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: 10

2002 IMO, 2
Tags: geometry , incenter , circumcircle , perpendicular bisector , IMO 2002 , hojoo lee , Angle Chasing
The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

2002 IMO Shortlist, 2
Tags: inequalities , number theory , Divisors , IMO , IMO 2002 , IMO Shortlist
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.

2002 IMO, 4
Tags: inequalities , number theory , Divisors , IMO , IMO 2002 , IMO Shortlist
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.

2002 IMO Shortlist, 4
Tags: functional equation , IMO , IMO 2002 , IMO Shortlist , algebra
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.

2002 IMO Shortlist, 6
Tags: algebra , polynomial , IMO , IMO 2002 , IMO Shortlist , Laurentiu Panaitopol
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer. [i]Laurentiu Panaitopol, Romania[/i]

2002 IMO Shortlist, 6
Tags: geometry , combinatorial geometry , geometric inequality , IMO 2002 , IMO Shortlist , Convex hull
Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]

2002 IMO, 3
Tags: algebra , polynomial , IMO , IMO 2002 , IMO Shortlist , Laurentiu Panaitopol
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer. [i]Laurentiu Panaitopol, Romania[/i]

2002 IMO, 5
Tags: functional equation , IMO , IMO 2002 , IMO Shortlist , algebra
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.

2002 IMO Shortlist, 3
Tags: geometry , incenter , circumcircle , perpendicular bisector , IMO 2002 , hojoo lee , Angle Chasing
The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

2002 IMO, 6
Tags: geometry , combinatorial geometry , geometric inequality , IMO 2002 , IMO Shortlist , Convex hull
Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]