MathDB

2

Part of 2015 Saudi Arabia IMO TST

Problems(4)

perpendicular wanted, circle with diameter AP, P random, orthocenter related

Source: 2015 Saudi Arabia IMO TST I p2

7/24/2020
Let ABCABC be a triangle with orthocenter HH. Let PP be any point of the plane of the triangle. Let Ω\Omega be the circle with the diameter APAP . The circle Ω\Omega cuts CACA and ABAB again at EE and FF , respectively. The line PHPH cuts Ω\Omega again at GG. The tangent lines to Ω\Omega at E,FE, F intersect at TT. Let MM be the midpoint of BCBC and LL be the point on MGMG such that ALAL and MTMT are parallel. Prove that LALA and LHLH are orthogonal.
Lê Phúc Lữ
geometryperpendicularcirclrorthocenter
2-player game on a horizontal 3 x 2015 white board

Source: 2015 Saudi Arabia IMO TST II p2

7/24/2020
Hamza and Majid play a game on a horizontal 3×20153 \times 2015 white board. They alternate turns, with Hamza going first. A legal move for Hamza consists of painting three unit squares forming a horizontal 1×31 \times 3 rectangle. A legal move for Majid consists of painting three unit squares forming a vertical 3×13\times 1 rectangle. No one of the two players is allowed to repaint already painted squares. The last player to make a legal move wins. Which of the two players, Hamza or Majid, can guarantee a win no matter what strategy his opponent chooses and what is his strategy to guarantee a win?
Lê Anh Vinh
combinatoricsgamegame strategy
collinear wanted, arc midpoint related

Source: 2015 Saudi Arabia IMO TST III p2

7/24/2020
Let ABCABC be a triangle and ω\omega its circumcircle. Point DD lies on the arc BCBC (not containing AA) of ω\omega and is different from B,CB, C and the midpoint of arc BCBC . The tangent line to ω\omega at DD intersects lines BC,CA,ABBC, CA,AB at A′,B′,C′A', B',C' respectively. Lines BB′BB' and CC′CC' intersect at EE. Line AA′AA' intersects again circle ω\omega at FF. Prove that the three points D,E,FD,E,F are colinear.
Malik Talbi
geometryarc midpointcollinear
total number of languages used in KAUST is n

Source: 2015 Saudi Arabia IMO TST IV p2

7/24/2020
The total number of languages used in KAUST is nn. For each positive integer k≤nk \le n, let AkA_k be the set of all those people in KAUST who can speak at least kk languages; and let BkB_k be the set of all people PP in KAUST with the property that, for any kk pairwise different languages (used in KAUST), PP can speak at least one of these kk languages. Prove that (a) If 2k≥n+12k \ge n + 1 then Ak⊆BkA_k \subseteq B_k (b) If 2k≤n+12k \le n + 1 then Ak⊇Bk.A_k \supseteq B_k.
Nguyễn Duy Thái Sơn
combinatorics