MathDB

1

Part of 2012 Iran MO (2nd Round)

Problems(2)

Parallel and three circles

Source: Iran 2nd round 2012-Day1-P1

4/30/2012
Consider a circle C1C_1 and a point OO on it. Circle C2C_2 with center OO, intersects C1C_1 in two points PP and QQ. C3C_3 is a circle which is externally tangent to C2C_2 at RR and internally tangent to C1C_1 at SS and suppose that RSRS passes through QQ. Suppose XX and YY are second intersection points of PRPR and OROR with C1C_1. Prove that QXQX is parallel with SYSY.
geometryincentergeometric transformationgeometry proposed
Partition of natural numbers by 2-element sets

Source: Iran 2nd round 2012-Day2-P4

5/1/2012
a) Do there exist 22-element subsets A1,A2,A3,...A_1,A_2,A_3,... of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number nn, sum of the elements of AnA_n equals 1391+n1391+n?
b) Do there exist 22-element subsets A1,A2,A3,...A_1,A_2,A_3,... of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number nn, sum of the elements of AnA_n equals 1391+n21391+n^2?
Proposed by Morteza Saghafian
inequalitiescombinatorics proposedcombinatorics