MathDB

3

Part of 2011 China Second Round Olympiad

Problems(2)

maximum of {i,j,k}

Source: China high school competition 2011

10/17/2011
Given n≥4n\ge 4 real numbers an>...>a1>0a_{n}>...>a_{1} > 0. For r>0r > 0, let fn(r)f_{n}(r) be the number of triples (i,j,k)(i,j,k) with 1≤i<j<k≤n1\leq i<j<k\leq n such that aj−aiak−aj=r\frac{a_{j}-a_{i}}{a_{k}-a_{j}}=r. Prove that fn(r)<n24{f_{n}(r)}<\frac{n^{2}}{4}.
floor functioncombinatorics proposedcombinatorics
find log-a b

Source: China Second Round Test 1

1/30/2012
Let a,ba,b be positive reals such that 1a+1b≤22\frac{1}{a}+\frac{1}{b}\leq2\sqrt2 and (a−b)2=4(ab)3(a-b)^2=4(ab)^3. Find log⁡ab\log_a b.
logarithmsinequalitiesalgebra unsolvedalgebra