This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 1311

2012 Indonesia TST, 1
Tags: induction , algebra proposed , algebra , binary representation , Sequence , recurrence relation
The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

2005 Junior Balkan Team Selection Tests - Romania, 18
Tags: algebra proposed , algebra
Consider two distinct positive integers $a$ and $b$ having integer arithmetic, geometric and harmonic means. Find the minimal value of $|a-b|$. [i]Mircea Fianu[/i]

2010 Singapore MO Open, 2
Tags: algebra proposed , algebra
Let $(a_n), (b_n)$, $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$ $a_{n+1} = 7a_n + 12b_n + 6$ $b_{n+1} = 4a_n + 7b_n + 3$ Prove that $a_n^2$ is the difference of two consecutive cubes.

2007 Serbia National Math Olympiad, 1
Tags: function , induction , algebra proposed , algebra
Let $k$ be a natural number. For each function $f : \mathbb{N}\to \mathbb{N}$ define the sequence of functions $(f_{m})_{m\geq 1}$ by $f_{1}= f$ and $f_{m+1}= f \circ f_{m}$ for $m \geq 1$ . Function $f$ is called $k$-[i]nice[/i] if for each $n \in\mathbb{N}: f_{k}(n) = f (n)^{k}$. (a) For which $k$ does there exist an injective $k$-nice function $f$ ? (b) For which $k$ does there exist a surjective $k$-nice function $f$ ?

2006 Austrian-Polish Competition, 8
Tags: induction , absolute value , algebra proposed , algebra
Let $A\subset \{x|0\le x<1\}$ with the following properties: 1. $A$ has at least 4 members. 2. For all pairwise different $a,b,c,d\in A$, $ab+cd\in A$ holds. Prove: $A$ has infinetly many members.

2009 Ukraine National Mathematical Olympiad, 1
Tags: analytic geometry , algebra proposed , algebra
Build the set of points $( x, y )$ on coordinate plane, that satisfies equality: \[ \sqrt{1-x^2}+\sqrt{1-y^2}=2-x^2-y^2.\]

2005 Romania National Olympiad, 4
Tags: function , limit , algebra proposed , algebra
Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function. a) Prove that $f$ is continous; b) Prove that there exists an unique function $g:[0,\infty)\to\mathbb{R}$ such that for all $x\geq 0$ we have \[ f(x+g(x)) = f(g(x)) - g(x) . \]

2001 Hungary-Israel Binational, 3
Tags: function , algebra proposed , algebra
Find all continuous functions $f : \mathbb{R}\to\mathbb{R}$ such that for all $x \in\mathbb{ R}$, \[f (f (x)) = f (x)+x.\]

1991 Brazil National Olympiad, 3
Tags: algebra , polynomial , function , rational function , algebra proposed
Given $k > 0$, the sequence $a_n$ is defined by its first two members and \[ a_{n+2} = a_{n+1} + \frac{k}{n}a_n \] a)For which $k$ can we write $a_n$ as a polynomial in $n$? b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$).

2014 Korea National Olympiad, 2
Tags: function , algebra proposed , algebra , functional equation , KMO , Korea
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

2003 India IMO Training Camp, 3
Tags: function , algebra proposed , algebra
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]

2005 Bulgaria Team Selection Test, 3
Tags: function , algebra proposed , algebra
Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.

2007 Moldova Team Selection Test, 2
Tags: algebra , polynomial , calculus , derivative , induction , algebra proposed
If $b_{1}, b_{2}, \ldots, b_{n}$ are non-negative reals not all zero, then prove that the polynomial \[x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0\] has only one positive root $p$, which is simple. Moreover prove that any root of the polynomial does not exceed $p$ in absolute value.

1986 India National Olympiad, 8
Tags: blogs , algebra proposed , algebra
Suppose $ A_1,\dots, A_6$ are six sets each with four elements and $ B_1,\dots,B_n$ are $ n$ sets each with two elements, Let $ S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n$. Given that each elements of $ S$ belogs to exactly four of the $ A$'s and to exactly three of the $ B$'s, find $ n$.

2009 ISI B.Math Entrance Exam, 4
Tags: trigonometry , algebra proposed , algebra
Find the values of $x,y$ for which $x^2+y^2$ takes the minimum value where $(x+5)^2+(y-12)^2=14$.

2010 Bosnia Herzegovina Team Selection Test, 3
Tags: function , algebra proposed , algebra
Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds: $a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$ $b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$

2011 Pre - Vietnam Mathematical Olympiad, 2
Tags: function , algebra proposed , algebra
Find all function $f,g: \mathbb{Q} \to \mathbb{Q}$ such that \[\begin{array}{l} f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y \\ g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y \\ \end{array}\] for all $x,y \in \mathbb{Q}$.

2000 Iran MO (3rd Round), 2
Tags: function , algebra proposed , algebra
Let $A$ and $B$ be arbitrary finite sets and let $f: A\longrightarrow B$ and $g: B\longrightarrow A$ be functions such that $g$ is not onto. Prove that there is a subset $S$ of $A$ such that $\frac{A}{S}=g(\frac{B}{f(S)})$.

2005 Kazakhstan National Olympiad, 4
Tags: function , algebra proposed , algebra , Kazakhstan
Find all functions $f :\mathbb{R}\to\mathbb{R}$, satisfying the condition $f(f(x)+x+y)=2x+f(y)$ for any real $x$ and $y$.

1997 All-Russian Olympiad, 1
Tags: quadratics , modular arithmetic , algebra , polynomial , algebra proposed
Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? [i]N. Agakhanov[/i]

1998 Spain Mathematical Olympiad, 1
Tags: search , algebra proposed , algebra
Find the tangents of the angles of a triangle knowing that they are positive integers.

1988 IberoAmerican, 5
Tags: quadratics , algebra , system of equations , algebra proposed
Consider all the numbers of the form $x+yt+zt^2$, with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$. Prove that if $x+yt+zt^2\not= 0$, then there exist rational numbers $u,v,w$ such that \[(x+yt+z^2)(u+vt+wt^2)=1\]

1986 Iran MO (2nd round), 3
Tags: function , algebra proposed , algebra
Prove that \[\arctan \frac 12 +\arctan \frac 13 = \frac{\pi}{4}.\]

2014 Contests, 2
Tags: function , induction , algebra proposed , algebra
Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that : \[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\] [i]Proposed by Mohammad Ahmadi[/i]

2013 Romania National Olympiad, 2
Tags: function , algebra proposed , algebra
Given $f:\mathbb{R}\to \mathbb{R}$ an arbitrary function and $g:\mathbb{R}\to \mathbb{R}$ a function of the second degree, with the property: for any real numbers m and n equation $f\left( x \right)=mx+n$ has solutions if and only if the equation $g\left( x \right)=mx+n$ has solutions Show that the functions $f$ and $g$ are equal.