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 $2$ ]

#Algebra

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

Yajat Shamji - 7 months 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 $\boxed{6}$

A Former Brilliant Member - 7 months ago

@Yajat Shamji - Done :)

A Former Brilliant Member - 7 months ago

Correct!

But... different method.

Yajat Shamji - 7 months ago

@Yajat Shamji – Yet again I see...

A Former Brilliant Member - 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

UKMT Special (Problem 555)

Comments

UKMT Special (Problem $5$ )