Olympiad problems

1989 IMO Longlists, 95

Let $ n$ be a positive integer, $ X \equal{} \{1, 2, \ldots , n\},$ and $ k$ a positive integer such that $ \frac{n}{2} \leq k \leq n.$ Determine, with proof, the number of all functions $ f : X \mapsto X$ that satisfy the following conditions: [b](i)[/b] $ f^2 \equal{} f;$ [b](ii)[/b] the number of elements in the image of $ f$ is $ k;$ [b](iii)[/b] for each $ y$ in the image of $ f,$ the number of all points $ x \in X$ such that $ f(x)\equal{}y$ is at most $ 2.$

2012 Tuymaada Olympiad, 2

Tags: algebra , polynomial , quadratics , inequalities , sum of roots , algebra unsolved

Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$. [i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]

2005 Finnish National High School Mathematics Competition, 3

Tags: absolute value , algebra unsolved , algebra

Solve the group of equations: \[\begin{cases} (x + y)^3 = z \\ (y + z)^3 = x \\ (z + x)^3 = y \end{cases}\]

1993 India National Olympiad, 2

Tags: quadratics , algebra , polynomial , algebra unsolved

Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.

2009 Ukraine National Mathematical Olympiad, 4

Tags: function , algebra , functional equation , algebra unsolved

Find all functions $f : \mathbb R \to \mathbb R$ such that \[f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.\]

2006 Bulgaria Team Selection Test, 1

Tags: induction , algebra unsolved , algebra

Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and \[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\] for all natural $n \geq 2$. [i]Peter Boyvalenkov[/i]

2008 Bulgaria National Olympiad, 3

Tags: inequalities , trigonometry , modular arithmetic , induction , ceiling function , pigeonhole principle , algebra unsolved

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.

2015 Mathematical Talent Reward Programme, MCQ: P 9

Tags: combinatorics , algebra unsolved

How many $5 \times 5$ grids are possible such that each element is either 1 or 0 and each row sum and column sum is $4 ?$ [list=1] [*] 64 [*] 32 [*] 120 [*] 96 [/list]

1992 Polish MO Finals, 2

Tags: function , algebra unsolved , algebra

Find all functions $f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$, where $\mathbb{Q}^{+}$ is the set of positive rationals, such that $f(x+1) = f(x) + 1$ and $f(x^3) = f(x)^3$ for all $x$.

2003 China Team Selection Test, 2

Tags: function , algebra , functional equation , algebra unsolved

Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

2006 China Team Selection Test, 1

Tags: induction , ratio , inequalities , limit , algebra unsolved , algebra

Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy: (a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$. (b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$. Find the general term of $\{ a_{n}\}$.

2003 China Team Selection Test, 2

Tags: function , induction , algebra unsolved , algebra

Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.

1989 IMO Longlists, 51

Tags: algebra , polynomial , algebra unsolved

Let $ f(x) \equal{} \prod^n_{k\equal{}1} (x \minus{} a_k) \minus{} 2,$ where $ n \geq 3$ and $ a_1, a_2, \ldots,$ an are distinct integers. Suppose that $ f(x) \equal{} g(x)h(x),$ where $ g(x), h(x)$ are both nonconstant polynomials with integer coefficients. Prove that $ n \equal{} 3.$

2003 Czech-Polish-Slovak Match, 6

Tags: function , algebra unsolved , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the condition \[f(f(x) + y) = 2x + f(f(y) - x)\quad \text{ for all } x, y \in\mathbb{R}.\]

2008 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: algebra unsolved , algebra

Consider a collection of stones whose total weight is $65$ pounds and each of whose stones is at most $w$ pounds. Find the largest number $w$ for which any such collection of stones can be divided into two groups whose total weights differ by at most one pound. Note: The weights of the stones are not necessarily integers.

1994 Vietnam National Olympiad, 3

Tags: algebra , polynomial , algebra unsolved

Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?

2010 Greece Team Selection Test, 1

Tags: algebra unsolved , algebra

Solve in positive reals the system: $x+y+z+w=4$ $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}=5-\frac{1}{xyzw}$

2009 Iran MO (3rd Round), 4

Tags: inequalities , function , algebra unsolved , algebra

Does there exists two functions $f,g :\mathbb{R}\rightarrow \mathbb{R}$ such that: $\forall x\not =y : |f(x)-f(y)|+|g(x)-g(y)|>1$ Time allowed for this problem was 75 minutes.

1996 Romania Team Selection Test, 3

Tags: trigonometry , algebra unsolved , algebra

Let $ x,y\in \mathbb{R} $. Show that if the set $ A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\} $ is finite then $ x,y \in \mathbb{Q} $. [i]Vasile Pop[/i]

2006 Hong kong National Olympiad, 2

Tags: function , logarithms , modular arithmetic , algebra unsolved , algebra

For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1}$ = $f_1 \circ f_n$ . Evaluate $f_{2007}(2^{2006} )$.

1996 Hungary-Israel Binational, 4

Tags: algebra unsolved , algebra

$ a_1, a_2, \cdots, a_n$ is a sequence of real numbers, and $ b_1, b_2, \cdots, b_n$ are real numbers that satisfy the condition $ 1 \ge b_1 \ge b_2 \ge \cdots \ge b_n \ge 0$. Prove that there exists a natural number $ k \le n$ that satisifes $ |a_1b_1 \plus{} a_2b_2 \plus{} \cdots \plus{} a_nb_n| \le |a_1 \plus{} a_2 \plus{} \cdots \plus{} a_k|$

2006 Czech-Polish-Slovak Match, 5

Tags: calculus , integration , algebra unsolved , algebra

Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and \[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\] for each $n \in \mathbb{N}$.

1984 IMO Longlists, 58

Tags: inequalities , limit , algebra unsolved , algebra

Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.

1970 IMO Longlists, 47

Tags: algebra , polynomial , calculus , derivative , algebra unsolved

Given a polynomial \[P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),\] where $a, b, c \neq 0$, prove that $P(x)$ is divisible by \[Q(x) = abx^2 + (a^2 + b^2)x + ab\] and conclude that $P(x_0)$ is divisible by $(a + b)^3$ for $x_0 = (a + b + 1)^n, n \in \mathbb N$.

1995 APMO, 1

Tags: algebra unsolved , algebra

Determine all sequences of real numbers $a_1$, $a_2$, $\ldots$, $a_{1995}$ which satisfy: \[ 2\sqrt{a_n - (n - 1)} \geq a_{n+1} - (n - 1), \ \mbox{for} \ n = 1, 2, \ldots 1994, \] and \[ 2\sqrt{a_{1995} - 1994} \geq a_1 + 1. \]