Subcontests
(12)Putnam 2005 B6
Let Sn denote the set of all permutations of the numbers 1,2,…,n. For π∈Sn, let σ(π)=1 if π is an even permutation and σ(π)=−1 if π is an odd permutation. Also, let v(π) denote the number of fixed points of π. Show that
π∈Sn∑v(π)+1σ(π)=(−1)n+1n+1n. Putnam 2005 B5
Let P(x1,…,xn) denote a polynomial with real coefficients in the variables x1,…,xn, and suppose that
(a) (∂x12∂2+⋯+∂xn2∂2)P(x1,…,xn)=0 (identically)
and that
(b) x12+⋯+xn2 divides P(x1,…,xn).
Show that P=0 identically. Putnam 2005 B4
For positive integers m and n, let f(m,n) denote the number of n-tuples (x1,x2,…,xn) of integers such that \left|x_1\right| \plus{} \left|x_2\right| \plus{} \cdots \plus{} \left|x_n\right|\le m. Show that f\left(m,n\right) \equal{} f\left(n,m\right). Putnam 2005 A6
Let n be given, n≥4, and suppose that P1,P2,…,Pn are n randomly, independently and uniformly, chosen points on a circle. Consider the convex n-gon whose vertices are the Pi. What is the probability that at least one of the vertex angles of this polygon is acute.? Putnam 2005 A2
Let S={(a,b)∣a=1,2,…,n,b=1,2,3}. A rook tour of S is a polygonal path made up of line segments connecting points p1,p2,…,p3n is sequence such that
(i) pi∈S,
(ii) pi and pi+1 are a unit distance apart, for 1≤i<3n,
(iii) for each p∈S there is a unique i such that pi=p.
How many rook tours are there that begin at (1,1) and end at (n,1)?
(The official statement includes a picture depicting an example of a rook tour for n=5. This example consists of line segments with vertices at which there is a change of direction at the following points, in order: (1,1),(2,1),(2,2),(1,2),(1,3),(3,3),(3,1),(4,1),(4,3),(5,3),(5,1).)