MathDB

2

Part of 2006 Estonia National Olympiad

Problems(4)

circle of radius 2 can be covered by 7 unit circles

Source: 2001 Estonia National Olympiad Final Round grade 9 p2

3/12/2020
Prove that the circle with radius 22 can be completely covered with 77 unit circles
circlescombinatoricscombinatorial geometry
ab + 1, bc + 1, ca + 1 all perfect squares

Source: 2006 Estonia National Olympiad Final Round grade 10 p2

3/12/2020
Let a,ba, b and cc be positive integers such that ab+1,bc+1ab + 1, bc + 1 and ca+1ca + 1 are all integer squares. a) Give an example of such numbers a,ba, b and cc. b) Prove that at least one of the numbers a,ba, b and cc is divisible by 44
number theoryPerfect Squaredivisible
Estonian Math Competitions 2005/2006

Source: Final Round Grade 11 Pro 2

7/30/2008
In a right triangle, the length of one side is a prime and the lengths of the other side and the hypotenuse are integral. The ratio of the triangle perimeter and the incircle diameter is also an integer. Find all possible side lengths of the triangle.
ratiogeometryperimetergeometry unsolved
Estonian Math Competitions 2005/2006

Source: Final Round Grade 12 Pro 2

7/30/2008
Find the smallest possible distance of points P P and Q Q on a xy xy-plane, if P P lies on the line y \equal{} x and Q Q lies on the curve y \equal{} 2^x.
algebra unsolvedalgebra