MathDB

1

Part of 2013 Pan African

Problems(2)

An integer for which n(n+2013) is a square

Source: Pan African MO 2013 Q1

6/30/2013
A positive integer nn is such that n(n+2013)n(n+2013) is a perfect square. a) Show that nn cannot be prime. b) Find a value of nn such that n(n+2013)n(n+2013) is a perfect square.
algebradifference of squaresspecial factorizationsnumber theory unsolvednumber theory
Equal sets of angles in a trapezium

Source: Pan African MO 2013 Q4

6/30/2013
Let ABCDABCD be a convex quadrilateral with ABAB parallel to CDCD. Let PP and QQ be the midpoints of ACAC and BDBD, respectively. Prove that if ∠ABP=∠CBD\angle ABP=\angle CBD, then ∠BCQ=∠ACD\angle BCQ=\angle ACD.
geometrytrapezoidparallelogramratiogeometry unsolved