AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
with the following property: for 
but not 
consecutive positive integers, each one is a divisor of .
be the incenter of triangle 
, and 
be the perpendicular bisector of 
. Suppose that 
is on the circumcircle of triangle , and line 
and intersect at point 
. Point 
is on such that 
.Suppose that line 
and the midsegment of that is parallel to 
intersect at 
. Show that 
is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](http://latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.
be positive real numbers. Prove that:![$$\sqrt[3]{a^3+b^3}+\sqrt[3]{b^3+c^3}+\sqrt[3]{c^3+a^3}\geq \sqrt[3]{2}(a+b+c)$$](http://latex.artofproblemsolving.com/2/6/e/26ebcdae35153bee73abbcfaf4a767d3a32b8912.png)
, is the incenter. Line 
intersects circumcircle of at 
, and it is given that 
. 
are points chosen on 
such that 
. Prove that 
be a positive integer. In an 
table, an upright path is a sequence of adjacent cells starting from the southwest corner to the northeast corner such that the next cell is either on the top or on the right of the previous cell. Find the smallest number of grids one needs to color in an table such that there exists only one possible upright path not containing any colored cells.
of positive integers satisfying the following two conditions:
, if 
is in 
, then so are 
and 
; and contains an integer 
such that 
is not divisible by 
.
such that 
and 
are all perfect square.
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points and lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
such that 
and 
.
has perpendicular diagonals. The perpendicular bisectors of the sides and 
meet at a unique point inside . Prove that the quadrilateral is cyclic if and only if triangles 
and 
have equal areas.
is continious. WLOG 
. First we prove by induction that 
and 
is increasing:
. By induction hypothesis 
and thus 
which implies 
. Also, 
and 
was increasing, thus 
which means 
.
for all , then it is bounded from above and hence has a limit, say 
. Then 
Now we prove that 
Suppose it isn't, then 
has a limit, say 
. Then 
And so 
But 
, and so 
goes to infinity, a contradiction. Thus, goes to infinity, and so does 
. As 
, we know that 
diverges.
. Take to be a big positive integer, and choose to be large enough that 
and 
. Then 
Great, now split the sum 
into sums depending on 
. We will consider one of them because the other are the same. 
because 
for 
. Thus, when 
goes to infinity we get 
Other sums can be considered in the same way. Thus, we get 
But now remember how we were choosing and . We could take to be large enough that 
and to be the floor of 
, and so 
. Then we can bound the sum above 
where 
is a small number bigger than . And so the series 
diverges, which is a contradiction. Thus there exists a positive integer such that 
.
such that 
and 
such that
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.