Preparation Marathon

[Labib]

$4.$ Write it as a product of $3$ numbers in as many ways you can, and find all their sums. It turned out that one sum was twice one of the others and a perfect square.

What does "find all their sums" mean? I mean, do I have to find the sum of the 3 divisors??

[sm.joty]

Yes, I also want to know about the question that Labib said.[\b]

@Thanks Labib.

@Rafi, ২ দিন সময় দিলে সমস্যার সংখ্যা আরও বাড়াতে হবে। তবে আমি একমত যে একদিনে সকল সমস্যা সমাধান করা প্রায় দুঃসাধ্য ব্যাপার। আর সমস্যাগুলো কাঠিন্যের ক্রমানুসারে সাজালে আসলেই ভালো হয়।

[Masum]

@Jyoti, $26\equiv -1 (mod 27)$
@ Masum vai,

$8.$ There are $10$ horizontal lines and $8$ vertical lines. Connect all the points.
How many intersections are there?$

Which points are we talking about, here??
I mean, are these the points of intersection of the lines?? (lines don't have ends, so they should be the only ones I think... )

Oh! When setting this problem, I was thinking two similar problem. So I mixed up. It will be:
There are $10$ points in a line and $8$ more in another line parallel to the previous one. Connect all of them. How many intersections are there?

[Masum]

Yes, I also want to know about the question that Labib said.[\b]

@Thanks Labib.

@Rafi, ২ দিন সময় দিলে সমস্যার সংখ্যা আরও বাড়াতে হবে। তবে আমি একমত যে একদিনে সকল সমস্যা সমাধান করা প্রায় দুঃসাধ্য ব্যাপার। আর সমস্যাগুলো কাঠিন্যের ক্রমানুসারে সাজালে আসলেই ভালো হয়।

In fact they according to me, they are in order.

[Labib]

I agree. They also seemed ordered to me.

[sm.joty]

@Labib & Masum vai,
That means you think 2 no. is very easy problem. (for those who don't know multinomial theorem.)

[Shihab]

But I want to participate in this marathon. @masum vy

[Masum]

@Labib & Masum vai,
That means you think 2 no. is very easy problem. (for those who don't know multinomial theorem.)

Indeed. If I am right then some years ago when I was in HSC first year final, but with no idea what mathematics or problem is, I heard from Milon vai that a similar problem was solved by a junior student in Noakhali or something like that. Because this problem requires nothing more than thinking. Does it? So I put this in $2$. Hope you understand.

[Jini]

$4.$ Write it as a product of $3$ numbers in as many ways you can, and find all their sums. It turned out that one sum was twice one of the others and a perfect square.

What does "find all their sums" mean? I mean, do I have to find the sum of the 3 divisors??

Same question.

[Masum]

$4.$ Write it as a product of $3$ numbers in as many ways you can, and find all their sums. It turned out that one sum was twice one of the others and a perfect square.

What does "find all their sums" mean? I mean, do I have to find the sum of the 3 divisors??

The question says that assume you already have the number. Then write it as product of $3$ positive integers and find their sum, and the rest.. Hope it is clear