MathDB

4

Part of 2016 Czech-Polish-Slovak Junior Match

Problems(2)

AK + KM = BL + LM wanted, AB = CK = CL given, perp. bisectors related

Source: Czech-Polish-Slovak Junior Match 2016, individual p4 CPSJ

1/20/2020
We are given an acute-angled triangle ABCABC with AB<AC<BCAB < AC < BC. Points KK and LL are chosen on segments ACAC and BCBC, respectively, so that AB=CK=CLAB = CK = CL. Perpendicular bisectors of segments AKAK and BLBL intersect the line ABAB at points PP and QQ, respectively. Segments KPKP and LQLQ intersect at point MM. Prove that AK+KM=BL+LMAK + KM = BL + LM.
Poland
geometryequal segmentsperpendicular bisector
6 square tile in a 11x11 square table

Source: Czech-Polish-Slovak Junior Match 2016, team p4 CPSJ

1/20/2020
Several tiles congruent to the one shown in the picture below are to be fit inside a 11×1111 \times 11 square table, with each tile covering 66 whole unit squares, no sticking out the square and no overlapping. (a) Determine the greatest number of tiles which can be placed this way. (b) Find, with a proof, all unit squares which have to be covered in any tiling with the maximal number of tiles. https://cdn.artofproblemsolving.com/attachments/c/d/23d93e9d05eab94925fc54006fe05123f0dba9.png Poland
Tilingcombinatoricscombinatorial geometry