MathDB

Problem 6

Part of 1998 Ukraine National Math Olympiad

Problems(4)

writing numbers in 5x120 board (Ukraine 1998 Grade 8 P6)

Source:

6/4/2021
Show that it is possible to write a number from the set {1,2,3,4,5}\{1,2,3,4,5\} in each square of a 5×1205\times120 board (55 columns and 120120 rows) so that the following conditions are satisfied:
(i) all numbers in the same row are distinct; (ii) all rows are distinct; (iii) the board can be partitioned into 2424 5×55\times5 boards which can be reassembled (without rotating and turning over) into a 120×5120\times5 board whose 120120 columns are distinct.
combinatorics
geo ineq, medians^2<=s (Ukraine 1998 Grade 9 P6)

Source:

6/5/2021
Prove that the sum of squared lengths of the medians of a triangle does not exceed the square of its semiperimeter.
geometrygeometric inequalityinequalities
triangles defined on circle, area sums

Source: Ukraine 1998 Grade 10 P6

6/5/2021
Let ABAB and CDCD be diameters of a circle with center OO. For a point MM on a shorter arc CBCB, lines MAMA and MDMD meet the chord BCBC at points PP and QQ respectively. Prove that the sum of the areas of the triangles CPMCPM and MQBMQB equals the area of triangle DPQDPQ.
geometryareasTrianglecircles
replace x by x^2-3yz with x,y,z written on board

Source: Ukraine 1998 Grade 11 P6

6/6/2021
The numbers 1,1998,19991,1998,1999 are written on the board. In each step it is allowed to replace one of the numbers by its square decreased by three times the product of the other two numbers. Is it possible to obtain three numbers with the sum 00 after several steps?
combinatoricsnumber theory