MathDB

2

Part of 2015 Tournament of Towns

Problems(3)

Reflection of a Point lying on Circumcircle

Source: Tournament of Towns Spring 2015 Senior A-level

2/24/2017
A point XX is marked on the base BCBC of an isosceles △ABC\triangle ABC, and points PP and QQ are marked on the sides ABAB and ACAC so that APXQAPXQ is a parallelogram. Prove that the point YY symmetrical to XX with respect to line PQPQ lies on the circumcircle of the △ABC\triangle ABC. (55 points)
geometrygeometric transformationreflectioncircumcircle
Junior A-Level Question Two (Fall 2015)

Source: Tournament of Towns Fall 2015 Junior A-Level Question Two

4/22/2017
From a set of integers {1,...,100}\{1,...,100\}, kk integers were deleted. Is it always possible to choose kk distinct integers from the remaining set such that their sum is 100100 if
(a) k=9k=9? (b) k=8k=8?
Tournament of Towns
Dividing a Grid into Polygons

Source: Tournament of Towns Fall 2015 Senior A-level

2/23/2017
A 10×1010 \times 10 square on a grid is split by 8080 unit grid segments into 2020 polygons of equal area (no one of these segments belongs to the boundary of the square). Prove that all polygons are congruent. (66 points)
combinatoricscombinatorial geometry