Serbian selection contest for the IMO 2025 - P6
by OgnjenTesic, May 22, 2025, 4:07 PM
For an 
table filled with natural numbers, we say it is a divisor table if:
- the numbers in the
-th row are exactly all the divisors of some natural number 
,
- the numbers in the
-th column are exactly all the divisors of some natural number 
,
-
for every 
.
A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
Proposed by Pavle Martinović
table filled with natural numbers, we say it is a divisor table if:- the numbers in the
-th row are exactly all the divisors of some natural number 
,- the numbers in the
-th column are exactly all the divisors of some natural number 
,-
for every 
.A prime number
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.Proposed by Pavle Martinović
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 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
such that 
is divisible by 
and 
.![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
is an arbitrary positive integer.
satisfying
Prove
,
,
,
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 
be real numbers such that 
. Find the value of 
and find at least one triplet 
that satisfy those conditions.
May 2025
April 2025