MathDB

Problem 4

Part of 2004 Croatia National Olympiad

Problems(4)

frog jumping on lattice points

Source: Croatian MO 2004 2nd Grade P4

4/8/2021
A frog jumps on the coordinate lattice, starting from the point (1,1)(1,1), according to the following rules:
(i) From point (a,b)(a,b) the frog can jump to either (2a,b)(2a,b) or (a,2b)(a,2b); (ii) If a>ba>b, the frog can also jump from (a,b)(a,b) to (a−b,b)(a-b,b), while for a<ba<b it can jump from (a,b)(a,b) to (a,b−a)(a,b-a). Can the frog get to the point: (a) (24,40)(24,40); (b) (40,60)(40,60); (c) (24,60)(24,60); (d) (200,4)(200,4)?
gamenumber theory
Occurences of subsequences in sequence (Croatian MO 2004 1st Grade P4)

Source:

4/8/2021
The sequence 1,2,3,4,0,9,6,9,4,8,7,…1,2,3,4,0,9,6,9,4,8,7,\ldots is formed so that each term, starting from the fifth, is the units digit of the sum of the previous four. (a) Do the digits 2,0,0,42,0,0,4 occur in the sequence in this order? (b) Will the initial digits 1,2,3,41,2,3,4 ever occur again in this order?
algebraSequence
coloring cells black in a board

Source: Croatian MO 2004 3rd Grade P4

4/9/2021
Finitely many cells of an infinite square board are colored black. Prove that one can choose finitely many squares in the plane of the board so that the following conditions are satisfied:
(i) The interiors of any two different squares are disjoint; (ii) Each black cell lies in one of these squares; (iii) In each of these squares, the black cells cover at least 15\frac15 and at most 45\frac45 of the area of that square.
combinatorics
cos(a),cos(2a),cos(4a),... negative

Source: Croatian MO 2004 4th Grade P4

4/9/2021
Determine all real numbers α\alpha with the property that all numbers in the sequence cos⁡α,cos⁡2α,cos⁡22α,…,cos⁡2nα,…\cos\alpha,\cos2\alpha,\cos2^2\alpha,\ldots,\cos2^n\alpha,\ldots are negative.
trigonometryalgebra