MathDB

3

Part of 2021 Francophone Mathematical Olympiad

Problems(2)

every point in the plane was colored in red or blue

Source: 2021 Francophone MO Juniors p3

4/3/2021
Every point in the plane was colored in red or blue. Prove that one the two following statements is true: ∙\bullet there exist two red points at distance 11 from each other; ∙\bullet there exist four blue points B1B_1, B2B_2, B3B_3, B4B_4 such that the points BiB_i and BjB_j are at distance ∣i−j∣|i - j| from each other, for all integers ii and jj such as 1≤i≤41 \le i \le 4 and 1≤j≤41 \le j \le 4.
combinatorial geometrycombinatoricsColoringFrancophoneRamsey Theory
tangent line to incircle of square wanted

Source: 2021 Francophone MO Seniors p3

4/3/2021
Let ABCDABCD be a square with incircle Γ\Gamma. Let MM be the midpoint of the segment [CD][CD]. Let P≠BP \neq B be a point on the segment [AB][AB]. Let E≠ME \neq M be the point on Γ\Gamma such that (DP)(DP) and (EM)(EM) are parallel. The lines (CP)(CP) and (AD)(AD) meet each other at FF. Prove that the line (EF)(EF) is tangent to Γ\Gamma
geometrysquareincircletangentFrancophone