Olympiad problems

2013 Turkmenistan National Math Olympiad, 1

Find the product $ \cos a \cdot \cos 2a\cdot \cos 3a \cdots \cos 1006a$ where $a=\frac{2\pi}{2013}$.

1983 Bulgaria National Olympiad, Problem 5

Tags: algebra , polynomial , algebra unsolved , Solve equation , complex numbers

Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?

2013 Finnish National High School Mathematics Competition, 1

Tags: algebra , polynomial , quadratics , algebra unsolved

The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.

2006 Federal Competition For Advanced Students, Part 2, 2

Tags: function , algebra , functional equation , algebra unsolved

Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation: \[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]

2014 Saudi Arabia BMO TST, 5

Tags: induction , inequalities , algebra unsolved , algebra

Find all positive integers $n$ such that \[3^n+4^n+\cdots+(n+2)^n=(n+3)^n.\]

2012 China Second Round Olympiad, 6

Tags: function , inequalities , algebra unsolved , algebra

Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$.

2004 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: algebra , polynomial , algebra unsolved

Find all polynomials $P(x)$ with real coefficients such that \[xP\bigg(\frac{y}{x}\bigg)+yP\bigg(\frac{x}{y}\bigg)=x+y\] for all nonzero real numbers $x$ and $y$.

2005 China Team Selection Test, 3

Tags: function , algebra unsolved , algebra

Let $\alpha$ be given positive real number, find all the functions $f: N^{+} \rightarrow R$ such that $f(k + m) = f(k) + f(m)$ holds for any positive integers $k$, $m$ satisfying $\alpha m \leq k \leq (\alpha + 1)m$.

2006 Estonia National Olympiad, 2

Tags: algebra unsolved , algebra

Find the smallest possible distance of points $ P$ and $ Q$ on a $ xy$-plane, if $ P$ lies on the line $ y \equal{} x$ and $ Q$ lies on the curve $ y \equal{} 2^x$.

2000 China Team Selection Test, 3

Tags: function , LaTeX , algebra unsolved , algebra

Let $n$ be a positive integer. Denote $M = \{(x, y)|x, y \text{ are integers }, 1 \leq x, y \leq n\}$. Define function $f$ on $M$ with the following properties: [b]a.)[/b] $f(x, y)$ takes non-negative integer value; [b] b.)[/b] $\sum^n_{y=1} f(x, y) = n - 1$ for $1 \eq x \leq n$; [b]c.)[/b] If $f(x_1, y_1)f(x2, y2) > 0$, then $(x_1 - x_2)(y_1 - y_2) \geq 0.$ Find $N(n)$, the number of functions $f$ that satisfy all the conditions. Give the explicit value of $N(4)$.

2006 Moldova National Olympiad, 10.5

Tags: logarithms , inequalities , calculus , derivative , function , algebra unsolved , algebra

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

2002 Finnish National High School Mathematics Competition, 2

Tags: algebra unsolved , algebra

Show that if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a + b + c},$ then also \[\frac{1}{a^n} +\frac{1}{b^n} +\frac{1}{c^n} =\frac{1}{a^n + b^n + c^n},\] provided $n$ is an odd positive integer.

1999 China Second Round Olympiad, 2

Tags: trigonometry , complex numbers , algebra unsolved , algebra

Let $a$,$b$,$c$ be real numbers. Let $z_{1}$,$z_{2}$,$z_{3}$ be complex numbers such that $|z_{k}|=1$ $(k=1,2,3)$ $~$ and $~$ $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1$ Find $|az_{1}+bz_{2}+cz_{3}|$.

2002 China Team Selection Test, 1

Tags: algebra unsolved , algebra

Given a positive integer $ n$, for all positive integers $ a_1, a_2, \cdots, a_n$ that satisfy $ a_1 \equal{} 1$, $ a_{i \plus{} 1} \leq a_i \plus{} 1$, find $ \displaystyle \sum_{i \equal{} 1}^{n} a_1a_2 \cdots a_i$.

2009 Bundeswettbewerb Mathematik, 2

Tags: algebra , algebra unsolved

Let $a,b$ be positive real numbers. Define $m(a,b)$ as the minimum of $\[ a,\frac{1}{b} \text{ and } \frac{1}{a}+b.\]$ Find the maximum of $m(a,b).$

1993 Vietnam Team Selection Test, 2

Tags: algebra unsolved , algebra

A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$. Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$, $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity. THERE MIGHT BE A TYPO!

1991 IMTS, 3

Tags: number theory , relatively prime , algebra unsolved , algebra

Prove that if $x,y$ and $z$ are pairwise relatively prime positive integers, and if $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$, then $x+y, x-z, y-z$ are perfect squares of integers.

2008 Moldova National Olympiad, 11.2

Tags: algebra unsolved , algebra

Let $ (a_{n})_{n\ge 1} $ be a sequence such that: $ a_{1}=1; a_{n+1}=\frac{n}{a_{n}+1}.$ Find $ [a_{2008}] $

2004 Regional Competition For Advanced Students, 2

Tags: algebra unsolved , algebra

Solve the following equation for real numbers: $ \sqrt{4\minus{}x\sqrt{4\minus{}(x\minus{}2)\sqrt{1\plus{}(x\minus{}5)(x\minus{}7)}}}\equal{}\frac{5x\minus{}6\minus{}x^2}{2}$ (all square roots are non negative)

1989 Federal Competition For Advanced Students, P2, 6

Tags: function , algebra , polynomial , algebra unsolved

Determine all functions $ f: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that $ f(f(n))\plus{}f(n)\equal{}2n\plus{}6$ for all $ n \in \mathbb{N}_0$.

2000 South africa National Olympiad, 5

Tags: function , algebra unsolved , algebra

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ (where $\mathbb{Z}$ is the set of all integers) such that \[ 2000f(f(x)) - 3999f(x) + 1999x = 0\textrm{ for all }x \in \mathbb{Z}. \]

2005 Kurschak Competition, 1

Tags: probability , algebra unsolved , algebra

Let $N>1$ and let $a_1,a_2,\dots,a_N$ be nonnegative reals with sum at most $500$. Prove that there exist integers $k\ge 1$ and $1=n_0<n_1<\dots<n_k=N$ such that \[\sum_{i=1}^k n_ia_{n_{i-1}}<2005.\]

2009 Ukraine National Mathematical Olympiad, 4

Tags: algebra , polynomial , algebra unsolved

Find all polynomials $P(x)$ with real coefficients such that for all pairwise distinct positive integers $x, y, z, t$ with $x^2 + y^2 + z^2 = 2t^2$ and $\gcd(x, y, z, t ) = 1,$ the following equality holds \[2P^2(t ) + 2P(xy + yz + zx) = P^2(x + y + z) .\] [b]Note.[/b] $P^2(k)=\left( P(k) \right)^2.$

1997 India National Olympiad, 3

Tags: inequalities , function , algebra unsolved , algebra

If $a,b,c$ are three real numbers and \[ a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t \] for some real number $t$, prove that $abc + t = 0 .$

2008 Nordic, 1

Tags: function , algebra unsolved , algebra

Find all reals $A,B,C$ such that there exists a real function $f$ satisfying $f(x+f(y))= Ax+By+C$ for all reals $x,y$.