Question

Nothing horrible looking equations this time -

Math Challenge - 3

Answer 1

Let n be a given non negative integer. Determine the number S(n) of solutions of the equation:

\[\large{x+2y+2z = n}\]

in non negative integers x,y, z

Answer 2

@ikram002p @Miracrown

Answer 3

@Zale101

Answer 4

I will post the answer to this question tomorrow :D

Till then best of luck

Answer 5

@hartnn @JungHyunRan

Answer 6

so let's see if we can find a pattern for this one
for example if n = 0, how many different solutions can there be?
so n can be any non-negative integer
0,1,2,3,4,5,...
and based on what n is, we want to know how many possible solutions there are for x,y,z
given that x,y,z are also non-negative integers
so, for example
if n = 0, then there would be exactly 1 solution
x = 0, y = 0, and z = 0
if n = 1 there is also exactly 1 solution
x = 1, y = 0, and z = 0
if y or z is not zero, then it would be too big

so what if n = 2
do you see how many solutions we would have?
can you think of any possibilities for x,y, and z if n = 2?

Answer 7

Well tried @Miracrown but we would be lost very soon if we used that method. I speak from experience :(
Or, may be I did something wrong *shrug*

Answer 8

you did something wrong, lol :)

Answer 9

Hahaha may be ? Uhm.. I solved this question at night with my eyelids closed :P There is a strong possibility *_*

Answer 10

* I solved using your approach at midnight

Answer 11

good on you!

Answer 12

Haha :D

Answer 13

I am not asking you to calculate all possible answers. I want you to find out a general number of solutions depending on the value of n. That is why I asked for S(n).

There *are* infinitely many solutions :)

Answer 14

Hey no need of deleting it. It was perfectly correct till that step :) But, does not match my solution. But, there are many possible solutions

Answer 15

I'm getting this :
\[\large S(n) = \lfloor\dfrac{n}{2} \rfloor + 1 + \dfrac{ \lfloor\dfrac{n}{2} \rfloor ( \lfloor\dfrac{n}{2} \rfloor +1)}{2} \]

Answer 16

Okay I am not going to check that but your answer and mine answer (the one given in the source) don't match. But that doesn't mean that your answer is wrong.

I will post the solution tomorrow and we would discuss it then.

Answer 17

hmm this is a trivial linear diophantine equation with constraints, so solutions can be represented in several ways but verifying them should be easy enough...

Answer 18

i guess a nice word problem can be cooked up from the constraints :)