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3

Part of 2015 Iran MO (2nd Round)

Problems(2)

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Source: Iran Second Round 2015 - Problem 3 Day 1

5/7/2015
Consider a triangle ABCABC . The points D,ED,E are on sides AB,ACAB,AC such that BDECBDEC is a cyclic quadrilateral. Let PP be the intersection of BEBE and CDCD. HH is a point on ACAC such that ∠PHA=90∘\angle PHA = 90^{\circ}. Let M,NM,N be the midpoints of AP,BCAP,BC. Prove that: ACD∼MNH ACD \sim MNH .
geometry2015
Iran Second Round 2015 - Problem 6 day2

Source:

5/8/2015
Let n≥50n \ge 50 be a natural number. Prove that nn is expressible as sum of two natural numbers n=x+yn=x+y, so that for every prime number pp such that p∣x p\mid x or p∣yp\mid y we have n≥p \sqrt{n} \ge p . For example for n=94n=94 we have x=80,y=14x=80, y=14.
number theoryIranprime numbersalgebra