MathDB

8

Part of 2013 Tuymaada Olympiad

Problems(2)

divisibility the number of ways

Source: Tuymaada 2013, Day 2, Problem 8 Seniors

7/26/2013
Cards numbered from 1 to 2n2^n are distributed among kk children, 1≤k≤2n1\leq k\leq 2^n, so that each child gets at least one card. Prove that the number of ways to do that is divisible by 2k−12^{k-1} but not by 2k2^k.
M. Ivanov
linear algebramatrixinductioncombinatorics proposedcombinatorics
inequality with areas

Source: Tuymaada 2013, Day 2, Problem 8 Juniors

7/26/2013
The point A1A_1 on the perimeter of a convex quadrilateral ABCDABCD is such that the line AA1AA_1 divides the quadrilateral into two parts of equal area. The points B1B_1, C1C_1, D1D_1 are defined similarly. Prove that the area of the quadrilateral A1B1C1D1A_1B_1C_1D_1 is greater than a quarter of the area of ABCDABCD.
L. Emelyanov
inequalitiesgeometryperimeteranalytic geometrytrigonometrygeometry proposed