# ISI2014-PCB-A-1a

1 vote
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Let $x=(x_1, x_2, \dots x_n) \in \{0,1\}^n$ By $H(x)$ we mean the number of 1's in $(x_1, x_2, \dots x_n)$. Prove that $H(x) = \frac{1}{2} (n-\Sigma^n_{i=1} (-1)^{x_i})$.
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What do we mean by 'x'? Is it the set of n, n-bit binary numbers? How do we build set x? How do we consider the elements in set x?

## Related questions

1 vote
1
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Let $A$ be a $30 \times 40$ matrix having $500$ non-zero entries. For $1 \leq i \leq 30$, let $r_i$ be the number of non-zero entries in the $i$-th row, and for $1 \leq j \leq 40$, let $m_j$ be the number of non-zero entries in the $j$-th column. Show that there is a k such that $1 \leq k \leq 30$, $r_k \geq 17$ and there is an $l$ such that $1 \leq l \leq 40$, $m_l \leq 12$.
1 vote
Let $x, y$ be two non-negative integers $< 2^{32}$. By $x \wedge y$ we mean the integer represented by the bitwise logical $AND$ of the 32- bit binary representations of $x$ and $y$. For example, if $x = 13$ and $y = 6$, then $x \wedge y$ ... of the pseudo-code for the input $x = 13$? What will be the output of the pseudo-code for an arbitrary non-negative integer $x < 2^{32}$?
Let $A$ be a 30 40 matrix having 500 non-zero entries. For $1 \leq i \leq 30$, let $r_i$ be the number of non-zero entries in the $i$-th row, and for $1 \leq j \leq 40$, let $m_j$ be the number of non-zero entries in the $j$-th column. Suppose ... top of the stack contains the value $max_{1\leq i \leq 30} r_i$. Write pseudo-code for creating such a stack using a single scan of the matrix $A$.
A group of $15$ boys plucked a total of $100$ apples. Prove that two of those boys plucked the same number of apples.