MathDB
P28 [Combinatorics] - Turkish NMO 1st Round - 2013

Source:

April 20, 2013
ceiling function

Problem Statement

In the beginning, there is a pair of positive integers (m,n)(m,n) written on the board. Alice and Bob are playing a turn-based game with the following move. At each turn, a player erases one of the numbers written on the board, and writes a different positive number not less than the half of the erased one. If a player cannot write a new number at some turn, he/she loses the game. For how many starting pairs (m,n)(m,n) from the pairs (7,79)(7,79), (17,71)(17,71), (10,101)(10,101), (21,251)(21,251), (50,405)(50,405), can Alice guarantee to win when she makes the first move?
<spanclass=latexbold>(A)</span> 4<spanclass=latexbold>(B)</span> 3<spanclass=latexbold>(C)</span> 2<spanclass=latexbold>(D)</span> 1<spanclass=latexbold>(E)</span> None of above <span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}