2018 NZMOC Camp Selection Problems

1

Suppose that a,b,c and d are four different integers. Explain why (ab)(ac)(ad)(bc)(bd)(cd)must be a multiple of 12.

2

Find all pairs of integers (a,b) such that a2+abb=2018.

3

Show that amongst any 8 points in the interior of a 7×12 rectangle, there exists a pair whose distance is less than 5. Note: The interior of a rectangle excludes points lying on the sides of the rectangle.

4

Let P be a point inside triangle ABC such that CPA=90o and CBP=CAP. Prove that PXY=90o, where X and Y are the midpoints of AB and AC respectively.

5

Let a,b and c be positive real numbers satisfying 1a+2019+1b+2019+1c+2019=12019.Prove that abc40383.

6

The intersection of a cube and a plane is a pentagon. Prove the length of at least oneside of the pentagon differs from 1 metre by at least 20 centimetres.

7

Let N be the number of ways to colour each cell in a 2×50 rectangle either red or blue such that each 2×2 block contains at least one blue cell. Show that N is a multiple of 325, but not a multiple of 326

8

Let λ be a line and let M,N be two points on λ. Circles α and β centred at A and B respectively are both tangent to λ at M, with A and B being on opposite sides of λ. Circles γ and δ centred at C and D respectively are both tangent to λ at N, with C and D being on opposite sides of λ. Moreover A and C are on the same side of λ. Prove that if there exists a circle tangent to all circles α,β,γ,δ containing all of them in its interior, then the lines AC,BD and λ are either concurrent or parallel.

9

Let x,y,p,n,k be positive integers such that xn+yn=pk.Prove that if n>1 is odd, and p is an odd prime, then n is a power of p.

10

Find all functions f:RR such that f(x)f(y)=f(xy+1)+f(xy)2for all x,yR.