10.4

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(2)

(x-3)/(x-6)^2 -(x-6)/{(x-3)^2=a(b-9x+x^2) (I Soros Olympiad 1994-95 R2 10.4)

Source:

It is known that for some

a

b

\frac{x-3}{(x-6)^2} -\frac{x-6}{(x-3)^2} =a(b-9x+x^2)

has as its largest root the number

1995

. Find the smallest root of this equation for the same

a

b

1995 segments such that any 3 of them form a triangle

Source: I Soros Olympiad 1994-95 Ukraine R2 10.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

There are 1995 segments such that a triangle can be formed from any three of them. Prove that using these

1995

segments, it is possible to assemble

664

acute-angled triangles so that each segment is part of no more than one triangle.

combinatoricsgeometrycombinatorial geometry