MathDB

4

Part of 2019 Tournament Of Towns

Problems(5)

fixed midpoint of bases of isosceles triangles coincide, fixed angle

Source: Tournament of Towns, Junior O-Level , Spring 2019 p4

5/9/2020
Isosceles triangles with a fixed angle α\alpha at the vertex opposite to the base are being inscribed into a rectangle ABCDABCD so that this vertex lies on the side BCBC and the vertices of the base lie on the sides ABAB and CDCD. Prove that the midpoints of the bases of all such triangles coincide.
(Igor Zhizhilkin)
fixedFixed pointisoscelesgeometry
red and blue endpoints of segments in 100-gon, numbers on vertices

Source: Tournament of Towns, Junior A-Level , Spring 2019 p4

5/13/2020
Each segment whose endpoints are the vertices of a given regular 100100-gon is colored red, if the number of vertices between its endpoints is even, and blue otherwise. (For example, all sides of the 100100-gon are red.) A number is placed in every vertex so that the sum of their squares is equal to 11. On each segment the product of the numbers at its endpoints is written. The sum of the numbers on the blue segments is subtracted from the sum of the numbers on the red segments. What is the greatest possible result?
(Ilya Bogdanov)
combinatoricsColoringSumProduct
a magician's trick, with 13 empty closed boxes

Source: Tournament of Towns, Senior O-Level , Spring 2019 p4

5/11/2020
A magician and his assistant are performing the following trick. There is a row of 1313 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistant knows which boxes contain coins. The magician returns and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects four boxes, which are then simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always successfully perform the trick.
(Igor Zhizhilkin)
[url=https://artofproblemsolving.com/community/c6h1801447p11962869]junior version posted here
combinatoricsgamegame strategy
sum of any 41 consecutive numbers on this circle is a multiple of 41^2

Source: Tournament of Towns, Junior O-Level , Fall 2019 p4

4/19/2020
There are given 10001000 integers a1,...,a1000a_1,... , a_{1000}. Their squares a12,...,a10002a^2_1, . . . , a^2_{1000} are written in a circle. It so happened that the sum of any 4141 consecutive numbers on this circle is a multiple of 41241^2. Is it necessarily true that every integer a1,...,a1000a_1,... , a_{1000} is a multiple of 4141?
(Boris Frenkin)
number theoryconsecutivemultiplecircleSum of Squares
PQ bisects segment connecting midpoints of CB and AB

Source: Tournament of Towns, Junior A-Level , Fall 2019 p4

4/19/2020
Let OPOP and OQOQ be the perpendiculars from the circumcenter OO of a triangle ABCABC to the internal and external bisectors of the angle BB. Prove that the linePQ PQ divides the segment connecting midpoints of CBCB and ABAB into two equal parts.
(Artemiy Sokolov)
angle bisectorgeometrybisects segmentperpendicular