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.
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 
.
+1 w
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 positive integer and 
be a family of at most distinct subsets of the set 
with the following property: we can consider distinct points in the plane, labelled 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to subsets in for any 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
is the bisector of an isosceles triangle 
. A point 
is chosen on the Base 
and a point 
is chosen on the lateral side 
so that 
. Prove that 
.
be the midpoint of 
, let 
.
, we get 
. So, 
.
.
.
.
and 
.
lie on a circle, and so 
.
onto 
and 
at 
and 
.
, 
.
is equivalent to showing :
, for some with 
( where 
) , which is easy. , so 
, which means that 
. Done.
is the bisector of an isosceles triangle . A point is chosen on the Base and a point is chosen on the lateral side so that . Prove that .
. So 
.
: 
to intersect 
at 
Use chicken feet theorem. Everything becomes clear I will complete my post 
to intersect at Use chicken feet theorem. Everything becomes clear I will complete my post 
is cyclic. be the midpoint of . thats enough to prove that 
is cyclic which this is obvious(by little computing)
and 
Prove that
point on 
and 
are midpoints of 
.
and 
so 
so 
