1998 USAMO

Part of USAMO

Subcontests

(5)

1

Subtract and Divide

Prove that for each

n\geq 2

, there is a set

S

n

integers such that

(a-b)^2

ab

for every distinct

a,b\in S

1

Chessboards

A computer screen shows a

98 \times 98

chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

1

n Tans

a_0,a_1,\cdots ,a_n

be numbers from the interval

(0,\pi/2)

\tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1.

\tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}.

1

Disjoint Pairs

Suppose that the set

\{1,2,\cdots, 1998\}

has been partitioned into disjoint pairs

\{a_i,b_i\}

1\leq i\leq 999

) so that for all

i

|a_i-b_i|

1

6

. Prove that the sum

|a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}|

ends in the digit

9

1

Usamo 1998

n \geq 5

be an integer. Find the largest integer

k

(as a function of

n

) such that there exists a convex

n

A_{1}A_{2}\dots A_{n}

for which exactly

k

of the quadrilaterals

A_{i}A_{i+1}A_{i+2}A_{i+3}

have an inscribed circle. (Here

A_{n+j} = A_{j}