Hi guys,

I'm Paolo and this is the second round of the second edition of [i]Oliforum contest[/i]. You can see first round [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=1640043#1640043]here[/url].

I copy below all rules, as always  :wink: 

[b]1-[/b] The contest is made of 2 rounds, this is the last one. 

[b]2-[/b] Each round is made of some problems (from 3 to 5..), and i will give points 0 to 7: 6 for all correct solutions, 7 for [i]only[/i] elegants and correct solutions (similar to ML contest). 

[b]3-[/b] It is tought for high-school student, and also for undergraduate or higher student, but you cannot use non-elementary-theory, otherwise the solution is invalid.

[b]4-[/b] All begins from italian-oliforum and the classification is unique.

[b]5-[/b] The first round starts on 15:00 of 16 september and it ends on 15:00 of 18 seprember, so you have 48 hours :) .
        [b]NB.[/b] All times (date and hours) are according to [i]ROME MERIDIAN (+2GTM), [/i]please attention!

[b]6-[/b] Enrollment is not necessary, it sufficies to send me solution.

[b]7-[/b] Time of arriving solution is not important.

[b]8-[/b] [i]Way of sending solutions[/i].
-          You need to send solutions to both email address: [u]pao_leo88 ML hotmail ML[/u] it and [u]leonettipaolo ML gmail ML com[/u] in a [i]unique pdf file[/i] (so, with attachment)  of a right size;
-          The pdf file must be written with [i]Latex [/i] (or a way that I can understand);
-          You [i]rename the pdf file [/i]as your nickname (the same of yours in ML);
-          You need to write nothing in the email;
-          Try to be [i]clear[/i] in solutions!
 
{If [i](and only if)[/i] you are not able to create a pdf in latex model, you can send me a email in .doc format with solutions, as latex or in a way that i (and all correctors) can understand, or also with private message to me}.

Later, together with texts, I'll make a list below this topic where i'll write all names of who sent me solutions in right way.

For all doubts and questions, ask me in private messages :wink: 

 

Thanks for the attentions and good luck

[i]Paolo Leonetti[/i]
1. **Define the functions \( h \) and \( k \):**
   - From condition (i), we have \( k(x) = h(0,x) \).
   - From condition (ii), we have \( k(0) = 0 \).
   - From condition (iii), we have \( h(k(x_1), x_2) = x_1 \).

2. **Analyze the conditions:**
   - Substitute \( k(x) \) from (i) into (iii): \( h(h(0,x_1), x_2) = x_1 \).
   - This implies that \( h(0, x_1) \) must be an inverse function in some sense.

3. **Consider the simplest case:**
   - Let \( S = \{0\} \). Then \( h(0,0) = 0 \) and \( k(0) = 0 \), which satisfy all conditions.

4. **Consider larger sets:**
   - Suppose \( S = \{0, a\} \) for some \( a \in \mathbb{N} \).
   - Then \( k(a) = h(0, a) \) and \( k(0) = 0 \).
   - For \( h(k(a), 0) = a \), we need \( h(h(0, a), 0) = a \).

5. **Generalize for any \( S \):**
   - If \( S \) contains more elements, we need \( h \) and \( k \) to satisfy the conditions for all pairs.
   - The simplest solution is \( h(x, y) = y \) and \( k(x) = x \).

6. **Verify the solution:**
   - \( k(x) = h(0, x) = x \) satisfies (i).
   - \( k(0) = 0 \) satisfies (ii).
   - \( h(k(x_1), x_2) = h(x_1, x_2) = x_1 \) satisfies (iii).

Conclusion:
The subset \( S \) can be any non-empty subset of \( \mathbb{N} \) with the functions \( h(x, y) = y \) and \( k(x) = x \).

The final answer is any non-empty subset \( \boxed{ S } \) of \( \mathbb{N} \) with \( h(x, y) = y \) and \( k(x) = x \).