MathDB

3

Part of 2014 Saudi Arabia IMO TST

Problems(4)

Circumcircle of RPS is tangent to BC

Source: Saudi Arabia IMO TST Day I Problem 3

7/22/2014
Let ABCABC be a triangle and let PP be a point on BCBC. Points MM and NN lie on ABAB and ACAC, respectively such that MNMN is not parallel to BCBC and AMPNAMP N is a parallelogram. Line MNMN meets the circumcircle of ABCABC at RR and SS. Prove that the circumcircle of triangle RPSRP S is tangent to BCBC.
geometrycircumcirclegeometry unsolved
Writing and n by n array of non-negative numbers

Source: Saudi Arabia IMO TST Day II Problem 3

7/22/2014
Show that it is possible to write a n×nn \times n array of non-negative numbers (not necessarily distinct) such that the sums of entries on each row and each column are pairwise distinct perfect squares.
inductionalgebrasystem of equationsnumber theory unsolvednumber theory
Turning coins on a table

Source: Saudi Arabia IMO TST Day III Problem 3

7/22/2014
There are 20152015 coins on a table. For i=1,2,…,2015i = 1, 2, \dots , 2015 in succession, one must turn over exactly ii coins. Prove that it is always possible either to make all of the coins face up or to make all of the coins face down, but not both.
combinatorics unsolvedcombinatorics
Switching pebbles in a lattice

Source: Saudi Arabia IMO TST Day IV Problem 3

7/22/2014
We are given a lattice and two pebbles AA and BB that are placed at two lattice points. At each step we are allowed to relocate one of the pebbles to another lattice point with the condition that the distance between pebbles is preserved. Is it possible after finite number of steps to switch positions of the pebbles?
analytic geometryvectorinductionmodular arithmeticcombinatorics unsolvedcombinatorics