MathDB

5

Part of 2020 Tournament Of Towns

Problems(4)

Tot Spring O level Problem 5

Source:

6/3/2020
On the 8×8 chessboard there are two identical markers in the squares a1 and c3. Alice and Bob in turn make the following moves (the first move is Alice’s): a player picks any marker and moves it horizontally to the right or vertically upwards through any number of squares. The aim of each player is to get tothe square h8. Which player has a winning strategy no matter what does his opponent? (There may be only one marker on a square,the markers may not go through each other.)
The 8x8 chessboard consists of columns lettered a to h from left to right and rows numbered 1-8 from bottom to top
https://cdn.artofproblemsolving.com/attachments/1/f/4c5548f606fda915e0a50a8cf886ff93e1f86d.png
ToT
right angle wanted, bases of inscirbed trapezoid AB =3 CD

Source: Tournament of Towns, Junior A-Level Paper, Spring 2020 , p5

6/4/2020
Let ABCDABCD be an inscribed trapezoid. The base ABAB is 33 times longer than CDCD. Tangents to the circumscribed circle at the points AA and CC intersect at the point KK. Prove that the angle KDAKDA is a right angle.
Alexandr Yuran
right angletrapezoidequal segmentstrapeziumgeometry
all possible D for any choice of line l lie on a single circle.

Source: Tournament of Towns, Senior O- Level Paper, Spring 2020 , p5

6/4/2020
Given are two circles which intersect at points PP and QQ. Consider an arbitrary line ℓ\ell through QQ, let the second points of intersection of this line with the circles be AA and BB respectively. Let CC be the point of intersection of the tangents to the circles in those points. Let DD be the intersection of the line ABAB and the bisector of the angle CPQCPQ. Prove that all possible DD for any choice of ℓ\ell lie on a single circle.
Alexey Zaslavsky
geometrycirclesLocus
4 copies of triangle on sphere can cover entire sphere

Source: Tournament of Towns 2020 oral p5 (15 March 2020)

5/17/2020
A triangle is given on a sphere of radius 11, the sides of which are arcs of three different circles of radius 11 centered in the center of a sphere having less than π\pi in length and an area equal to a quarter of the area of the sphere. Prove that four copies of such a triangle can cover the entire sphere.
A. Zaslavsky
geometry3D geometryspherecombinatorial geometryTilingcoveringcombinatorics