Games
🔗
(Belarus National Olympiad 2023 Grade 8 Day 1 P1, AoPS) A move on an unordered triple of numbers \((a,b,c)\) changes the triple to either \((a,b,2a+2b-c)\), \((a,2a+2c-b,c)\) or \((2b+2c-a,b,c)\). Can you perform a finite sequence of moves on the triple \((3,5,14)\) to get the triple \((3,13,6)\)?
🔗
Click here for the spoiler!
- Show that if a sequence of moves is performed on a triple \((a,b,c)\), then the resulting unordered triple is congruent to one of the triples \[ (a, b, - (a + b + c)), (a, - (a + b + c), c), (- (a + b + c), b, c), (a, b, c) \] modulo \(3\), that is, the residues of the entries of the resulting triple modulo \(3\) coincide with the residues of the entries of one of the above triples in some order.
🔗
(Belarus National Olympiad 2023 Grade 9 Day 1 P2, AoPS) A move on an unordered triple of numbers \((a,b,c)\) changes the triple to either \((a,b,2a+2b-c)\), \((a,2a+2c-b,c)\) or \((2b+2c-a,b,c)\). Can you perform a finite sequence of moves on the triple \((3,5,14)\) to get the triple \((9, 8, 11)\)?
🔗
Click here for the spoiler!
- Show that the mod \(4\) congruence class of the sum of the entries of such a triple remains invariant under the allowed moves if the sum of the entries of the initial triple is congruent to \(2\) modulo \(4\).
Lecture notes for Math Olympiad (IOQM, RMO, INMO)
Click on the icons below to download.
Topics Links Algebra Combinatorics Geometry Number Theory INMO Training Camp IMO Training Camp MOPSS Simon Marais Mathematics Competition Resources