[/quote]
,
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 
.
and 
Prove that
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects 
Let 
be a point on 
Let 
be the intersection of 
and 
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 
,
.
such that: 
.
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 ![\[
d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1}
\]](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 
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 
.
be a positive integer and 
be a family of at most 
with the following property: we can consider 
and draw segments connecting these points such that points 
and 
are connected if and only if 
belongs to 
Determine the maximal value that the sum of the cardinalities of the subsets in 
can take.
Y by Adventure10, Mango247
In a triangle 
. Given is a point 
on side 
such that 
. In addition, point 
outside the triangle satisfies 
. Given that all angles of triangles 
and 
, measured in degrees, are integers, determine the angles of these two triangles.
. Given is a point 
on side 
such that 
. In addition, point 
outside the triangle satisfies 
. Given that all angles of triangles 
and 
, measured in degrees, are integers, determine the angles of these two triangles.This post has been edited 1 time. Last edited by parmenides51, Aug 11, 2018, 9:42 AM
Reason: name change
Reason: name change
