# ISI2011-PCB-A-4a

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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 $M/m \geq \sqrt{3}$.

## Related questions

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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 pseudo-code that calculates (without using ... a given input $x \in (5.8, 25.8)$ belongs to, i.e., your pseudo-code should calculate $i \in \{1, 2, 3, 4\}$ such that $x \in A_i$.
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.
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 $D_{m,n} = \Sigma_k \begin{pmatrix} m \\ k \end{pmatrix} \begin{pmatrix} n+k \\ m \end{pmatrix}$
An $n \times n$ matrix is said to be tridiagonal if its entries $a_{ij}$ are zero except when $|i&minus;j| \leq 1$ for $1 \leq i, \: j \leq n$. Note that only $3n &minus; 2$ entries of a tridiagonal matrix are non-zero. Thus, an array $L$ of size $3n &minus; 2$ ... a tridiagonal matrix. Given $i, j$, write pseudo-code to store $a_{ij}$ in $L$, and get the value of $a_{ij}$ stored earlier in $L$.