P5

Part of 2020 Dürer Math Competition (First Round)

Problems(2)

Self-describing tables

Source: 2019-20 International Dürer Competition , Category E, P5

We call a table of size

n \times n

self-describing if each cell of the table contains the total number of even numbers in its row and column other than itself. How many self-describing tables of size a)

3 \times 3

4 \times 4

5 \times 5

exist? Two tables are different if they differ in at least one cell.

combinatoricscountingTables

polynomial attains every possible value mod p

Source: 2019-20 International Dürer Competition,Category E+, P5

p

k > 1

be a divisor of

p-1

. Show that if a polynomial of degree

k

with integer coefficients attains every possible value modulo

p

(0,1,\dots, p-1)

at integer inputs then its leading coefficient must be divisible by

p

. Note: the leading coefficient of a polynomial of degree d is the coefficient of the

x_d

abstract algebraalgebrapolynomial