UKMT Special (Problem 55)

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

a+b+c=2a + b + c = 2

b+c+d=2b + c + d = 2

c+d+e=2c + d + e = 2

d+e+f=2d + e + f = 2

e+f+g=2e + f + g = 2

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

[UKMT Cayley Olympiad 20152015, Q22]

Note by Yajat Shamji
8 months, 2 weeks ago

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1 vote

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You have until this Sunday, 3:003:00pm!

Yajat Shamji - 8 months, 2 weeks ago

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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+ga+b+c=b+c+d=c+d+e=d+e+f=e+f+g

From the above equations, we can simplify to get a=d=g,\to a=d=g, b=eb=e and c=fc=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+aa+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 00 or greater than 22, and that it must be an integer. This leaves the possible options of 0,10, 1 and 22. All of these options will work, but 22 is the greatest value here.

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

A Former Brilliant Member - 8 months, 2 weeks ago

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@Yajat Shamji - Done :)

A Former Brilliant Member - 8 months, 2 weeks ago

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Correct!

But... different method.

Yajat Shamji - 8 months, 2 weeks ago

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@Yajat Shamji Yet again I see...

A Former Brilliant Member - 8 months, 2 weeks ago

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