MathDB

3

Part of 2003 Turkey MO (2nd round)

Problems(2)

for all real a_1 , a_2 , ... , a_2004

Source: Turkey NMO 2003 Problem 3

2/24/2009
Let f:R→R f: \mathbb R \rightarrow \mathbb R be a function such that f(tx_1\plus{}(1\minus{}t)x_2)\leq tf(x_1)\plus{}(1\minus{}t)f(x_2) for all x1,x2∈R x_1 , x_2 \in \mathbb R and t∈(0,1) t\in (0,1). Show that \sum_{k\equal{}1}^{2003}f(a_{k\plus{}1})a_k \geq \sum_{k\equal{}1}^{2003}f(a_k)a_{k\plus{}1} for all real numbers a1,a2,...,a2004 a_1,a_2,...,a_{2004} such that a1≥a2≥...≥a2003 a_1\geq a_2\geq ... \geq a_{2003} and a_{2004}\equal{}a_1
functioninequalities unsolvedinequalities
mxn chessboard largest beautiful number

Source: Turkey NMO 2003 Problem 6

2/24/2009
An assignment of either a 0 0 or a 1 1 to each unit square of an m mxn n chessboard is called fair fair if the total numbers of 0 0s and 1 1s are equal. A real number a a is called beautiful beautiful if there are positive integers m,n m,n and a fair assignment for the m mxn n chessboard such that for each of the m m rows and n n columns , the percentage of 1 1s on that row or column is not less than a a or greater than 100\minus{}a. Find the largest beautiful number.
combinatorics unsolvedcombinatorics