15th Eötvös Competition Problems 1908

1.  m and n are odd. Show that 2^k divides m³ - n³ iff it divides m - n.

2.  Let a right angled triangle have side lengths a > b > c. Show that for n > 2, aⁿ > bⁿ + cⁿ.

3.  Let the vertices of a regular 10-gon be A₁, A₂, ... , A₁₀ in that order. Show that A₁A₄ - A₁A₂ is the radius of the circumcircle. 

Solutions

Problem 1

m and n are odd. Show that 2^k divides m³ - n³ iff it divides m - n.

Solution

m³ - n³ = (m-n)(m²+mn+n²) = (m-n) x odd. 

Problem 2

Let a right angled triangle have side lengths a > b > c. Show that for n > 2, aⁿ > bⁿ + cⁿ.

Solution

Induction on n. We have a² = b² + c². Hence a³ = ab² + ac² > b³ + c³, so it is true for n = 3. Suppose it is true for n. Then aⁿ⁺¹ > abⁿ + acⁿ > bⁿ⁺¹ + cⁿ⁺¹. 

Problem 3

Let the vertices of a regular 10-gon be A₁, A₂, ... , A₁₀ in that order. Show that A₁A₄ - A₁A₂ is the radius of the circumcircle.

Solution

AB is parallel to DG and AD is parallel to BC, so ABCD is a parallelogram. Similarly DEFG is a parallelogram. Obviously BC and EF are (part of) long diagonals and hence intersect at O. Hence DG = EF = EO + EF = AB + radius, which is the required result.