MathDB

3

Part of 2001 IMO Shortlist

Problems(4)

IMO ShortList 2001, geometry problem 3

Source: IMO ShortList 2001, geometry problem 3

9/30/2004
Let ABCABC be a triangle with centroid GG. Determine, with proof, the position of the point PP in the plane of ABCABC such that AP⋅AG+BP⋅BG+CP⋅CGAP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG is a minimum, and express this minimum value in terms of the side lengths of ABCABC.
geometryparallelogramminimizationTriangleIMO Shortlist
IMO ShortList 2001, number theory problem 3

Source: IMO ShortList 2001, number theory problem 3

9/30/2004
Let a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}, and a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4. Determine a1414 a_{14^{14}}.
modular arithmeticIMO Shortlistnumber theoryInteger sequencerecurrence relation
IMO ShortList 2001, algebra problem 3

Source: IMO ShortList 2001, algebra problem 3

9/30/2004
Let x1,x2,…,xnx_1,x_2,\ldots,x_n be arbitrary real numbers. Prove the inequality x11+x12+x21+x12+x22+⋯+xn1+x12+⋯+xn2<n. \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}.
inequalitiestrigonometryalgebraIMO Shortlist
IMO ShortList 2001, combinatorics problem 3

Source: IMO ShortList 2001, combinatorics problem 3, HK 2009 TST 2 Q.2

9/30/2004
Define a k k-clique to be a set of k k people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.
combinatoricsgraph theoryClique numberIMO Shortlist