[/quote]
,
table filled with natural numbers, we say it is a divisor table if:
-
,
-
,
for every 
.
is given. Determine the smallest natural number 
,
such that there exists a sequence of positive real numbers 
,
,![\[
a_1 + \cdots + a_{n+1} < \alpha \cdot a_n.
\]](http://latex.artofproblemsolving.com/9/b/7/9b73deb46032844b324288b02470ff0623ac0ff1.png)
Proposed by Pavle Martinović
of the set 
,
such that:
,
,
.
?
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
be an acute triangle. Let 
be the reflection of point 
.
and 
be the circumcenter and the orthocenter of triangle 
be the midpoint of segment 
.
and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle 
is the foot of the perpendicular from 
,
.
be a prime number and 
.![\[\left| p^m - (p - 2)! \right| > p^2.\]](http://latex.artofproblemsolving.com/2/1/c/21ca8bb6e5727d48b18b7ec3b127029c4c97694f.png)
Proposed by Miloš Milićev
,
,
,
of real numbers satisfies![\[
a_{2n-1} + a_{2n} > a_{2n+1} + a_{2n+2} \qquad \mbox{and} \qquad a_{2n} + a_{2n+1} < a_{2n+2} + a_{2n+3}
\]](http://latex.artofproblemsolving.com/4/d/a/4da3b88a20c42c1798141b8db086de341cfb9a67.png)
for every positive integer 
such that 
for every positive integer 
such that 
is divisible by 
and 
.
satisfying
Prove
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
is an arbitrary positive integer.
Prove that
Where 
such that for all real numbers 
and 
,
b
its circumscribed circle. Let 
with 
be the midpoint of the segment 
and let 
be the point of intersection of 
with 
.
Y by
A point 
is chosen on the side 
of a rectangle 
. From the vertex 
, the perpendicular 
is dropped to the segment 
. The segments and divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to are equal then the third circle is also equal to them.
is chosen on the side 
of a rectangle 
. From the vertex 
, the perpendicular 
is dropped to the segment 
. The segments and divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to are equal then the third circle is also equal to them.Attachments:
This post has been edited 1 time. Last edited by QueenArwen, Mar 11, 2025, 10:52 AM
