2012 Indonesia MO Problems/Problem 7

Problem

Let $n$ be a positive integer. Show that the equation $\sqrt{x}+\sqrt{y}=\sqrt{n}$ have solution of pairs of positive integers $(x,y)$ if and only if $n$ is divisible by some perfect square greater than $1$ .

Solution

Since iff is a double implication, we can prove that if there exists a positive integer solution $(x,y)$ to $\sqrt{x}+\sqrt{y}=\sqrt{n}, then$ n $is divisible by some perfect square greater than$ 1 $, and if$ n $is divisible by some perfect square greater than$ 1 $then there exists a positive integer solution (x,y) fo$ \sqrt{x}+\sqrt{y}=\sqrt{n}$.

Lets tackle the latter first, let$ (Error compiling LaTeX. Unknown error_msg)n=m^2p $where$ m>1 $and$ p $is not divisible by any perfect square greater than$ 1 $, let$ x=p(m-1)^2 $and$ y=p(1)^2 $. Substituting back in we can get$ \sqrt{p(m-1)^2}+\sqrt{p(1)^2}=\sqrt{m^2p}\implies (m-1)\sqrt{p€+\aqrt{p}=m\sqrt{p}\implies m\sqrt{p}=m\sqrt{p}$which is true, thus it is proven

For the first, let$ (Error compiling LaTeX. Unknown error_msg)x=a^2b $and$ y=c^2d $where$ b,d $are not divisible by a perfect square greater than$ 1 $,$ \sqrt{x}+\sqrt{y}=\sqrt{n}\implies x+y+2\sqrt{xy}=n $. Since$ \sqrt{xy} $has to be an integer, then$ xy $must be a perfect square, that means$ a^2c^2bd $is a perfect square which means$ bd $is a percect square, let$ b=p_1p_2\dots p_i $where$ p $are distinct primes, for$ bd $to be a perfect square,$ d $must be exactly$ p_1p_2\dots p_i $, as if it were less there exists a$ p_k $that divides$ b $but not$ d $and thus would not be a perfect square, the same logic would apply if$ d $was bigger than$ b $, thus$ b=d$. <cmath>x+y+2\sqrt{xy}=n</cmath> <cmath>a^2b+c^2b+2\sqrt{a^2b^2c^2=n</cmath> <cmath>a^2b+c^2b+2abc=n</cmath> <cmath>b(a^2+c^2+2ac)=n</cmath> <cmath>b(a+c)^2=n</cmath> since$ (Error compiling LaTeX. Unknown error_msg)a,c\geq 1\implies a+c\geq 2$, thus n is divisible by a perfect square greater than 1

2012 Indonesia MO (Problems)
Preceded by Problem 6	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 8
All Indonesia MO Problems and Solutions

Page

Toolbox

Search

2012 Indonesia MO Problems/Problem 7

Problem

Solution

See Also