UKMT Special (Problem $5$)

The integers $a, b, c, d, e, f, g$, none of which is negative, satisfy the following system of simultaneous equations:

$a + b + c = 2$

$b + c + d = 2$

$c + d + e = 2$

$d + e + f = 2$

$e + f + g = 2$

Find the maximum possible value of $a + b + c + d + e + f + g$

[UKMT Cayley Olympiad $2015$, Q$2$]

Note by Yajat Shamji
8 months, 2 weeks ago

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You have until this Sunday, $3:00$pm!

- 8 months, 2 weeks ago

As all the equations are equal to the same thing, we know they must be equal by Euclid's Axiom (Things which are equal to the same thing are also equal to one another)

$a+b+c=b+c+d=c+d+e=d+e+f=e+f+g$

From the above equations, we can simplify to get $\to a=d=g,$ $b=e$ and $c=f$

So the equation we want to find the maximum possible value of gets simplified

$a+b+c+d+e+f+g=a+b+c+a+b+c+a=2(a+b+c)+a=4+a$

We know that a can't be lesser than $0$ or greater than $2$, and that it must be an integer. This leaves the possible options of $0, 1$ and $2$. All of these options will work, but $2$ is the greatest value here.

Thus, the maximum value of the equation is $\boxed{6}$

- 8 months, 2 weeks ago

@Yajat Shamji - Done :)

- 8 months, 2 weeks ago

Correct!

But... different method.

- 8 months, 2 weeks ago

Yet again I see...

- 8 months, 2 weeks ago