MathDB

4

Part of 2002 Estonia National Olympiad

Problems(4)

5 numbers on blackboard each replaced by x + y - z, can all be equal ?

Source: 2002 Estonia National Olympiad Final Round grade 9 p4

3/16/2020
Mary writes 55 numbers on the blackboard. On each step John replaces one of the numbers on the blackboard by the number x+y−zx + y - z, where x,yx, y and zz are three of the four other numbers on the blackboard. Can John make all five numbers on the blackboard equal, regardless of the numbers initially written by Mary?
combinatorics
max length of broken line on surface of unit cube

Source: 2002 Estonia National Olympiad Final Round grade 10 p4

3/16/2020
Find the maximum length of a broken line on the surface of a unit cube, such that its links are the cube’s edges and diagonals of faces, the line does not intersect itself and passes no more than once through any vertex of the cube, and its endpoints are in two opposite vertices of the cube.
geometrymaxLinecube3D geometry
max N such not all of the sums a_i+a_j are integers

Source: 2002 Estonia National Olympiad Final Round grade 11 p4

3/14/2020
Let a1,...,a5a_1, ... ,a_5 be real numbers such that at least NN of the sums ai+aja_i+a_j (i<ji < j) are integers. Find the greatest value of NN for which it is possible that not all of the sums ai+aja_i+a_j are integers.
Integersnumber theorySummax
tangent circumcircles result in tangent circumcircles

Source: 2002 Estonia National Olympiad Final Round grade 12 p4

3/14/2020
A convex quadrilateral ABCDABCD is inscribed in a circle ω\omega. The rays ADAD and BCBC meet in point KK and the rays ABAB and DCDC meet in LL. Prove that the circumcircle of triangle AKLAKL is tangent to ω\omega if and only if so is the circumcircle of triangle CKLCKL.
geometrycircumcircletangent circles