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Part of 2007 Estonia National Olympiad

Problems(4)

max number divisible ny it's own digits and each digit one less than previous

Source: 2007 Estonia National Olympiad Final Round grade 9 p1

Find the largest integer such that every number after the first is one less than the previous one and is divisible by each of its own numbers.

Digitsnumber theory

7digit number, divided by it's own digits

Source: 2007 Estonia National Olympiad Final Round grade 10 p1

The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

number theorydividedDigits

Estonian Math Competitions 2006/2007

Source: Finalround Problem 7

Find all real numbers a such that all solutions to the quadratic equation x^2 \minus{} ax \plus{} a \equal{} 0 are integers.

quadraticsnumber theory unsolvednumber theory

Estonian Math Competitions 2006/2007

Source: Finalround Problem 12

Consider a cylinder and a cone with a common base such that the volume of the part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.

geometry3D geometryratiogeometry unsolved