Olympiad problems

1990 Vietnam National Olympiad, 1

The sequence $ (x_n)$, $ n\in\mathbb{N}^*$ is defined by $ |x_1|<1$, and for all $ n \ge 1$, \[ x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2}\] (a) Find the necessary and sufficient condition for $ x_1$ so that each $ x_n > 0$. (b) Is this sequence periodic? And why?

1993 All-Russian Olympiad, 3

Tags: function , algebra , domain , functional equation , algebra unsolved

Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.

1988 IMO Longlists, 27

Tags: trigonometry , algebra , polynomial , Vieta , function , algebra unsolved

Assuming that the roots of $x^3 + p \cdot x^2 + q \cdot x + r = 0$ are real and positive, find a relation between $p,q$ and $r$ which gives a necessary condition for the roots to be exactly the cosines of the three angles of a triangle.

2004 India IMO Training Camp, 2

Tags: function , algebra unsolved , algebra

Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule: (a) $g$ is nondecrasing (b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$, Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$

2011 China Second Round Olympiad, 4

Tags: trigonometry , algebra unsolved , algebra

If ${\cos^5 x}-{\sin^5 x}<7({\sin^3 x}-{\cos ^3 x}) $ (for $x\in [ 0,2\pi) $), then find the range of $x$.

2005 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: algebra unsolved , algebra

Consider all finite sequences of positive real numbers each of whose terms is at most $3$ and the sum of whose terms is more than $100$. For each such sequence, let $S$ denote the sum of the subsequence whose sum is the closest to $100$, and define the [i]defect[/i] of this sequence to be the value $|S-100|$. Find the maximum possible value of the defect.

2005 China Team Selection Test, 2

Tags: logarithms , algebra unsolved , algebra

Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

2006 Federal Competition For Advanced Students, Part 2, 1

Tags: algebra unsolved , algebra

For which rational $ x$ is the number $ 1 \plus{} 105 \cdot 2^x$ the square of a rational number?

2002 China Team Selection Test, 2

Tags: function , algebra unsolved , algebra

Given an integer $k$. $f(n)$ is defined on negative integer set and its values are integers. $f(n)$ satisfies \[ f(n)f(n+1)=(f(n)+n-k)^2, \] for $n=-2,-3,\cdots$. Find an expression of $f(n)$.

2001 China Team Selection Test, 2

Tags: algebra unsolved , algebra , combinatorics , number theory

$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?

2005 MOP Homework, 5

Tags: inequalities , algebra unsolved , algebra

Let $a_1$, $a_2$, ..., $a_{2004}$ be non-negative real numbers such that $a_1+...+ a_{2004} \le 25$. Prove that among them there exist at least two numbers $a_i$ and $a_j$ ($i \neq j$) such that $|\sqrt{a_i}-\sqrt{a_j}| \le \frac{5}{2003}$.

2009 Polish MO Finals, 6

Tags: algebra unsolved , algebra

Let $ n$ be a natural number equal or greater than 3 . A sequence of non-negative numbers $ (c_0,c_1,\ldots,c_n)$ satisfies the condition: $ c_{p}c_{s}\plus{}c_{r}c_{t}\equal{} c_{p\plus{}r}c_{r\plus{}s}$ for all non-negative $ p,q,r,s$ such that $ p\plus{}q\plus{}r\plus{}s\equal{}n$. Determine all possible values of $ c_2$ when $ c_1\equal{}1$.

2001 Romania National Olympiad, 1

Tags: algebra , polynomial , algebra unsolved

a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots. b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.

2009 Tuymaada Olympiad, 4

Tags: algebra , polynomial , floor function , functional equation , algebra unsolved

Determine the maximum number $ h$ satisfying the following condition: for every $ a\in [0,h]$ and every polynomial $ P(x)$ of degree 99 such that $ P(0)\equal{}P(1)\equal{}0$, there exist $ x_1,x_2\in [0,1]$ such that $ P(x_1)\equal{}P(x_2)$ and $ x_2\minus{}x_1\equal{}a$. [i]Proposed by F. Petrov, D. Rostovsky, A. Khrabrov[/i]

2012 Indonesia TST, 1

Tags: function , algebra unsolved , algebra

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that \[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\] for all $x,y \in \mathbb{R}$.

2010 Argentina Team Selection Test, 3

Tags: function , limit , algebra unsolved , algebra

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that \[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\] holds for all real numbers $x,y$.

1994 Irish Math Olympiad, 3

Tags: algebra , polynomial , induction , algebra unsolved

Find all real polynomials $ f(x)$ satisfying $ f(x^2)\equal{}f(x)f(x\minus{}1)$ for all $ x$.

2006 Lithuania National Olympiad, 3

Tags: inequalities , algebra unsolved , algebra

Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.

2001 Polish MO Finals, 3

Tags: algebra unsolved , algebra

A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$. Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.

2004 Finnish National High School Mathematics Competition, 1

Tags: quadratics , algebra unsolved , algebra

The equations $x^2 +2ax+b^2 = 0$ and $x^2 +2bx+c^2 = 0$ both have two different real roots. Determine the number of real roots of the equation $x^2 + 2cx + a^2 = 0.$

2013 Iran MO (3rd Round), 3

Tags: function , algebra , polynomial , algebra unsolved

Real function $f$ [b]generates[/b] real function $g$ if there exists a natural $k$ such that $f^k=g$ and we show this by $f \rightarrow g$. In this question we are trying to find some properties for relation $\rightarrow$, for example it's trivial that if $f \rightarrow g$ and $g \rightarrow h$ then $f \rightarrow h$.(transitivity) (a) Give an example of two real functions $f,g$ such that $f\not = g$ ,$f\rightarrow g$ and $g\rightarrow f$. (b) Prove that for each real function $f$ there exists a finite number of real functions $g$ such that $f \rightarrow g$ and $g \rightarrow f$. (c) Does there exist a real function $g$ such that no function generates it, except for $g$ itself? (d) Does there exist a real function which generates both $x^3$ and $x^5$? (e) Prove that if a function generates two polynomials of degree 1 $P,Q$ then there exists a polynomial $R$ of degree 1 which generates $P$ and $Q$. Time allowed for this problem was 75 minutes.

1994 Romania TST for IMO, 4:

Tags: algebra unsolved , algebra

Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that $(1)$ $f(n) \leq n^8$ for any $n \geq 2$, $(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that $$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$

2021 Vietnam National Olympiad, 5

Tags: algebra unsolved , inequalities

Let the polynomial $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0$ where $1011\leq a_i\leq 2021$ for all $i=0,1,2,...,21.$ Given that $P(x)$ has an integer root and there exists an positive real number$c$ such that $|a_{k+2}-a_k|\leq c$ for all $k=0,1,...,19.$ a) Prove that $P(x)$ has an only integer root. b) Prove that $$\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.$$

2006 China Team Selection Test, 3

Tags: algebra , polynomial , algebra unsolved

Find all second degree polynomial $d(x)=x^{2}+ax+b$ with integer coefficients, so that there exists an integer coefficient polynomial $p(x)$ and a non-zero integer coefficient polynomial $q(x)$ that satisfy: \[\left( p(x) \right)^{2}-d(x) \left( q(x) \right)^{2}=1, \quad \forall x \in \mathbb R.\]

2008 Germany Team Selection Test, 3

Tags: algebra , polynomial , inequalities , algebra unsolved

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