MathDB

2

Part of 2014 Spain Mathematical Olympiad

Problems(2)

Rational Square Root

Source: OME 2014 2

7/16/2014
Given the rational numbers rr, qq, and nn, such that 1r+qn+1q+rn=1r+q\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}, prove that n−3n+1\displaystyle\sqrt{\frac{n-3}{n+1}} is a rational number.
number theory unsolvednumber theory
a^2+13b^2

Source: OME 2014 5

7/16/2014
Let MM be the set of all integers in the form of a2+13b2a^2+13b^2, where aa and bb are distinct itnegers.
i) Prove that the product of any two elements of MM is also an element of MM.
ii) Determine, reasonably, if there exist infinite pairs of integers (x,y)(x,y) so that x+y∉Mx+y\not\in M but x13+y13∈Mx^{13}+y^{13}\in M.
modular arithmeticnumber theory unsolvednumber theory