MathDB

6

Part of 2008 Saint Petersburg Mathematical Olympiad

Problems(3)

Inscribed quadrilateral (again)

Source: St Petersburg 2008 10th grade #5

7/28/2011
In cyclic quadrilateral ABCDABCD rays ABAB and DCDC intersect at point EE, while segments ACAC and BDBD intersect at FF. Point PP is on ray EFEF such that angles BPEBPE and CPECPE are congruent. Prove that angles APBAPB and DPCDPC are also equal.
geometrycyclic quadrilateralgeometry proposed
Good diagonals

Source: St Petersburg 2008 10th grade #6

7/29/2011
A diagonal of a 100-gon is called good if it divides the 100-gon into two polygons each with an odd number of sides. A 100-gon was split into triangles with non-intersecting diagonals, exactly 49 of which are good. The triangles are colored into two colors such that no two triangles that border each other are colored with the same color. Prove that there is the same number of triangles colored with one color as with the other.
Fresh translation; slightly reworded.
geometrygeometric transformationcombinatorics proposedcombinatorics
Maximum GCD

Source: St Petersburg Olympiad 2008, Grade 11, P6

8/30/2017
a+b+c≤3000000a+b+c \leq 3000000 and a≠b≠c≠aa\neq b \neq c \neq a and a,b,ca,b,c are naturals. Find maximum GCD(ab+1,ac+1,bc+1)GCD(ab+1,ac+1,bc+1)
number theorygreatest common divisor