MathDB

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Part of 1993 All-Russian Olympiad

Problems(4)

Can 5n + 3 be a prime?

Source: All Russian Olympiads 1993 - Grade 9 - Day 1 - Problem 1

7/13/2011
For a positive integer nn, numbers 2n+12n+1 and 3n+13n+1 are both perfect squares. Is it possible for 5n+35n+3 to be prime?
number theory proposednumber theory
The area of triangle can't be an integer number

Source: All Russian Olympiads 1993 - Grade 10 - Day 1 - Problem 1

7/13/2011
The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.
geometryperimeterinequalitiesnumber theoryprime numbersarea of a triangleHeron's formula
If (x-y)(y-z)(z-x)=x+y+z then 27|x+y+z

Source: All Russian Olympiads 1993 - Grade 9 - Day 2 - Problem 1

7/13/2011
For integers xx, yy, and zz, we have (x−y)(y−z)(z−x)=x+y+z(x-y)(y-z)(z-x)=x+y+z. Prove that 27∣x+y+z27|x+y+z.
modular arithmeticparameterizationquadraticsnumber theory proposednumber theory
Find all quadruples of real numbers

Source: All Russian Olympiads 1993 - Grade 11 - Day 2 - Problem 1

7/13/2011
Find all quadruples of real numbers such that each of them is equal to the product of some two other numbers in the quadruple.
geometrygeometric transformationalgebra proposedalgebra