Hard Math

I remember some math questions from the math Olympiads, so can you do better than I can? Good luck!

1) a-squared +b-squared=2018-2a, what is a and b?

2) What is the last non-0 digit of 50!

3a) a and b are positive integers, where a= the sum of all the digits of a and b, and b= the product of all the digits of a and b. Find all pairs that satisfy this equation, under 100.

3b) FInd all pairs under 1000, and one of the numbers has to be over 99.

Really, I only got an answer for 2 (probably wrong), and nothing for the rest. Please tell me what you got, and please note, I might have remembered this wrong. But, it should still be right. How did you go? :)

Easy Math Editor

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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

Q1) The answer is that there are no possible INTEGRAL a and b satisfying the equation..................otherwise there are infinitely many pairs of a and b..........

Aaghaz Mahajan - 2 years, 8 months ago

What I found that it may seem like that, but when I did it, it would always be short. So, (Idk if this is right) there should be only one answer.

Caleb Kum - 2 years, 8 months ago

I rechecked my calculations and found no error..........Check again..........There are no Integral a and b satisfying the condition...........If you have found a solution, then tell me the answer....... :)

Aaghaz Mahajan - 2 years, 8 months ago

@Aaghaz Mahajan – Ok, I'll take your word for it. Hey, anyone, can YOU find an answer? (I can't be stuffed to do so).

Caleb Kum - 2 years, 8 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}`	$\boxed{123}$