AMTI Bhaskara Contest Final Junior (October 2013) question no. 8(b)

Do there exist 10 distinct integers such that sum of any 9 of which is a perfect square?

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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

Suppose that the numbers are $x_1,x_2,\ldots,x_{10}$ , and suppose that the sum of all the numbers except the $j$ th is the square $y_j^2$ . If $N = x_1+x_2+\cdots+x_{10}$ , we deduce that $x_j \; = \; N - y_j^2 \qquad 1 \le j \le 10$ For this to be consistent, we need $N \; = \; x_1 + x_2 + \cdots + x_{10} \; = \; 10N - (y_1^2 + y_2^2 + \cdots + y_{10}^2)$ so that $y_1^2 + y_2^2 + \cdots + y_{10}^2 \; = \; 9N$ We need to choose $y_1,y_2,\cdots,y_{10}$ to be distinct integers such that the sum of their squares is divisible by $9$ , for example $2,3,4,5,6,7,8,9,11,12$ . This gives $N=61$ and $x_1,x_2,\ldots,x_{10} \; = \; 57,52,45,36,25,12,-3,-20,-60,-83$ There is no rule that says that the integers have to be positive.

Mark Hennings - 7 years, 7 months ago

Ahhh....loved it....the method of solving was awesome.......but you did take time to solve the integers $2,3,4,5,6,7,8,9,11,12$ ....right? If not,could you please tell me how you found those integers??

Vaibhav Reddy - 7 years, 7 months ago

Not really. There are lots of options. I could have chosen $0,9,18,27,\ldots$ , but I wanted small numbers.

Mark Hennings - 7 years, 7 months ago

@Mark Hennings – so you arithmetically solved it on a piece of paper

Vaibhav Reddy - 7 years, 7 months ago

@Vaibhav Reddy – Well, I played with the values of squares modulo $9$ .

Mark Hennings - 7 years, 7 months ago

@Mark Hennings – Thank you....I realized it later....

Vaibhav Reddy - 7 years, 7 months ago

Wow..thanks a lot.

Piyushkumar Palan - 7 years, 7 months ago

This may be simple or difficult ..i tried for a while...need help!

Piyushkumar Palan - 7 years, 7 months ago

Why graph of quadratic equation is parabola ?

Vaibhav Gandhapwad - 6 years 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}`	$\boxed{123}$