2018 Singapore Senior Math Olympiad

2nd Round

1

You are given some equilateral triangles and squares, all with side length 1, and asked to form convex n sided polygons using these pieces. If both types must be used, what are the possible values of n, assuming that there is sufficient supply of the pieces?

2

In a convex quadrilateral ABCD,A<90o,B<90o and AB>CD. Points P and Q are on the segments BC and AD respectively. Suppose the triangles APD and BQC are similar. Prove that AB is parallel to CD.

3

Determine the largest positive integer n such that the following statement is true: There exists n real polynomials, P1(x),,Pn(x) such that the sum of any two of them have no real roots but the sum of any three does.

4

Let a,b,c,d be positive integers such that a+c=20 and ab+cd<1. Find the maximum possible value of ab+cd.

5

Starting with any n-tuple R0, n1, of symbols from A,B,C, we define a sequence R0,R1,R2,, according to the following rule: If Rj=(x1,x2,,xn), then Rj+1=(y1,y2,,yn), where yi=xi if xi=xi+1 (taking xn+1=x1) and yi is the symbol other than xi,xi+1 if xixi+1. Find all positive integers n>1 for which there exists some integer m>0 such that Rm=R0.