Subcontests
(5)NT is dead once again
Prove that for every positive integer t there is a unique permutation a0,a1,…,at−1 of 0,1,…,t−1 such that, for every 0≤i≤t−1, the binomial coefficient (2ait+i) is odd and 2ai=t+i. k-knight visits cells on a grid
Consider a 100×100 table, and identify the cell in row a and column b, 1≤a,b≤100, with the ordered pair (a,b). Let k be an integer such that 51≤k≤99. A k-knight is a piece that moves one cell vertically or horizontally and k cells to the other direction; that is, it moves from (a,b) to (c,d) such that (∣a−c∣,∣b−d∣) is either (1,k) or (k,1). The k-knight starts at cell (1,1), and performs several moves. A sequence of moves is a sequence of cells (x0,y0)=(1,1), (x1,y1),(x2,y2), …,(xn,yn) such that, for all i=1,2,…,n, 1≤xi,yi≤100 and the k-knight can move from (xi−1,yi−1) to (xi,yi). In this case, each cell (xi,yi) is said to be reachable. For each k, find L(k), the number of reachable cells.