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Consider the following $$3 \times 2$$ array formed by using the numbers $$1 , 2 , 3 ,4 ,5 , 6 \ :$$ $$\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} = \begin{pmatrix} 1 &6 \\ 2 & 5 \\ 3 & 4 \end{pmatrix}$$ . Observe that all row sums are equal , but the sum of the squares is not the same for each row .Extend the above arrar to a $$3 \times k$$ array $$(a_{ij}){3 \times k}$$ for a suitable k , adding more columns , using the numbers $$7 , 8 , 9 , ….,3k$$ such that $$\displaystyle \sum_{j=1}^{k} a_{1j} = \sum_{j=1}^{k} a_{2j}= \sum_{j=1}^{k}a_{3j} \ and \ \sum_{j=1}^{k} (a_{1j})^2= \sum_{j=1}^{k} (a_{2j})^2 = \sum_{j=1}^{k} (a_{3j})^2$$

Source of the problem

Regional Math Olympiad, 2019 Problem 4

Inequality

8/10

Suggested Book
Challenges and Thrills in Pre College Mathematics

Start with hints

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1+2+3+ …  3k = $$\frac{3k(3k+1)}{2}$$ So, $$\sum_{j=1}^{k} a_{1j} = \sum_{j=1}^{k} a_{2j}= \sum_{j=1}^{k}a_{3j} = \frac{k(3k+1)}{2}$$ $$1^2 + 2^2 +… (3k)^2 = \frac{k(3k+1)(6k+1)}{2}$$ So, $$\sum_{j=1}^{k} (a_{1j})^2= \sum_{j=1}^{k} (a_{2j})^2 = \sum_{j=1}^{k} (a_{3j})^2 = \frac{k(3k+1)(6k+1)}{6}$$.  This means that 3 | k.

Step 1: Try out with k = 3. Prove that it is not possible to arrange them in the desired order as already some numbers are fixed. We will now try for k = 6.  Claim: If 3|k and k > 3 then it is always possible. (This is our conjecture too)
n + (n + 5) + (m + 1) + (m + 4) + (l + 2) + (l + 3) =  2n + 2m + 2l + 15 =  (n + 1) + (n + 4) + (m + 2) + (m + 3) + (l) + (l+ 5) $$n^2 + (n + 5)^2 – (n + 1)^2 – (n + 4)^2 = 8$$ $$(m + 1)^2 + (m + 4)^2 – (m + 2)^2 – (m + 3)^2 = 4$$ $$(l + 2)^2 + (l + 3)^2 – (l)^2 – (l+ 5)^2 = – 12$$ So, we get  $$n^2 + (n + 5)^2 + (m + 1)^2 + (m + 4)^2 + (l + 2)^2 + (l + 3)^2 = (n + 1)^2 + (n + 4)^2 + (m + 2)^2 +(m + 3)^2 + (l)^2 + (l+ 5)^2$$. Also,
n + (n + 5) + (m + 1) + (m + 4) + (l + 2) + (l + 3) =  2n + 2m + 2l + 15 =  (n + 1) + (n + 4) + (m + 2) + (m + 3) + (l) + (l+ 5) Hence putting suitable values of l, m, and n, we get an array like the one below: $$\begin{pmatrix} 1 & 6 & 8 & 11 & 15 & 16 \\ 2 & 5 & 9 & 10 & 13 & 18 \\ 3 & 4 & 7 & 12 & 14 & 17\end{pmatrix}$$ $$(n + 1)^2 + (n + 6)^2 + (n + 8)^2 + (n + 11)^2 + (n + 15)^2 + (n + 16)^2 = (n + 2)^2 + (n + 5)^2+ (n + 9)^2 + (n + 10)^2+ (n + 13)^2+ (n + 18)^2 = (n + 3)^2 + (n + 4)^2 + (n + 7)^2+ (n + 12)^2 + (n+14)^2 + (n+17)^2$$. Using this and the above two matrices, you can prove by induction that the claim holds!
Using the array found similarly :  $$\begin{pmatrix} 1 & 6 & 8 & 11 & 18 & 13 & 21 & 23 & 25 \\ 2 & 5 & 7 & 12 & 15 & 17 & 19 & 22 & 27 \\ 3 & 4 & 9 & 10 & 14 & 16 & 20 & 24 & 26\end{pmatrix}$$

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