(a) 28 integers are chosen from the interval $$[104, 208]$$. Show that there exist two of them having a common prime divisor.
(b) AB is a line segment. C is a point on AB. ACPQ and CBRS are squares drawn on the same side of AB. Prove the S is the orthocentre of the triangle APB.

For part (a), there are 19 primes in the interval $$[104,208]$$, and only 6 more integers can be found (one multiple of each of 2, 3, 5, 7, 11 and 13) that share no common prime factor with any other chosen number, giving a maximum of $$19+6=25$$ pairwise coprime-in-this-sense numbers; choosing 28 numbers forces at least two to share a common prime factor by the pigeonhole principle. For part (b), drawing a line from B through S to meet AP at M, since PC (the diagonal of square ACPQ) and BM both pass through S, tracking the angles around S using the square's $$45^\circ$$ diagonal angles shows $$\angle PMS = 90^\circ$$, meaning BM is perpendicular to AP; since PC and BM are both altitudes of triangle APB meeting at S, S is the orthocenter.