MathDB

4

Part of 2011 Romania Team Selection Test

Problems(2)

disjoint cycles

Source: 2011 Romania TST,problem 4

2/4/2012
Given an integer n≥2n\ge 2, compute ∑σsgn(σ)nℓ(σ)\sum_{\sigma} \textrm{sgn}(\sigma) n^{\ell(\sigma)}, where all nn-element permutations are considered, and where ℓ(σ)\ell(\sigma) is the number of disjoint cycles in the standard decomposition of σ\sigma.
linear algebramatrixcombinatorics proposedcombinatorics
Sum of consecutive squares

Source: American Mathematical Monthly - Romanian TST 2011

5/31/2011
Show that: a) There are infinitely many positive integers nn such that there exists a square equal to the sum of the squares of nn consecutive positive integers (for instance, 22 is one such number as 52=32+425^2=3^2+4^2). b) If nn is a positive integer which is not a perfect square, and if xx is an integer number such that x2+(x+1)2+...+(x+n−1)2x^2+(x+1)^2+...+(x+n-1)^2 is a perfect square, then there are infinitely many positive integers yy such that y2+(y+1)2+...+(y+n−1)2y^2+(y+1)^2+...+(y+n-1)^2 is a perfect square.
number theory proposednumber theory