OliForum Matematico

A prime number $ p $ is given. A sequence of positive integers $ a_{1},a_{2},a_{3},... $ is determined by this conditon:
$ a_{n+1}=a_{n}+p \lfloor \sqrt[p]{a_{n}} \rfloor $
Prove that there is a term in this sequence, which is a $ p $-power of an integer number.

There is integer number $ x\geqslant 2 $ . Find the smallest integer number$ y\geqslant x $, such that for every division of set$ \lbrace x, x+1, ..., y\rbrace $ into two subsets at least one of thease subsets contains such numbers $ m, n, p $ (not necessarily different), such that $ mn=p $ .

no one could do it?

no ideas at all?? I have one solution but it's horrible and I'll post it after someone else will prove this

Now I know this solution is wrong. Cause n must be related to m (which is given)
So if someone has any solution please post it!

I have the same but i'm not sure if it's correct

Could you show how you've got a = b ?? I dont know what CM and AM are.

reals but total {-1,-2,1,5, etc.}

A pyramid $ ABCDS $ is given (the base is convex quadrilateral). A sphere is inscribed in this pyramid and it is tangent to side ABCD at point P.
Prove that
$ \angle APB + \angle CPD = 180^{o} $

but why is a=b?? Could you explain?

We have a total number $ m \geq 2 $ We must find the smallest total number n which is:
$ n \geq m $
and for which:

for all divisions of the set {m, m+1, ..., n} into two subsets at least one of the subsets contains such number a, b, c (not necessary different)
that ab = c

Points $ P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6}, P_{7} $ are placed correspondingly on sides $ BC, CA, AB, BC, CA, AB, BC $ of triangle $ ABC $ and thease angles are equal:
$ P_{1}P_{2}C = AP_{2}P_{3} = P_{3}P_{4}B = CP_{4}P_{5} = P_{5}P_{6}A = BP_{6}P_{7} = 60^{o} $
Prove that
$ P_{1} = P_{7} $

I know that the answer is -5 but can't prove it

Find the smallest real number x , such that: For arbitrary real numbers $ a, b, c \geqslant x $ and $ a + b + c = 3 $ this inequality is correct:$ a^{3} + b^{3} + c^{3} \geqslant 3 $

So any ideas so far? I know a bit itallian but i prefer english. I havent't noticed too many people speaking english on OliForum