This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3

2012 India Regional Mathematical Olympiad, 6
Tags: subset , set theory , combinatorics of set , combinatorics
Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$. We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$. Define [i]WSUM [/i] for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms and the second the even numbered terms. (For example, if $A = \{2, 5, 7, 8\}$, [i]WSUM [/i] is $3(2 + 7) + 2(5 + 8)$.) Find the sum of [i]WSUMs[/i] over all the subsets of S. (Assume that WSUM for the null set is $0$.)

2012 India Regional Mathematical Olympiad, 4
Tags: subset , combinatorics of set , combinatorics
Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7\}$.

2012 India Regional Mathematical Olympiad, 4
Tags: subset , combinatorics of set , combinatorics
Let $X=\{1,2,3,...,12\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7,8\}$.