MathDB

6

Part of 2007 IMO Shortlist

Problems(3)

a^2_1 + a^2_2 + ... + a^2_100 = 1

Source: IMO Shortlist 2007, A6, AIMO 2008, TST 7, P1

7/13/2008
Let a1,a2,…,a100 a_1, a_2, \ldots, a_{100} be nonnegative real numbers such that a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1. Prove that a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. Author: Marcin Kuzma, Poland
inequalitiesIMO Shortlist
Find the smallest positive real number

Source: IMO Shortlist 2007, G6 / USA TST 2008, Day 2, Problem 6

7/13/2008
Determine the smallest positive real number k k with the following property. Let ABCD ABCD be a convex quadrilateral, and let points A1 A_1, B1 B_1, C1 C_1, and D1 D_1 lie on sides AB AB, BC BC, CD CD, and DA DA, respectively. Consider the areas of triangles AA1D1 AA_1D_1, BB1A1 BB_1A_1, CC1B1 CC_1B_1 and DD1C1 DD_1C_1; let S S be the sum of the two smallest ones, and let S1 S_1 be the area of quadrilateral A1B1C1D1 A_1B_1C_1D_1. Then we always have kS1≥S kS_1\ge S.
Author: Zuming Feng and Oleg Golberg, USA
geometrycircumcirclesymmetryratioIMO Shortlist
Select an easier sub-problem and you get an IMO problem

Source: IMO Shortlist 2007, N6

7/13/2008
Let k k be a positive integer. Prove that the number (4 \cdot k^2 \minus{} 1)^2 has a positive divisor of the form 8kn \minus{} 1 if and only if k k is even. [url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above. Author: Kevin Buzzard and Edward Crane, United Kingdom
quadraticsnumber theoryIMOIMO 2007Vieta JumpingDivisibility