2006 IMAR Test

Part of IMAR Test

Subcontests

(3)

1

soccer tournament

n > m \geq 1

soccer teams play a full tournament, each team meeting (once) each other. Points are awarded:

2

1

0

for a loss. At the end, each team has won half of its points against the

m

teams placed last (including each of these teams, who won half of its points against the other

m-1

). Find all possible values for

n

m

, supported with examples of such tournaments.

1

equation

Consider the equation

\frac{xy-C}{x+y}= k ,

where all symbols used are positive integers. 1. Show that, for any (fixed) values

C, k

this equation has at least a solution

x, y

; 2. Show that, for any (fixed) values

C, k

this equation has at most a finite number of solutions

x, y

; 3. Show that, for any

C, n

k = k(C,n)

such that the equation has more than

n

x, y

1

isosceles triangles and a nice locus

Consider the isosceles triangle

ABC

AB = AC

M

the midpoint of

BC

. Find the locus of the points

P

interior to the triangle, for which

\angle BPM+\angle CPA = \pi