MathDB

1

Part of 2001 IMC

Problems(2)

integer matrix

Source: IMC 2001 day 1 problem 1

10/29/2005
Let n n be a positive integer. Consider an n×n n\times n matrix with entries 1,2,...,n2 1,2,...,n^2 written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for n \equal{} 3 we get [123456789] \left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]) We choose n n entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?
linear algebramatrixlinear algebra unsolved
IMC 2001 Problem 7

Source: IMC 2001 Day 2 Problem 1

10/30/2020
Let r,s≥1r, s \geq 1 be integers and a0,a1,...,ar−1,b0,b1,...,bs−1a_{0}, a_{1}, . . . , a_{r-1}, b_{0}, b_{1}, . . . , b_{s-1} be real non-negative numbers such that (a0+a1x+a2x2+...+ar−1xr−1+xr)(b0+b1x+b2x2+...+bs−1xs−1+xs)=1+x+x2+...+xr+s−1+xr+s(a_0+a_1x+a_2x^2+. . .+a_{r-1}x^{r-1}+x^r)(b_0+b_1x+b_2x^2+. . .+b_{s-1}x^{s-1}+x^s) =1 +x+x^2+. . .+x^{r+s-1}+x^{r+s}. Prove that each aia_i and each bjb_j equals either 00 or 11.
Polynomials