,
,
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
,
is an integer. Prove that there is a positive integer 
such that 
for all 
.
be a positive integer. On the table, we have 
ornaments in 
different colours.
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in 
,
where 
denotes the Fibonacci sequence.**
is defined by the conditions
and 
for a set of indices 
.
satisfying the equation![\[
(I_p + A)^n = B^n + C^n.
\]](http://latex.artofproblemsolving.com/2/d/7/2d75d47203353160ccac356251659bd16a4cb5c8.png)
be a function that is at least twice differentiable on an open interval containing ![\( [0, 2\pi] \)](http://latex.artofproblemsolving.com/9/7/7/977adaa4f83dc8164975a3e78539f27cbb354adf.png)
.![\[
f(0) = f(2\pi) = f'(0) = f'(2\pi) = 0
\]](http://latex.artofproblemsolving.com/d/9/9/d998bdd53ffec1fa4c88688d7402ad25efa31112.png)
and![\[
f(x) + f''(x) \geq 0, \quad \forall x \in [0,2\pi].
\]](http://latex.artofproblemsolving.com/a/1/c/a1c24260aa02cd989630295d9de8741af659e958.png)
Prove that 
for all ![\( x \in [0,2\pi] \)](http://latex.artofproblemsolving.com/d/c/5/dc59f105fb3b9f6be175d2c035ba2ead4b9d984a.png)
.
,
and Moivre's formula.
.
,
Y by
Our teacher asked this problem: (i translated it from turkish)
Reyyan's car tire bursts on the road. When Reyyan changes the tire, she sees that there are 5 screws in the tire. How many times is there a situation where none of the screws fit into their place?
I directly understood that it is about inclusion exclusion principle but i couldnt understand what means Screw1 ∩ Screw2 etc. means exactly. Can anybody help?
Reyyan's car tire bursts on the road. When Reyyan changes the tire, she sees that there are 5 screws in the tire. How many times is there a situation where none of the screws fit into their place?
I directly understood that it is about inclusion exclusion principle but i couldnt understand what means Screw1 ∩ Screw2 etc. means exactly. Can anybody help?
,
and 
.
