4

Part of 2011 China Second Round Olympiad

Problems(2)

matrix and elements

Source: 2011-2012 china second round,problem 4

A

3 \times 9

matrix. All elements of

A

are positive integers. We call an

m\times n

A

"ox" if the sum of its elements is divisible by

10

, and we call an element of

A

"carboxylic" if it is not an element of any "ox" submatrix. Find the largest possible number of "carboxylic" elements in

A

linear algebramatrixcombinatorics proposedcombinatorics

Source: China Second Round Test 1

{\cos^5 x}-{\sin^5 x}<7({\sin^3 x}-{\cos ^3 x})

x\in [ 0,2\pi)

), then find the range of

x

trigonometryalgebra unsolvedalgebra