2 prob from my friend....summoning Prophet sir!

1) Find the minimum of (a+b)4+(b+c)4+(c+a)4-47(a4+b4+c4) a,b,c are reals.

2) Solve the Diophantine equation for integers x3+2y3+4z3-6xyz=1

Beleive me, number 2 is really hard...

11 Answers

341
Hari Shankar ·

By minimum I take it you mean a numerical value and also you dont mean a lower bound.

Note that f(0,0,0) = 0

Now if the minimum m = f(p,q,r)

Suppose m>0, then for |k|>1 f(p/k, q/k, r/k) = 1/k4 f(p,q,r) = m/k4<m

Similarly if m<0, we choose |k|>1 and consider f(pk,qk,rk) and exhibit a lower minimum contradicting the fact that m is the minimum value

Hence m = 0

66
kaymant ·

Those arguments also apply if you replace that factor of 4/7 to something else like 1000....

66
kaymant ·

in fact the factor 4/7 is quite strict.

341
Hari Shankar ·

That's why I want to know if there's an expression on the RHS like some f(a,b,c)?

66
kaymant ·

This was one of olympiads postal coaching of last month and the problem statement is exactly what soumik has written.

11
Mani Pal Singh ·

Is Diophantine equation in Olympiads [5]

341
Hari Shankar ·

hmm, i've got to check if it is bounded below

341
Hari Shankar ·

Well I have just seen the solution for your 2nd problem in a book. It is, as you said, a tough one. I dont think I should give the source, since the olympiad aspirants are supposed to have a crack at it. But like the 1st problem it seemed like I have seen it somewhere, which seems to be the trend for our olympiad and JEE coaching - the trainers are smug with the solutions in their pockets, ready to be drawn out the moment a student asks them. No real training takes place this way. This is, of course, just my opinion.

341
Hari Shankar ·

Ah! I have got very good at detective work - i have tracked down the 1st problem too, again from a well known source. So much for original problem setting and solving!

66
kaymant ·

hmm...so these are copied too... lets hope the students at least are honestly doing them..

341
Hari Shankar ·

most of them tend to be like this. Even some INMO probs I have seen are from old Hungarian competitions.

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