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?
2018 Singapore Senior Math Olympiad
2nd Round
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.
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.
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.
Starting with any n-tuple R0, n≥1, 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 xi≠xi+1. Find all positive integers n>1 for which there exists some integer m>0 such that Rm=R0.