MathDB

3

Part of 2016 Tournament Of Towns

Problems(4)

Cutting a Square

Source: Tournament of Towns Spring 2016

2/22/2017
Given a square with side 1010. Cut it into 100100 congruent quadrilaterals such that each of them is inscribed into a circle with diameter 3\sqrt{3}. (5 points)
Ilya Bogdanov
combinatoricscombinatorial geometry
Determining angle in a Triangle

Source: Tournament of Towns Spring 2016

2/22/2017
Let MM be the midpoint of the base ACAC of an isosceles △ABC\triangle ABC. Points EE and FF on the sides ABAB and BCBC respectively are chosen so that AE≠CFAE \neq CF and ∠FMC=∠MEF=α\angle FMC = \angle MEF = \alpha. Determine ∠AEM\angle AEM. (6 points)
Maxim Prasolov
geometry
Combinatorial geometry with rectangle

Source: Tournament of Towns oral round p3

3/21/2016
Rectangle p∗q,p*q, where p,qp,q are relatively coprime positive integers with p<qp <q is divided into squares 1∗11*1.Diagonal which goes from lowest left vertice to highest right cuts triangles from some squares.Find sum of perimeters of all such triangles.
rectanglecombinatoricscombinatorial geometrygeometryTournament of Towns
Circle contains midpoint of diagonal

Source: Tournament of Towns 2016 Fall Tour, A Senior, Problem #3

4/22/2017
The quadrilateral ABCDABCD is inscribed in circle Ω\Omega with center OO, not lying on either of the diagonals. Suppose that the circumcircle of triangle AOCAOC passes through the midpoint of the diagonal BDBD. Prove that the circumcircle of triangle BODBOD passes through the midpoint of diagonal ACAC.
(A. Zaslavsky)
(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
geometrycircumcircle