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Part of 2003 China Western Mathematical Olympiad

Problems(2)

Find the smallest possible sum of the 4 numbers on a side

Source: CWMO 2003, Problem 1

Place the numbers

1, 2, 3, 4, 5, 6, 7, 8

at the vertices of a cuboid such that the sum of any

3

numbers on a side is not less than

10

. Find the smallest possible sum of the 4 numbers on a side.

combinatorics unsolvedcombinatorics

Prove that all the terms of the sequence are integral

Source: CWMO 2003, Problem 5

\{a_n\}

satisfies a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots, where

k

is a fixed positive integer. Prove that all the terms of the sequence are integral and that

2k

divides a_{2n}, n \equal{} 0, 1, 2, \ldots.

calculusintegrationinductionfunctionquadraticsalgebra unsolvedalgebra