MathDB

3

Part of 2005 Ukraine National Mathematical Olympiad

Problems(4)

board

Source: Ukraine 2005 grade 8

7/24/2009
A 6×6 6 \times 6 board is filled out with positive integers. Each move consists of selecting a square larger than 1×1 1 \times 1, consisting of entire cells, and increasing all numbers inside the selected square by 1 1. Is it always possible to perform several moves so as to reach a situation where all numbers on the board are divisible by 3 3?
combinatorics proposedcombinatorics
n points

Source: Ukraine 2005 grade 9

7/26/2009
On the plane are given n≥3 n \ge 3 points, not all on the same line. For any point M M on the same plane, f(M) f(M) is defined to be the sum of the distances from M M to these n n points. Suppose that there is a point M1 M_1 such that f(M1)≤f(M) f(M_1)\le f(M) for any point M M on the plane. Prove that if a point M2 M_2 satisfies f(M_1)\equal{}f(M_2), then M1≡M2. M_1 \equiv M_2.
inequalitiestriangle inequalitygeometry unsolvedgeometry
divisibility

Source: Ukraine 2005 grade 10

7/26/2009
An integer n>101 n>101 is divisible by 101 101. Suppose that every divisor d d of n n with 1<d<n 1<d<n equals the difference of two divisors of n n. Prove that n n is divisble by 100 100.
number theory unsolvednumber theory
intersection

Source: Ukraine 2005 grade 11

7/28/2009
In an acute-angled triangle ABC ABC, ω \omega is the circumcircle and O O its center, ω1 \omega _1 the circumcircle of triangle AOC AOC, and OQ OQ the diameter of ω1 \omega _1. Let M M and N N be points on the lines AQ AQ and AC AC respectively such that the quadrilateral AMBN AMBN is a parallelogram. Prove that the lines MN MN and BQ BQ intersect on ω1 \omega _1.
geometrycircumcirclegeometry proposed