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((n-1)!-n)(n-2)!=m(m-2)
NO_SQUARES 0
Source: Regional Stage of ARO 2025 9.5=11.4
Find all pairs of integer numbers 
and 
such that 
.
A. Kuznetsov
and 
such that 
.A. Kuznetsov
0 replies
Perfect squares imply GCD is a perfect square
MathMystic33 0
Source: 2024 Macedonian Team Selection Test P6
Let 
be positive integers such that 
, 
, and 
are perfect squares. Prove that 
is also a perfect square.
be positive integers such that 
, 
, and 
are perfect squares. Prove that 
is also a perfect square.
0 replies
Maximum number of edge‐colors for strong monochromatic connectivity
MathMystic33 0
Source: 2024 Macedonian Team Selection Test P5
Let 
be a convex polyhedron with the following properties:
1) has exactly 
edges.
2) The degrees of all vertices of differ by at most 
.
3) There is an edge‐coloring of with 
colors such that for each color 
and any two distinct vertices 
, there exists a path from 
to 
all of whose edges have color .
Determine the largest positive integer for which such a polyhedron exists.
be a convex polyhedron with the following properties:1)
has exactly 
edges.2) The degrees of all vertices of
differ by at most 
.3) There is an edge‐coloring of
with 
colors such that for each color 
and any two distinct vertices 
, there exists a path from 
to 
all of whose edges have color .Determine the largest positive integer
for which such a polyhedron exists.
0 replies
Functional equation with extra divisibility condition
MathMystic33 1
N
by grupyorum
Source: 2025 Macedonian Team Selection Test P4
Find all functions 
such that
1)
divides 
for every 
, and
2) for all
we have
![\[
f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr).
\]](//latex.artofproblemsolving.com/c/f/f/cff41352422c8765978b8fe6dc966ba934a9df49.png)
such that1)
divides 
for every 
, and2) for all
we have![\[
f\bigl(f(a)+kb\bigr)\;=\;f\bigl(a + k\,f(b)\bigr).
\]](http://latex.artofproblemsolving.com/c/f/f/cff41352422c8765978b8fe6dc966ba934a9df49.png)
1 reply
1 viewing
Concurrency of tangent touchpoint lines on thales circles
MathMystic33 0
Source: 2024 Macedonian Team Selection Test P4
Let 
be an acute scalene triangle. Denote by 
the circle with diameter 
, and let 
be the contact points of the tangents from 
to , chosen so that 
and 
lie on opposite sides of 
and 
and 
lie on opposite sides of 
. Similarly, let 
be the circle with diameter 
, with tangents from touching at 
, and 
the circle with diameter , with tangents from touching at 
.
Prove that the lines
are concurrent.
be an acute scalene triangle. Denote by 
the circle with diameter 
, and let 
be the contact points of the tangents from 
to , chosen so that 
and 
lie on opposite sides of 
and 
and 
lie on opposite sides of 
. Similarly, let 
be the circle with diameter 
, with tangents from touching at 
, and 
the circle with diameter , with tangents from touching at 
.Prove that the lines
are concurrent.
0 replies
Equal areas of the triangles on the parabola
NO_SQUARES 0
Source: Regional Stage of ARO 2025 10.10; also Kvant 2025 no. 3 M2837
On the graphic of the function 
were selected 
pairwise distinct points, abscissas of which are integer numbers from the segment ![$[0; 100000]$](//latex.artofproblemsolving.com/0/5/e/05e1d2855ef6a6dfaf9b24f2713455ab1017a30d.png)
. Prove that it is possible to choose six different selected points , , , 
, 
, 
such that areas of triangles 
and 
are equals.
A. Tereshin
were selected 
pairwise distinct points, abscissas of which are integer numbers from the segment ![$[0; 100000]$](http://latex.artofproblemsolving.com/0/5/e/05e1d2855ef6a6dfaf9b24f2713455ab1017a30d.png)
. Prove that it is possible to choose six different selected points , , , 
, 
, 
such that areas of triangles 
and 
are equals.A. Tereshin
0 replies
Al-Khwarizmi birth year in a combi process
Assassino9931 1
N
by Assassino9931
Source: Al-Khwarizmi International Junior Olympiad 2025 P3
On a circle are arranged 
baskets, each containing at least one candy. The total number of candies is 
. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from 
consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least 
candies, regardless of the initial distribution of candies and Sevinch's actions.
Shubin Yakov, Russia
baskets, each containing at least one candy. The total number of candies is 
. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from 
consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least 
candies, regardless of the initial distribution of candies and Sevinch's actions.Shubin Yakov, Russia
1 reply
Anything real in this system must be integer
Assassino9931 6
N
by Assassino9931
Source: Al-Khwarizmi International Junior Olympiad 2025 P1
Determine the largest integer 
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
Marek Maruin, Slovakia
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.Marek Maruin, Slovakia
6 replies
Concurrency from symmetric points on the sides of a triangle
MathMystic33 0
Source: 2024 Macedonian Team Selection Test P3
Let be a triangle. On side take points 
and 
such that 
on side take points 
and 
such that 
and on side take points 
and 
such that 
Let 
and 
Prove that the lines
are concurrent.
be a triangle. On side take points 
and 
such that 
on side
take points 
and 
such that 
and on side take points 
and 
such that 
Let 
and 
Prove that the lines
are concurrent.
0 replies
Grouping angles in a pentagon with bisectors
Assassino9931 2
N
by Assassino9931
Source: Al-Khwarizmi International Junior Olympiad 2025 P2
Let 
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](//latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.
Miroslav Marinov, Bulgaria
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.Miroslav Marinov, Bulgaria
2 replies
