Junior Balkan MO 2014 1 Find all triples of primes (p,q,r) satisfying 3p4−5q4−4r2 = 26. 2 Consider an acute triangle ABC of area S. Let CD ⊥ AB (D ∈ AB), DM ⊥ AC (M ∈ AC) and DN ⊥ BC (N ∈ BC). Denote by H and H the orthocentres of the triangles MNC, 1 2 respectively MND. Find the area of the quadrilateral AH BH in terms of S. 1 2 3 For positive real numbers a,b,c with abc = 1 prove that (cid:0)a+ 1(cid:1)2 +(cid:0)b+ 1(cid:1)2 +(cid:0)c+ 1(cid:1)2 ≥ b c a 3(a+b+c+1) 4 Forapositiveintegern, twopayersAandB playthefollowinggame: Givenapileofsstones, the players take turn alternatively with A going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of n stones. The winner is the one who takes the last stone. Assuming both A and B play perfectly, for how many values of s the player A cannot win? This file was downloaded from the AoPS Math Olympiad Resources Page Page 1 http://www.artofproblemsolving.com/