A number theory problem by Dragan Marković

Number Theory Level 1

$\large{\begin{cases} x^2+y^2+z^2=18 \\ xy+yz+zx = 9 \end{cases}}$

Let $x,y$ and $z$ be integers satisfying the system of equations above. Find $|x| + |y| + |z|$ .

Notation : $| \cdot |$ denotes the absolute value function .

3 27 9 12 6 15

3 solutions

Aditya Sky
May 20, 2016

You should change your question to $|\,x+y+z\,|$ from $|x|+|y|+|z|$ , because if $xy+yz+zx=9$ ,then that doesn't necessarily implies $|xy|+|yz|+|zx|=9$ , which is required to solve the problem.

Yes ur right!

Rishabh Tiwari - 5 years ago

Not necessarily try to prove that x,y and z are the same sign and then you can find the wanted sum

Dragan Marković - 5 years ago

Jessica Drake
Apr 28, 2020

substitute z=0 , then you get , x2 + y2 = 18 and xy = 9 , now 3 is the solution for x and y , so x + y + z = 3 + 3 + 0 = 6

Alex Fullbuster
May 6, 2019

Observe that x+y+z=6, and WLOG let us assume x ≥ y ≥ z. Observe, if mod(x) + mod(y) + mod(z) = E≥ 7.34, then minimum value of E^2 ≥ 18, therefore, 7≥mod(x) + mod(y) + mod(z)≥5. Now, its easier.

A number theory problem by Dragan Marković

3 solutions

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