},\sqrt[p]{b}$ and $\sqrt[p]{c}$ are rational.

2006 JBMO ShortLists, 3

Tags: induction , algebra proposed , algebra

Let $ n\ge 3$ be a natural number. A set of real numbers $ \{x_1,x_2,\ldots,x_n\}$ is called [i]summable[/i] if $ \sum_{i\equal{}1}^n \frac{1}{x_i}\equal{}1$. Prove that for every $ n\ge 3$ there always exists a [i]summable[/i] set which consists of $ n$ elements such that the biggest element is: a) bigger than $ 2^{2n\minus{}2}$ b) smaller than $ n^2$

1990 India National Olympiad, 5

Tags: search , inequalities , triangle inequality , algebra proposed , algebra , limit , india

Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity \[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\] must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?

2001 Romania Team Selection Test, 1

Tags: trigonometry , complex numbers , algebra proposed , algebra

Show that if $a,b,c$ are complex numbers that such that \[ (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a\] then $a,b,c$ are real numbers.

2010 Slovenia National Olympiad, 4

Tags: inequalities , algebra proposed , algebra

Find all non-zero real numbers $x$ such that \[\min \left\{ 4, x+ \frac 4x \right\} \geq 8 \min \left\{ x,\frac 1x\right\} .\]

2005 Romania National Olympiad, 3

Tags: logarithms , function , algebra proposed , algebra

a) Prove that there are no one-to-one (injective) functions $f: \mathbb{N} \to \mathbb{N}\cup \{0\}$ such that \[ f(mn) = f(m)+f(n) , \ \forall \ m,n \in \mathbb{N}. \] b) Prove that for all positive integers $k$ there exist one-to-one functions $f: \{1,2,\ldots,k\}\to\mathbb{N}\cup \{0\}$ such that $f(mn) = f(m)+f(n)$ for all $m,n\in \{1,2,\ldots,k\}$ with $mn\leq k$. [i]Mihai Baluna[/i]

2009 Indonesia TST, 4

Tags: function , algebra proposed , algebra

Let $ S$ be the set of nonnegative real numbers. Find all functions $ f: S\rightarrow S$ which satisfy $ f(x\plus{}y\minus{}z)\plus{}f(2\sqrt{xz})\plus{}f(2\sqrt{yz})\equal{}f(x\plus{}y\plus{}z)$ for all nonnegative $ x,y,z$ with $ x\plus{}y\ge z$.

2013 District Olympiad, 1

Tags: inequalities , algebra proposed , algebra

a) Prove that, whatever the real number x would be, the following inequality takes place ${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$ b) Solve the following system in the set of real numbers: ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$. The Mathematical Gazette

2012 ELMO Shortlist, 4

Tags: floor function , induction , algebra proposed , algebra

Let $a_0,b_0$ be positive integers, and define $a_{i+1}=a_i+\lfloor\sqrt{b_i}\rfloor$ and $b_{i+1}=b_i+\lfloor\sqrt{a_i}\rfloor$ for all $i\ge0$. Show that there exists a positive integer $n$ such that $a_n=b_n$. [i]David Yang.[/i]

1997 Romania Team Selection Test, 4

Tags: algebra , polynomial , number theory , greatest common divisor , modular arithmetic , algebra proposed

Let $n\ge 2$ be an integer and let $P(X)=X^n+a_{n-1}X^{n-1}+\ldots +a_1X+1$ be a polynomial with positive integer coefficients. Suppose that $a_k=a_{n-k}$ for all $k\in 1,2,\ldots,n-1$. Prove that there exist infinitely many pairs of positive integers $x,y$ such that $x|P(y)$ and $y|P(x)$. [i]Remus Nicoara[/i]