MathDB

at least 2/3 of the squares of the m x n board are covered with dominos

Source: 2021 Dutch IMO TST 1.4

12/28/2021

On a rectangular board with

m \times n

squares (

m, n \ge 3

) there are dominoes (

2 \times 1

or

1\times 2

tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least

2/3

of the squares of the board are covered with dominos.

combinatoricsdominos

an^k, bn^{\ell} , cn^m are sidelengths of a triangle

Source: 2021 Dutch IMO TST 2.4

12/28/2021

Determine all positive integers

n

with the following property: for each triple

(a, b, c)

of positive real numbers there is a triple

(k, \ell, m)

of non-negative integer numbers so that

an^k

,

bn^{\ell}

and

cn^m

are the lengths of the sides of a (non-degenerate) triangle shapes.

triangle inequalitygeometry

p is a divisor of 5^m7^n-1

Source: 2021 Dutch IMO TST 3.4

12/28/2021

Let

p > 10

be prime. Prove that there are positive integers

m

and

n

with

m + n < p

exist for which

p

is a divisor of

5^m7^n-1

.

number theorydivisor

4

Problems(3)