MathDB

3

Part of 2014 Saudi Arabia GMO TST

Problems(3)

divide a square into finitely many white and green rectangles

Source: 2014 Saudi Arabia GMO TST I p3

7/26/2020
Turki has divided a square into finitely many white and green rectangles, each with sides parallel to the sides of the square. Within each white rectangle, he writes down its width divided by its height. Within each green rectangle, he writes down its height divided by its width. Finally, he calculates SS, the sum of these numbers. If the total area of white rectangles equals the total area of green rectangles, determine the minimum possible value of SS.
rectanglecombinatoricscombinatorial geometrysquare
concurrency wanted, circle concentric with incircle

Source: 2014 Saudi Arabia GMO TST day III p3

7/31/2020
Let ABCABC be a triangle, II its incenter, and ω\omega a circle of center II. Points A′,B′,C′A',B', C' are on ω\omega such that rays IA′,IB′,IC′,IA', IB', IC', starting from II intersect perpendicularly sides BC,CA,ABBC, CA, AB, respectively. Prove that lines AA′,BB′,CC′AA', BB', CC' are concurrent.
geometryconcurrencyconcurrentincentermoving points
BC x DE = CD x PQ wanted, if BP xQE = PQ^ 2 inside a cyclic pentagon

Source: 2014 Saudi Arabia GMO TST day II p3

7/31/2020
Let ABCDEABCDE be a cyclic pentagon such that the diagonals ACAC and ADAD intersect BEBE at PP and QQ, respectively, with BP⋅QE=PQ2BP \cdot QE = PQ^2. Prove that BC⋅DE=CD⋅PQBC \cdot DE = CD \cdot PQ.
geometryCyclicpentagon