Pre RMO type practice questions

  1. Let x_{1}, x_{2}, \ldots, x_{100} be positive integers such that x_{i}+x_{i+1}=k for all i, where k is a constant. I x_{10}=1, find the value of x_{1}.
  2. If a_{0}=1, a_{1}=1 and a_{n}=a_{n-1}a_{n-2}+1 for n > 1, then find out if a_{465}, a_{466} are even or odd.
  3. Two trains of equal length L, travelling at speeds V_{1} and V_{2} miles per hour in opposite directions, take T seconds to cross each other. Then, find L in feet (1 mile 1280 feet).
  4. A salesman sold two pipes at Rs. 12 each. His profit on one was 20% and the loss on the other was 20%. Then, on the whole, what amount did he gain or lose or did he break even?
  5. What is the digit in the units position of the integer 1! +2! +3! + \ldots +99!?
  6. Find the value of the following expression:

(1+q)(1+q^{2})(1+q^{4})(1+q^{8})(1+q^{16})(1+q^{32})(1+q^{64}) where q \neq 1.

Good luck for the ensuing oct pre RMO 🙂

Nalin Pithwa

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