[/quote]
,
,
be positive real numbers, such that 
.![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
be an interior point of the isosceles triangle 
with 
.
prove that 
.
be positive real numbers such that 
.
are positive real numbers .Show that
and 
be touch points of the incenter of 
at 
and 
,
and 
be the circumcenter of triangles 
and 
,
and 
concurrent.
,
the number of positive divisors of 
is an odd integer.
holds for every positive divisor 
of 
be a positive integer such that ![\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]](http://latex.artofproblemsolving.com/e/2/d/e2daf7800da3f984cd761ed1fecf10083444db25.png)
Prove that 
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
and 
Prove that 
Let 
and 
Let 
Prove that 
.
,
terms in this sequence, say 
and 
,
is odd, then switch their placement, while the other terms stay in place. This creates a new sequence.
to 
,
board, such that Batman and Robin are on the same diagonal (picture). In each round, first The Joker moves to an adjacent cell (having a common side), without exiting the board. Then in the same round Batman and Robin move to an adjacent cell. The Joker wins if he reaches the fourth "target" vertex cell (marked T). Batman and Robin win if they catch The Joker i.e. at least one of them is on the same cell as The Joker.
be the foot of perpendicular from 
to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle 
with centre 
passes through 
at 
and 
respectively. Let 
,
be the midpoint of 
is equidistant from 
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
Y by FlakeLCR, Davi-8191, tenplusten, AlastorMoody, DofL, Adventure10, kimyager, Mango247, and 1 other user
Given a positive integer 
, we write 
for the total number 
of prime factors of 
, counted with multiplicity. Let 
(so, for example, 
).
Prove the following two claims:
i) There are infinitely many positive integers such that 
;
ii) There are infinitely many positive integers such that 
.
(Romania) Dan Schwarz
, we write 
for the total number 
of prime factors of 
, counted with multiplicity. Let 
(so, for example, 
).Prove the following two claims:
i) There are infinitely many positive integers
such that 
;ii) There are infinitely many positive integers
such that 
.(Romania) Dan Schwarz
we can construct solutions:
--> 
---> 
--- > ....
they are infinte and also
such that for all 
,
,
,
and 
have opposite signs. Continuing, we see that 
have opposite signs, and we can do that until we reach 
,
be an odd number, 
,
.
,
(respectively 
)
.
and 
.
or 
such that 
are infinite, this implies that there are infinitely many 
since otherwise one of the numbers 
or 
is a good number. However, this implies that 
.
,
is good. Since there are infinitely many 
which has infinitely many solutions implies the first part of the problem.
which also has infinitely many solutions takes care of the second part.
for all integers 
.
such that 
or there don't. If infinitely many such 
,
.
,
is clearly a multiplicative function. Then after a certain point, the values of 
etc., so we are done.
for all integers 
.
to the equation 
;
is a solution, and 
is a solution. Thus we are done.
.
is 1 for infinitely many positive integers 
holds for infinitely many 
.
,
,
,
and we use the Pell's equation.
then 
and opposite.Now,just pick ann odd integer such 
(
.
and 
,
,
and the rest is the same.
,
even, 
is false. Take any 
,
and 
are not both equal to 
)
has no 
,
for 
with 
odd. For the case where we prove (ii), take 
with 
and let 
This is a Pell type equation and kills a.)
let 
.
and for part b) 
and 
do not satisfy the condition then 
is even. Thus 
satisfies the conditions.
,
,
,
is a consecutive red pair. Clearly, 
and 
are both red. Then, if either one of 
and 
are red, then the claim would be true. However, if they are both blue, then their product, 
,
such that 
and 
are both red by our FTSOC claim. However, this means that 
and 
are both blue, and they are consecutive integers, which contradicts our FTSOC claim since they are both still larger than 
.
would be determined by the parity of 
(multiplying by 
.
.
(resp. 
)
(resp. 
.
.
Hildebrand [1] showed that all eight patters on 
occur infinitely often and much later Matomaki et al. [2] have shown that these patters occur with a positive frequency (I am talking about the generalised problem, but it turns our that the hardest part of the proof is dealing with the runs in question, namely 
are present infinitely often. Of course, it is expected that for any 
,
