MathDB

4

Part of 1998 Estonia National Olympiad

Problems(4)

x + y = 2 and xy = z^2 + 1, x^2 + y^2+ z^2 =?

Source: 1998 Estonia National Olympiad Final Round grade 9 p4

3/14/2020
For real numbers x,yx, y and zz it is known that {x+y=2xy=z2+1\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases} Find the value of the expression x2+y2+z2x^2 + y^2+ z^2.
system of equationsalgebra
if 5^n + 3^n + 1 is prime, then n is divided by 12.

Source: 1998 Estonia National Olympiad Final Round grade 10 p4

3/14/2020
Prove that if for a positive integer nn is 5n+3n+15^n + 3^n + 1 is prime number, then nn is divided by 1212.
primesdividednumber theorydivisible
if 1/a = a - [a] then a is irrational

Source: 1998 Estonia National Olympiad Final Round grade 11 p4

3/11/2020
A real number aa satisfies the equality 1a=a−[a]\frac{1}{a} = a - [a]. Prove that aa is irrational.
floor functionnumber theoryalgebrairrational
(2n)! = (n-2)!n!(n+2)! factorial diophantine

Source: 1998 Estonia National Olympiad Final Round grade 12 p4

3/11/2020
Find all integers n>2n > 2 for which (2n)!=(n−2)!n!(n+2)!(2n)! = (n-2)!n!(n+2)! .
factorialdiophantineDiophantine equationnumber theory