我也來一發

[JGU (逗簍有囉西咕)]

Time allowed: 9 hours                                 Version: English
Each question is worth 7 points.

Problem 1
Let n be a positive integer. Let T be the set of points (x, y) in the plane
where x and y are non-negative integers and x + y < n. Each point of T is
coloured red or blue. If a point (x, y) is red, then so are all points (x', y')
of T with both x'≦x and y'≦y. Define an X-set to be a set of n blue points
having distinct x-coordinates, and a Y-set to be a set of n blue points having
distinct y-coordinates. Prove that the number of X-sets is equal to the number
of Y-sets.

Problem 2
Let BC be a diameter of the circle Γ with centre O. Let A be a point on Γ
such that 0°<∠AOB < 120°. Let D be the midpoint of the arc AB not
containing C. The line through O parallel to DA meets the line AC at J. The
perpendicular bisector of OA meets Γ at E and at F. Prove that J is the
incentre of the triangle CEF.

Problem 3
Find all pairs of integers m,n≧3 such that there exist infinitely many
positive integers a for which
                      a^m + a - 1 / a^n + a^2 - 1
is an integer.

Problem 4
Let n be an integer greater than 1. The positive divisors of n are
d1, d2,…, dk where
                     1 = d1 < d2 < … < dk = n.
Define D = d1d2 + d2d3 + ... + dk-1dk .
   (a) Prove that D < n^2 .
   (b) Determine all n for which D is a divisor of n^2 .

Problem 5
Find all functions f from the set R of real numbers to itself such that
       ( f(x) + f(z) )( f(y) + f(t) ) = f( xy - zt ) + f( xt + yz )
for all x,y,z,t in R .

Problem 6
Let Γ1 , Γ2 , … , Γn be circles of radius 1 in the plane, where n≧3.
Denote their centres by O1,O2,…,On respectively. Suppose that no line meets
more than two of the circles. Prove that

                      Σ   ( 1 / OiOj )  ≦  (n-1)π/4
                  1≦i＜j≦n


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