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PrintTHE 68th ROMANIAN MATHEMATICAL OLYMPIAD
Romania algebra
Problem
Two children, Alex and Cristi, play several times a game, in which the winner receives points, and the loser points ( and are nonnegative integers, with , and in any game one of the children is the winner and the other is the loser). The final score is to , in Alex's favour. Cristi has won games. Determine the numbers and .
Bogdan Antohe
Bogdan Antohe
Solution
Denote by the number of games won by Alex. Then and . Subtracting the above equalities, we obtain , or .
and are positive integers, because , and are nonnegative integers, with and , because Alex has won more games. From follows that is an odd number, hence is odd, and is an odd divisor of .
We have two possible cases: and , where from , and , which is impossible; and , where from , and , hence and .
and are positive integers, because , and are nonnegative integers, with and , because Alex has won more games. From follows that is an odd number, hence is odd, and is an odd divisor of .
We have two possible cases: and , where from , and , which is impossible; and , where from , and , hence and .
Final answer
x = 13, y = 5
Techniques
Simple EquationsFactorization techniquesTechniques: modulo, size analysis, order analysis, inequalities