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.
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
have,where 
and 
are natural numbers.
be the set of the positive real numbers. Find all functions 
such that 
for all positive real numbers 
and 
.
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
with 
and the circle 
of radius centered at 
Let 
be the midpoint of 
The line 
intersects a second time at 
Let be a point on such that 
Let 
be the intersection of 
and Prove that 
such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers 
such that 
, there will be a colour coloring exactly one integer from the set 
.
such that: 
. Find the maximum: 
such that if![\[
1 = d_1 < d_2 < \cdots < d_{k-1} < d_k = n
\]](http://latex.artofproblemsolving.com/a/d/2/ad2cbfe9e2110de2fdb6308d93f58880bd79afbc.png)
are the divisors of , then the sequence![\[
d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1}
\]](http://latex.artofproblemsolving.com/d/2/3/d239210af0c2aa005ee1e5b820678d4aadc366c0.png)
forms a permutation of an arithmetic progression.(Kritesh Dhakal,Nepal)
be a finite set of integers. Prove that there exists a number 
depending on such that for each non-constant polynomial 
with integer coefficients the number of integers 
satisfying 
does not exceed 
.
such that 
and 
for all reals 
.
, 
be positive integers such that 
is a multiple of 
for all positive integers . Prove that 
.
be a triangle with 
and the circle with diameter and center . The incircle of touches 
at 
respectively. 
intersects at points 
and 
. Show that 
.
. Angle chasing gives 
and similarly, 
. Thus, 
.
and 
meet 
at 
and 
. Since 
it follows that 
is cyclic. Similarly 
is cyclic.
and 
, which implies 
is isosceles, 
lies on and similarly, lies on , which means 
and 
and hence .
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and 
Prove that
