A5. From the numbers 1 to 100, Jaap wants to write as many (different) numbers as possible on a piece of paper. No two numbers on the paper may add up to 125. What is the maximum number of numbers he can write on the paper?
A) 50
B) 61
C) 62
D) 63
E) 64
A5. C) 62 Jaap can certainly write down 62 numbers, for example, the numbers 1 through 62 (since $62+61<125$). Jaap cannot write down more than 62 numbers. Indeed, the numbers 25 to 100 can be divided into pairs that always add up to 125: $25+100=125$, $26+99=125$, and so on up to $62+63=125$. Therefore, he must miss at least one number from each of these 38 pairs. In total, he can thus write down no more than $100-38=62$ numbers.