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: 15

1997 Romania National Olympiad, 1
Tags: function , induction , algebra , functional equation , IMO 1981 , IMO , romania
function $f:\mathbb{N}^{\star} \times \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star}$ ($\mathbb{N}^{\star}=\mathbb{N}\cup \{0\}$)with these conditon: 1- $f(0,x)=x+1$ 2- $f(x+1,0)=f(x,1)$ 3- $f(x+1,y+1)=f(x,f(x+1,y))$(romania 1997) find $f(3,1997)$

1981 IMO Shortlist, 1
Tags: number theory , least common multiple , algebra , polynomial , IMO , IMO 1981
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? [b]b.)[/b] For which $n>2$ is there exactly one set having this property?

1981 IMO Shortlist, 8
Tags: combinatorics , IMO , binomial coefficients , pascal s triangle , Combinatorial Identity , IMO 1981
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

1981 IMO, 1
Tags: number theory , least common multiple , algebra , polynomial , IMO , IMO 1981
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? [b]b.)[/b] For which $n>2$ is there exactly one set having this property?

1981 IMO, 2
Tags: combinatorics , IMO , binomial coefficients , pascal s triangle , Combinatorial Identity , IMO 1981
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

1981 IMO Shortlist, 7
Tags: function , algebra , functional equation , IMO , IMO 1981
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

1981 IMO, 3
Tags: function , algebra , functional equation , IMO , IMO 1981
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

1981 IMO, 2
Tags: geometry , incenter , circumcircle , similar triangles , IMO , IMO 1981
Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.

2006 Hong Kong TST., 2
Tags: function , functional equation , IMO , IMO 1981 , Hong Kong
The function $f(x,y)$, defined on the set of all non-negative integers, satisfies (i) $f(0,y)=y+1$ (ii) $f(x+1,0)=f(x,1)$ (iii) $f(x+1,y+1)=f(x,f(x+1,y))$ Find f(3,2005), f(4,2005)

1981 IMO, 3
Tags: algebra , polynomial , Vieta , number theory , IMO , IMO 1981 , Diophantine equation
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

1981 IMO Shortlist, 12
Tags: algebra , polynomial , Vieta , number theory , IMO , IMO 1981 , Diophantine equation
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

1981 IMO Shortlist, 17
Tags: geometry , incenter , circumcircle , similar triangles , IMO , IMO 1981
Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.

2010 ISI B.Stat Entrance Exam, 8
Tags: combinatorics , IMO , binomial coefficients , pascal s triangle , Combinatorial Identity , IMO 1981
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]

1981 IMO Shortlist, 15
Tags: trigonometry , geometry , incenter , geometric inequality , minimization , IMO , IMO 1981
Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]

1981 IMO, 1
Tags: trigonometry , geometry , incenter , geometric inequality , minimization , IMO , IMO 1981
Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]