MathDB

P4

Part of 2019 Dürer Math Competition (First Round)

Problems(2)

writing numbers on the points of the first quadrant

Source: 12th Dürer Math Competition ,First Round, Category E , P4

8/18/2020
Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point (a,b)(a, b) is one greater than the average of the numbers written on points (a+1,b−1) (a+1 , b-1) and (a−1,b+1) (a-1,b+1) . Which numbers could he write on point (121,212)(121, 212)? Note: The elements of the first quadrant are points where both of the coordinates are non- negative.
combinatoricsIntegercoordinates
N-tuples irrationality

Source: 12th Dürer Math Competition ,First Round, Category E+, P4

8/18/2020
An nn-tuple (x1,x2,…,xn)(x_1, x_2,\dots, x_n) is called unearthly if q1x1+q2x2+⋯+qnxnq_1x_1 +q_2x_2 +\dots+q_nx_n is irrational for any non-negative rational coefficients q1,q2,…,qnq_1, q_2, \dots, q_n where qiq_i’s are not all zero. Prove that it is possible to select an unearthly nn-tuple from any 2n−12n-1 distinct irrational numbers.
irrationaln-tuplescombinatorics