Click on the image and use the password : ogeometrie .The QR code on page 2 contains a link to the Books in my Electronic Library. (Click on the year of publication. Same password if needed.) A collection of important magazines is at the letter G : GAZETA MATEMATICĂ seria B.
A problem to illustrate the love for numbers, proposed for 5th grade (page 159) :
"E : 17457 . Write the number $220^{2n+1}$ as the sum of 5 distinct
nonzero perfect squares.
{author : } Marin CHIRCIU, Pitești "
ANSWER CiP
$220=1^2+3^2+4^2+5^2+13^2$ hence
$220^{2n+1}=(220^n)^2+(3\cdot 220^n)^2+(4 ^2\cdot 220^n)^2+(5\cdot 220^n)^2+(13\cdot 220^n)^2$
Solution CiP
We have $220^{2n+1}=(220^n)^2\cdot 220^1$ , so let's try to write the number $220$ as the sum of five distinct nonzero perfect squares.
We write some sums, $\color{Green}{highlighting}$ a perfect square term:
$220=\color{Green}{196}+24$
$220=\color{Green}{169}+51$
$220=\color{Green}{144}+76$
..........................
But any number is written as the sum of four squares, according to Lagrange's Four-Square Theorem, so let's try this with the second term in the writings above.
Asking a friend I found out that :
<< Adevărul e că $24$ are foarte puține reprezentări valide.
Singura reprezentare cu pătrate întregi ne-negative este:
>>
In translation : <<The truth is that 24 has very few valid representations. The only representation with non-negative integer squares is: ... >> But the squares here are not all nonzero and not all distinct.
Moving on to term $51$ , the same friend gave me the answer you see in the first row above.
The problem is now solved.
$\blacksquare$
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