MathDB

3

Part of 2014 Saint Petersburg Mathematical Olympiad

Problems(3)

Some geometry

Source: St Petersburg Olympiad 2014, Grade 10, P3

10/26/2017
DD is inner point of triangle ABCABC. EE is on BDBD and CE=BDCE=BD. ∠ABD=∠ECD=10,∠BAD=40,∠CED=60\angle ABD=\angle ECD=10,\angle BAD=40,\angle CED=60 Prove, that AB>ACAB>AC
geometry
Coloring numbers

Source: St Petersburg Olympiad 2014, Grade 11, P3

10/24/2017
NN in natural. There are natural numbers from N3N^3 to N3+NN^3+N on the board. aa numbers was colored in red, bb numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that b∣ab|a
number theorycombinatoricsalgebrainequalities
Deputies in commisions

Source: St Petersburg Olympiad 2014, Grade 9, P3

10/27/2017
100100 deputies formed 450450 commissions. Each two commissions has no more than three common deputies, and every 55 - no more than one. Prove that, that there are 44 commissions that has exactly one common deputy each.
combinatorics