RMO or Pre-RMO training: Homibhabha Science Foundation exam

Sample questions based on logic only:

  1. Given the sixty-four squares of a chess board are filled with positive integers one on each in such a way that each integer is the average of the integers on the neighbouring squares. (Two squares are neighbours if they share a common edge or vertex. Thus, a square can have 8, 5 or 3 neighbours depending on its position). Show that alll the sixty four entries are in fact equal.
  2. Let T be the set of all triples (a,b,c) of integers such that 1 \leq a <b<c \leq b. For each triple (a,b,c) in T, take the product abc. Add all these products corresponding to all triples in F. Prove that the sum is divisible by 7.
  3. In a class of 25 students, there are 17 cyclists, 13 swimmers and 8 weight lifters and no one is all the three. In a certain mathematics examination, 6 students get grades D or E. If the cyclists, swimmers and weight lifters all got grade B or C determine the number of students, who got grade A. Also, find the number of cyclists who are swimmers.

Any ideas ? Please wait for solutions tomorrow.

Nalin Pithwa.

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