MathDB

P4

Part of 2021 Junior Balkan Team Selection Tests - Romania

Problems(3)

Romania Junior TST 2021 Day 1 P4

Source:

5/16/2021
Let n≥2n\geq 2 be a positive integer. On an n×nn\times n board, nn rooks are placed in such a manner that no two attack each other. All rooks move at the same time and are only allowed to move in a square adjacent to the one in which they are located. Determine all the values ​​of nn for which there is a placement of the rooks so that, after a move, the rooks still do not attack each other.
Note: Two squares are adjacent if they share a common side.
combinatoricsRomanian TSTTST
Romania Junior TST 2021 Day 2 P4

Source:

6/6/2021
Let n≥2n\geq 2 be a positive integer. Prove that there exists a positive integer mm, such that n∣m, m<n4n\mid m, \ m<n^4 and at most four distinct digits are used in the decimal representation of mm.
number theorydecimal representationromaniaRomanian TST
Romania Junior TST 2021 Day 3 P4

Source:

6/7/2021
Let MM be a set of 1313 positive integers with the property that ∀ m∈M, 100≤m≤999\forall \ m\in M, \ 100\leq m\leq 999. Prove that there exists a subset S⊂MS\subset M and a combination of arithmetic operations (addition, subtraction, multiplication, division – without using parentheses) between the elements of SS, such that the value of the resulting expression is a rational number in the interval (3,4)(3,4).
combinatoricsSetsOperationsromaniaRomanian TST