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: 14

2004 China Team Selection Test, 2
Tags: matrix , combinatorics , IMO , Ramsey Theory , Extremal combinatorics , IMO 2001 , IMO Shortlist
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2001 IMO, 2
Tags: IMO Shortlist , IMO 2001 , inequalities , hojoo lee , IMO , Hi
Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]

2001 IMO Shortlist, 6
Tags: IMO Shortlist , IMO 2001 , inequalities , hojoo lee , IMO , Hi
Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]

2001 IMO Shortlist, 2
Tags: geometry , Circumcenter , Triangle , IMO , IMO 2001 , hojoo lee , geometric inequality
Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

2001 IMO Shortlist, 8
Tags: matrix , combinatorics , IMO , Ramsey Theory , Extremal combinatorics , IMO 2001 , IMO Shortlist
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2001 IMO, 4
Tags: modular arithmetic , IMO 2001 , IMO , combinatorics , factorial , global
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

2001 IMO Shortlist, 2
Tags: modular arithmetic , IMO 2001 , IMO , combinatorics , factorial , global
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

2001 IMO, 1
Tags: geometry , Circumcenter , Triangle , IMO , IMO 2001 , hojoo lee , geometric inequality
Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

2001 IMO, 3
Tags: matrix , combinatorics , IMO , Ramsey Theory , Extremal combinatorics , IMO 2001 , IMO Shortlist
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2004 China Team Selection Test, 2
Tags: matrix , combinatorics , IMO , Ramsey Theory , Extremal combinatorics , IMO 2001 , IMO Shortlist
Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2001 IMO Shortlist, 8
Tags: geometry , ratio , trigonometry , similar triangles , angle bisector , IMO , IMO 2001
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?

2001 IMO, 5
Tags: geometry , ratio , trigonometry , similar triangles , angle bisector , IMO , IMO 2001
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?

2001 IMO Shortlist, 5
Tags: number theory , IMO , IMO 2001 , IMO Shortlist , composite numbers
Let $a > b > c > d$ be positive integers and suppose that \[ ac + bd = (b+d+a-c)(b+d-a+c). \] Prove that $ab + cd$ is not prime.

2001 IMO, 6
Tags: number theory , IMO , IMO 2001 , IMO Shortlist , composite numbers
Let $a > b > c > d$ be positive integers and suppose that \[ ac + bd = (b+d+a-c)(b+d-a+c). \] Prove that $ab + cd$ is not prime.