MathDB

5

Part of 2001 Estonia National Olympiad

Problems(4)

numbers 1,2,..., 2001 to a 9x2001 table

Source: 2001 Estonia National Olympiad Final Round grade 9 p5

3/12/2020
A table consisting of 99 rows and 20012001 columns is filfed with integers 1,2,...,20011,2,..., 2001 in such a way that each of these integers occurs in the table exactly 99 times and the integers in any column differ by no more than 33. Find the maximum possible value of the minimal column sum (sum of the numbers in one column).
combinatoricstablemax
a tribe called Ababab uses only letters A and B to create words

Source: 2001 Estonia National Olympiad Final Round grade 10 p5

3/12/2020
A tribe called Ababab uses only letters AA and BB, and they create words according to the following rules: (1) AA is a word; (2) if ww is a word, then wwww and ww‾w\overline{w} are also words, where w‾\overline{w} is obtained from ww by replacing all letters AA with BB and all letters BB with AA ( xyxy denotes the concatenation of xx and yy) (3) all words are created by rules (1) and (2). Prove that any two words with the same number of letters differ exactly in half of their letters.
Wordscombinatorics
trapezoids with interior angles of 90^o, 90^o, 45^o,135^o

Source: 2001 Estonia National Olympiad Final Round grade 11 p5

3/12/2020
Consider all trapezoids in a coordinate plane with interior angles of 90o,90o,45o90^o, 90^o, 45^o and 135o135^o whose bases are parallel to a coordinate axis and whose vertices have integer coordinates. Define the size of such a trapezoid as the total number of points with integer coordinates inside and on the boundary of the trapezoid. (a) How many pairwise non-congruent such trapezoids of size 20012001 are there? (b) Find all positive integers not greater than 5050 that do not appear as sizes of any such trapezoid.
trapezoidcombinatorial geometrycombinatoricslattice
variation of a 3x3 magic square

Source: 2001 Estonia National Olympiad Final Round grade 12 p5

3/12/2020
A 3×33\times 3 table is filled with real numbers in such a way that each number in the table is equal to the absolute value of the difference of the sum of numbers in its row and the sum of numbers in its column. (a) Show that any number in this table can be expressed as a sum or a difference of some two numbers in the table. (b) Show that there is such a table not all of whose entries are 00.
magic squarecombinatoricsDifferenceSumsquare table