[/quote]
,
,
.
be the point on 
such that 
.
.
,
are relatively prime positive integers and 
.
is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide 
,
.
and 
intersects segments 
and 
,
and 
intersect at 
,
and 
intersect at 
.
if and only if 
.
satisfy 
.
.
if 
,
,
and 
are lengths of altitudes in triangle 
,
be its incircle. Denote by 
and 
the points where 
and 
the points on sides 
and 
,
and 
.
.
.
,
be positive integers such that 
is a multiple of 
for all positive integers 
.
.
lies on side 
lies on side 
and 
,
is parallel to 
be a point on line 
,
.
be the point on line 
,
.
,
and 
be positive real numbers satisfying 
Prove that
,
be the intersection point of segments 
.
are 
is the center of the square. Prove that points 
such that:
is strictly increasing,
such that 
for all 
,
,
such that![\[
f(y) = \frac{f(x) + f(x + 2024)}{2}.
\]](http://latex.artofproblemsolving.com/b/e/2/be26213154bb74bd5a35b8d160011351871bfa9b.png)
Proposed by Pavle Martinović
. Prove that
Y by
Let 
be a prime number. Find the number of elements of the set
![\[
M_p = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2p^{n} x} \in \mathbb{Q} \right\}.
\]](//latex.artofproblemsolving.com/5/4/1/54152836e072da8d4ec5984fcb7fea428cda8e48.png)
Edit: I edited with a variation of the same problem because I intend to give this one to an exam for my pupils.
be a prime number. Find the number of elements of the set![\[
M_p = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2p^{n} x} \in \mathbb{Q} \right\}.
\]](http://latex.artofproblemsolving.com/5/4/1/54152836e072da8d4ec5984fcb7fea428cda8e48.png)
Edit: I edited with a variation of the same problem because I intend to give this one to an exam for my pupils.
This post has been edited 1 time. Last edited by td12345, Apr 14, 2025, 6:17 PM
Reason: variation of the problem
Reason: variation of the problem
,
comes from but I think when 
we get 2 extra: 
.
not the set of the square root expression no? I got a different number from yours and I don't know where I went wrong, I will lay down the 
:
where 
. This leads to ![\( [\frac{2025+1}{2}]\)](http://latex.artofproblemsolving.com/d/9/3/d93cfced933adf97b0b51a17da9558e88d0838fc.png)
values of 
where 
.
extra values for 
:
where 
. This leads to 
distinct non-zero values of 
where 
distinct non-zero values of 
?
it leads to 
for the very first family and 
we shall have 
.
,
goes from 1 to 2024. This leads to 
extra values for 
,
.
.
is natural numbers (positive integers) and 
is natural numbers without 
?
