MathDB

5

Part of 1999 Estonia National Olympiad

Problems(4)

5 x 24 x 30 tiles for a 23 x 19 hole

Source: 1999 Estonia National Olympiad Final Round grade 9 p5

3/13/2020
There is a hole in the roof with dimensions 23×1923 \times 19 cm. Can August fill the the roof with tiles of dimensions 5×24×305 \times 24 \times 30 cm?
Tilingcombinatorics
locus of midpoints, equilateral triangles related

Source: 1999 Estonia National Olympiad Final Round grade 10 p5

3/11/2020
Let CC be an interior point of line segment ABAB. Equilateral triangles ADCADC and CEBCEB are constructed to the same side from ABAB. Find all points which can be the midpoint of the segment DEDE.
geometryLocusmidpointEquilateral
2^k grains of oat on a 8x8 chessboard, no that a knight eats is divisible by 3

Source: 1999 Estonia National Olympiad Final Round grade 11 p5

3/11/2020
On the squares a1,a2,...,a8a1, a2,... , a8 of a chessboard there are respectively 20,21,...,272^0, 2^1, ..., 2^7 grains of oat, on the squares b8,b7,...,b1b8, b7,..., b1 respectively 28,29,...,2152^8, 2^9, ..., 2^{15} grains of oat, on the squares c1,c2,...,c8c1, c2,..., c8 respectively 216,217,...,2232^{16}, 2^{17}, ..., 2^{23} grains of oat etc. (so there are 2632^{63} grains of oat on the square h1h1). A knight starts moving from some square and eats after each move all the grains of oat on the square to which it had jumped, but immediately after the knight leaves the square the same number of grains of oat reappear. With the last move the knight arrives to the same square from which it started moving. Prove that the number of grains of oat eaten by the knight is divisible by 33.
Chessboardcombinatoricspower of 2divisibledividesnumber theory
0, 1, 2, ..., 9 are written in some order on the circumference

Source: 1999 Estonia National Olympiad Final Round grade 12 p5

3/11/2020
The numbers 0,1,2,...,90, 1, 2, . . . , 9 are written (in some order) on the circumference. Prove that a) there are three consecutive numbers with the sum being at least 1515, b) it is not necessarily the case that there exist three consecutive numbers with the sum more than 1515.
circlenumbersconsecutiveSumnumber theorycombinatorics