MathDB

Problem 2

Part of 2005 Federal Math Competition of S&M

Problems(3)

find angle in circle and triangle configuration (Serbia MO 2005 1st Grade P2)

Source:

4/11/2021
Let ABCABC be an acute triangle. Circle kk with diameter ABAB intersects ACAC and BCBC again at MM and NN respectively. The tangents to kk at MM and NN meet at point PP. Given that CP=MNCP=MN, determine ∠ACB\angle ACB.
geometry
Labeling 3x3 board with + or - (Serbia MO 2005 2nd Grade P2)

Source:

4/11/2021
Every square of a 3×33\times3 board is assigned a sign ++ or −-. In every move, one square is selected and the signs are changed in the selected square and all the neighboring squares (two squares are neighboring if they have a common side). Is it true that, no matter how the signs were initially distributed, one can obtain a table in which all signs are −- after finitely many moves?
game
in hexagon, lines connecting opposite midpts concurrent

Source: Serbia MO 2005 3&4th Grades P2

4/11/2021
Suppose that in a convex hexagon, each of the three lines connecting the midpoints of two opposite sides divides the hexagon into two parts of equal area. Prove that these three lines intersect in a point.
geometryhexagon