**Day I**

1. Prove that for each positive
integer

*n*, there are pairwise relatively prime integers*k*_{0}*; k*_{1}*, ...., k**, all strictly greater than 1, such that*_{n}*k*_{0}*k*_{1}*...**k*_{n }*-*1 is the product of two consecutive*integers.*
2. Let

*ABC*be an acute, scalene triangle, and let*M, N*, and*P*be the midpoints of*BC, CA*, and*AB*, respectively. Let the perpendicular bisectors of*AB*and*AC*intersect ray*AM*in points*D*and*E*respectively, and let lines*BD*and*CE*intersect in point*F*, inside of triangle*ABC*. Prove that points*A, N, F*, and*P*all lie on one circle.
3. Let

*n*be a positive integer. Denote by*S**n*the set of points (*x; y*) with integer coordinates such that A*path*is a sequence of distinct points (*x*_{1}*, y*_{1})*,*(*x*_{2}*, y*_{2})*, ... ,*(*x*_{l}*, y**) in*_{l}*S*_{n}*such that, for**i*= 2*, ...*,*l*the distance between (*x*_{i}*, y**) and (*_{i}*x*_{i}_{-}_{1}*, y*_{i}_{-}_{1}) is 1 (in other words, the points (*x*_{i}*, y**) and (*_{i}*x*_{i}_{-}_{1}*, y*_{i}_{-}_{1}) are neighbors in the lattice of points with integer coordinates).
Prove that the points in

*S*_{n}*cannot be partitioned into fewer than**n*paths (a partition of*S*_{n}*into**m*paths is a set*P*of*m*nonempty paths such that each point in*S*_{n}*appears in exactly one of the**m*paths in*P*).**Day II**

4. Let

*P*be a convex polygon with*n*sides,*n**≥*3. Any set of*n**-*3 diagonals of*P*that do not intersect in the interior of the polygon determine a*triangulation*of*P*into*n**-*2 triangles. If*P*is regular and there is a triangulation of*P*consisting of only isosceles triangles, find all the possible values of*n*.
5. Three nonnegative real numbers

*r*_{1},*r*_{2},*r*_{3}are written on a blackboard. These numbers have the property that there exist integers*a*_{1},*a*_{2},*a*_{3}, not all zero, satisfying*a*_{1}*r*_{1}+*a*_{2}*r*_{2}+*a*_{3}*r*_{3}= 0.
We are permitted to perform the
following operation: find two numbers

*x*,*y*on the blackboard with*x**≤**y*, then erase*y*and write*y**-**x*in its place. Prove that after a finite number of such operations, we can end up with at least one 0 on the blackboard.
6. At a certain mathematical
conference, every pair of mathematicians are either friends orstrangers. At
mealtime, every participant eats in one of two large dining rooms. Each mathematician
insists upon eating in a room which contains an even number of his or her friends.
Prove that the number of ways that the mathematicians may be split between the
two rooms is a power of two (i.e., is of the form 2

*k*for some positive integer*k*).
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