## 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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