1

Part of 2000 China National Olympiad

Problems(2)

Sign of a+b-2r-2r

Source: Chinese Mathematical Olympiad 2000 Problem 1

a,b,c

ABC

a\le b\le c

. The circumradius and inradius of triangle

ABC

R

r

respectively. Let

f=a+b-2R-2r

. Determine the sign of

f

by the measure of angle

C

geometrycircumcircleinradiusgeometry unsolved

'Innovated tuple' and 'innovated degree'

Source: Chinese Mathematical Olympiad 2000 Problem 4

Given an ordered

n

A=(a_1,a_2,\cdots ,a_n)

of real numbers, where

n\ge 2

b_k=\max{a_1,\ldots a_k}

for each k. We define

B=(b_1,b_2,\cdots ,b_n)

to be the “innovated tuple” of

A

. The number of distinct elements in

B

is called the “innovated degree” of

A

. Consider all permutations of

1,2,\ldots ,n

n

-tuple. Find the arithmetic mean of the first term of the permutations whose innovated degrees are all equal to

2

combinatoricspermutationspermutation statistics