MathDB

Problem 4

Part of 1999 Ukraine National Mathematical Olympiad

Problems(2)

compass+straightedge construction of triangle given circumcircle

Source: Ukraine 1999 Grade 9 P4

5/10/2021
The bisectors of angles A,B,CA,B,C of a triangle ABCABC intersect the circumcircle of the triangle at A1,B1,C1A_1,B_1,C_1, respectively. Let PP be the intersection of the lines B1C1B_1C_1 and ABAB, and QQ be the intersection of the lines B1A1B_1A_1 and BCBC. Show how to construct the triangle ABCABC by a ruler and a compass, given its circumcircle, points PP and QQ, and the halfplane determined by PQPQ in which point BB lies.
geometryconstructioncircumcircle
writing fibonacci-like sequence

Source: Ukraine 1999 Grade 10 P4

5/11/2021
Two players alternately write integers on a blackboard as follows: the first player writes a1a_1 arbitrarily, then the second player writes a2a_2 arbitrarily, and thereafter a player writes a number that is equal to the sum of the two preceding numbers. The player after whose move the obtained sequence contains terms such that ai−aja_i-a_j and ai+1−aj+1 (i≠j)a_{i+1}-a_{j+1}~(i\ne j) are divisible by 19991999, wins the game. Which of the players has a winning strategy?
number theorygame