MathDB

5

Part of 1999 All-Russian Olympiad

Problems(3)

Cutting up an equilateral triangle

Source: All-Russian MO 1999

12/31/2012
An equilateral triangle of side nn is divided into equilateral triangles of side 11. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.
complementary countingcombinatorics unsolvedcombinatorics
s(n) = 100 and s(44n) = 800, compute s(3n)

Source: All-Russian MO 1999

12/31/2012
The sum of the (decimal) digits of a natural number nn equals 100100, and the sum of digits of 44n44n equals 800800. Determine the sum of digits of 3n3n.
number theory unsolvednumber theory
cd | (a+b)^2 any permutation a,b,c,d

Source: All-Russian MO 1999

12/31/2012
Four natural numbers are such that the square of the sum of any two of them is divisible by the product of the other two numbers. Prove that at least three of these numbers are equal.
modular arithmeticnumber theory unsolvednumber theory