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ISI2011PCBA4a
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Consider six distinct points in a plane. Let $m$ and $M$ denote the minimum and maximum distance between any pair of points. Show that $M/m \geq \sqrt{3}$.
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ISI2011PCBA4b
Consider the following intervals on the real line: $A_1 = (13.3, 18.3) \: A_3 = (8.3, 23.3) − A_1 \cup A_2$ $A_2 = (10.8, 20.8) − A_1 \: A_4 = (5.8, 25.8) − A_1 \cup A_2 \cup A_3$ where $(a, b) = \{x : a < x < b\}$. Write pseudocode that ... given input $x \in (5.8, 25.8)$ belongs to, i.e., your pseudocode should calculate $i \in \{1, 2, 3, 4\}$ such that $x \in A_i$.
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Jun 3, 2016
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Algorithms
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jothee
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algorithms
algorithmdesign
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ISI2011PCBA3b
The numbers $1, 2, \dots , 10$ are arranged in a circle in some order. Show that it is always possible to find three adjacent numbers whose sum is at least $17$, irrespective of the ordering.
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Jun 3, 2016
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Combinatory
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jothee
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pigeonholeprinciple
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ISI2011PCBA3a
Consider an $m \times n$ integer lattice. A path from $(0, 0)$ to $(m, n)$ can use steps of $(1, 0)$, $(0, 1)$ or diagonal steps $(1, 1)$. Let $D_{m,n}$ be the number of such distinct paths. Prove that $D_{m,n} = \Sigma_k \begin{pmatrix} m \\ k \end{pmatrix} \begin{pmatrix} n+k \\ m \end{pmatrix}$
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Jun 3, 2016
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Combinatory
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jothee
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permutationandcombination
proof
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ISI2011PCBA2b
An $n \times n$ matrix is said to be tridiagonal if its entries $a_{ij}$ are zero except when $i−j \leq 1$ for $1 \leq i, \: j \leq n$. Note that only $3n − 2$ entries of a tridiagonal matrix are nonzero. Thus, an array $L$ of size ... matrix. Given $i, j$, write pseudocode to store $a_{ij}$ in $L$, and get the value of $a_{ij}$ stored earlier in $L$.
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Linear Algebra
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linearalgebra
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