MathDB

3

Part of 2012 Spain Mathematical Olympiad

Problems(2)

n-x balls in x+1 boxes

Source: Spanish MO 2012 Q3

6/7/2012
Let xx and nn be integers such that 1≤x≤n1\le x\le n. We have x+1x+1 separate boxes and n−xn-x identical balls. Define f(n,x)f(n,x) as the number of ways that the n−xn-x balls can be distributed into the x+1x+1 boxes. Let pp be a prime number. Find the integers nn greater than 11 such that the prime number pp is a divisor of f(n,x)f(n,x) for all x∈{1,2,…,n−1}x\in\{1,2,\ldots ,n-1\}.
combinatorics proposedcombinatorics
R,P,D and S are concyclic

Source: Spanish MO 2012 Q6

6/7/2012
Let ABCABC be an acute-angled triangle. Let ω\omega be the inscribed circle with centre II, Ω\Omega be the circumscribed circle with centre OO and MM be the midpoint of the altitude AHAH where HH lies on BCBC. The circle ω\omega be tangent to the side BCBC at the point DD. The line MDMD cuts ω\omega at a second point PP and the perpendicular from II to MDMD cuts BCBC at NN. The lines NRNR and NSNS are tangent to the circle Ω\Omega at RR and SS respectively. Prove that the points R,P,DR,P,D and SS lie on the same circle.
circumcirclegeometry proposedgeometry