MathDB

3

Part of 2013 BAMO

Problems(2)

Tromino tiling

Source: 2013 BAMO-8 #3

2/27/2016
Define a size-nn tromino to be the shape you get when you remove one quadrant from a 2n×2n2n \times 2n square. In the figure below, a size-11 tromino is on the left and a size-22 tromino is on the right. http://i.imgur.com/2065v7Y.png We say that a shape can be tiled with size-11 trominos if we can cover the entire area of the shape—and no excess area—with non-overlapping size-11 trominos. For example, a 2323 rectangle can be tiled with size-11 trominos as shown below, but a 3333 square cannot be tiled with size-11 trominos. http://i.imgur.com/UBPeeRw.png
a) Can a size-55 tromino be tiled by size-11 trominos?
b) Can a size-20132013 tromino be tiled by size-11 trominos? Justify your answers.
geometry
2013 BAMO12 p3 triangle with vertices circumcenters congruent to ABC

Source:

8/26/2019
Let HH be the orthocenter of an acute triangle ABCABC. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than 90o90^o.) Draw three circles: one passing through A,BA, B, and HH, another passing through B,CB, C, and HH, and finally, one passing through C,AC, A, and HH. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle ABCABC.
geometrycongruent trianglesCircumcenterorthocenter