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2

Part of 1990 Romania Team Selection Test

Problems(2)

geom ineq lp

Source: Laurentiu Panaitopol,Romania, TST 1990

8/10/2005
Prove that in any triangle ABCABC the following inequality holds: a2b+c−a+b2a+c−b+c2a+b−c≥33R. \frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R. Laurentiu Panaitopol
inequalitiesfunctiongeometrycircumcircleinequalities proposed
binomial sums equality

Source: Romania IMO TST 1990 p2

2/19/2020
Prove the following equality for all positive integers m,nm,n: ∑k=0n(m+kk)2n−k+∑k=0m(n+kk)2m−k=2m+n+1\sum_{k=0}^{n} {m+k \choose k} 2^{n-k} +\sum_{k=0}^m {n+k \choose k}2^{m-k}= 2^{m+n+1}
algebraBinomialSumcombinatorics