MathDB

3

Part of 2004 Estonia National Olympiad

Problems(4)

segment bisect area of convex quadrilateral , parallel wanted

Source: 2004 Estonia National Olympiad Final Round grade 9 p3

3/25/2020
On the sides AB,BCAB , BC of the convex quadrilateral ABCDABCD lie points MM and NN such that ANAN and CMCM each divide the quadrilateral ABCDABCD into two equal area parts. Prove that the line MNMN and ACAC are parallel.
geometrybisects areaconvexparallel
perfect square 44...488..89

Source: 2004 Estonia National Olympiad Final Round grade 10 p3

3/25/2020
The teacher had written on the board a positive integer consisting of a number of 44s followed by the same number of 88s followed . During the break, Juku stepped up to the board and added to the number one more 44 at the start and a 99 at the end. Prove that the resulting number is an a square. of an integer.
number theoryDigitsPerfect Square
AK+ BL+CM =0 in vectors => ABC equilateral

Source: 2004 Estonia National Olympiad Final Round grade 12 p3

3/25/2020
Let K,L,MK, L, M be the feet of the altitudes drawn from the vertices A,B,CA, B, C of triangle ABCABC, respectively. Prove that AK→+BL→+CM→=O→\overrightarrow{AK} + \overrightarrow{BL} + \overrightarrow{CM} = \overrightarrow{O} if and only if ABCABC is equilateral.
vectorEquilateralgeometry
17 points out of 25 lattice points, 3 collinear whose line does not pass center

Source: 2004 Estonia National Olympiad Final Round grade 11 p 3

3/25/2020
From 2525 points in a plane, both of whose coordinates are integers of the set {−2,−1,0,1,2}\{-2,-1, 0, 1, 2\}, some 1717 points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.
combinatoricscombinatorial geometry