MathDB

3

Part of 1995 Bundeswettbewerb Mathematik

Problems(2)

a+b = n , x/a + y/b = 1

Source: 1995 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

1/27/2020
A natural number nn is called breakable if there exist positive integers a,b,x,ya,b,x,y such that a+b=na+b = n and xa+yb=1\frac{x}{a}+\frac{y}{b}= 1. Find all breakable numbers.
number theory
each diagonal of convex pentagon is parallel to one side of the pentagon

Source: 1994 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p3

1/27/2020
Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.
convexpentagongeometryparalleldiagonal