MathDB

P1

Part of Russian TST 2016

Problems(13)

Humpty config

Source: Russian TST 2016, Day 10 P1 (Group B)

4/19/2023
The circles ω1\omega_1 and ω2\omega_2 intersect at KK{} and LL{}. The line ℓ\ell touches the circles ω1\omega_1 and ω2\omega_2 at the points XX{} and YY{}, respectively. The point KK{} lies inside the triangle XYLXYL. The line XKXK intersects ω2\omega_2 a second time at the point ZZ{}. Prove that LYLY is the bisector of the angle XLZXLZ.
geometry
Checkered plane divided into dominoes

Source: Russian TST 2016, Day 11 P1 (Group B)

4/19/2023
The infinite checkered plane is divided into dominoes. If we move any horizontal domino of the partition by 49 cells to the right or left, we will also get a domino of the partition. If we move any vertical domino of the partition up or down by 49 cells, we will also get a domino of the partition. Can this happen?
combinatoricsdominoes
Geo with two squares

Source: Russian TST 2016, Day 13 P1

4/20/2023
The squares ABCDABCD and AXYZAXYZ are given. It turns out that CDXYCDXY is a cyclic quadrilateral inscribed in the circle Ω\Omega, and the points A,BA, B and ZZ{} lie inside this circle. Prove that either AB=AXAB = AX or AC⊥XYAC\perp{}XY.
geometrysquare