Largest Possible Value | PRMO-2019 | Problem 17

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Largest Possible Value| PRMO | Problem-17
- Key Concepts
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Try this beautiful problem from PRMO, 2019 based on Largest Possible Value.

Largest Possible Value | PRMO | Problem-17

Let a, b, c be distinct positive integers such that $b + c – a$,$ c + a – b$ and $a + b – c$ are all perfect squares.
What is the largest possible value of $a + b + c$ smaller than $100$?

$20$
$91$
$13$

Key Concepts

Number theory

Perfect square

Integer

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Answer:$91$

Source

Try with Hints

First hint

Let $b + c – a = x^2$ … (i)
$c + a – b = y^2$ … (ii)
$a + b – c = z^2$ … (iii)

Now since $a$,$ b$, $c$ are distinct positive integers,
Therefore, $x$, $y$, $z$ will also be positive integers,
add (i), (ii) and (iii)
$a + b + c = x^2 + y^2 + z^2$
Now, we need to find largest value of $a + b + c or x^2 + y^2 + z^2$ less than $100$
Now, to get a, b, c all integers $x$,$ y$, $z$ all must be of same parity, i.e. either all three are even or all three
are odd.

Can you now finish the problem ..........

Second Hint

Let us maximize$x^2 + y^2 + z^2$, for both cases.
If $x$, $y$, $z $are all even.
Therefore,

$b + c – a = 8^2 = 64$
$c + a – b = 42 = 16$
$a + b – c = 22 = 4$
Which on solving, give$ a = 10$,$ b = 34$, $c = 40$ and $a + b + c = 84$
If x, y, z are all odd
$\Rightarrow b + c – a = 92 = 81$
$c + a – b = 32 = 9$
$a + b – c = 12 = 1$
Which on solving, give $a = 5$ ,$b = 41$, $c = 45$ and$ a + b + c = 91$

Can you finish the problem........

Final Step

Therefore Maximum value of $a + b + c < 100 = 91$