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Hexagonal lotus leaves
MathMystic33 0
Source: 2025 Macedonian Team Selection Test P2
A lake is in the shape of a regular hexagon of side length 
. Initially there is a single lotus leaf somewhere in the lake, sufficiently far from the shore. Each day, from every existing leaf a new leaf may grow at distance 
(measured between centers), provided it does not overlap any other leaf. If the lake is large enough that edge effects never interfere, what is the least number of days required to have 
leaves in the lake?
. Initially there is a single lotus leaf somewhere in the lake, sufficiently far from the shore. Each day, from every existing leaf a new leaf may grow at distance 
(measured between centers), provided it does not overlap any other leaf. If the lake is large enough that edge effects never interfere, what is the least number of days required to have 
leaves in the lake?
0 replies
Collinearity of intersection points in a triangle
MathMystic33 0
Source: 2025 Macedonian Team Selection Test P1
On the sides of the triangle 
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let
![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](//latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
lie the following points: 
and 
on 
, 
on 
, and 
on 
. Let![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet at 
. Prove that the points 
, , and 
are collinear.
0 replies
1 viewing
3 variable FE with divisibility condition
pithon_with_an_i 2
N
by ATM_
Source: Revenge JOM 2025 Problem 1, Revenge JOMSL 2025 N2, Own
Find all functions 
such that 
for all 
.
such that 
for all 
.
2 replies
Dissection into equal‐area pieces using diagonals
MathMystic33 0
Source: Macedonian Mathematical Olympiad 2025 Problem 5
Let 
be a natural number, and let be the square of side length 
subdivided into 
unit squares. Determine for which values of it is possible to dissect into connected regions of equal area using only the diagonals of those unit squares, subject to the condition that from each unit square at most one of its diagonals is used (some unit squares may have neither diagonal).
be a natural number, and let be the square of side length 
subdivided into 
unit squares. Determine for which values of it is possible to dissect into connected regions of equal area using only the diagonals of those unit squares, subject to the condition that from each unit square at most one of its diagonals is used (some unit squares may have neither diagonal).
0 replies
Brazilian Locus
kraDracsO 15
N
by Ilikeminecraft
Source: IberoAmerican, Day 2, P4
Let 
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
, let 
be the barycenter of the triangle 
. Determine the locus of points such that 
.
Note: The locus is the set of all points of the plane that satisfies the property.
and 
be two fixed points in the plane. For each point 
of the plane, outside of the line 
, let 
be the barycenter of the triangle 
. Determine the locus of points such that 
.Note: The locus is the set of all points of the plane that satisfies the property.
15 replies
Divisibility with the polynomial ax^{75}+b
MathMystic33 0
Source: Macedonian Mathematical Olympiad 2025 Problem 4
Let 
be a polynomial where 
and 
are coprime integers in the set 
, and suppose it satisfies the following condition: there exists at most one prime 
such that for every positive integer 
, 
. Prove that for every prime 
there exists a positive integer for which 
be a polynomial where 
and 
are coprime integers in the set 
, and suppose it satisfies the following condition: there exists at most one prime 
such that for every positive integer 
, 
. Prove that for every prime 
there exists a positive integer for which 
0 replies
1 viewing
Maximum reach of splitting tokens
MathMystic33 0
Source: Macedonian Mathematical Olympiad 2025 Problem 3
On a horizontally placed number line, a pile of 
tokens is placed on each number 
. As long as at least one pile contains at least two tokens, we repeat the following procedure: we choose such a pile (say, it consists of 
tokens), and move the top token from the selected pile 
unit positions to the right along the number line. What is the largest natural number 
on which a token can be placed? (Express as a function of 
.)
tokens is placed on each number 
. As long as at least one pile contains at least two tokens, we repeat the following procedure: we choose such a pile (say, it consists of 
tokens), and move the top token from the selected pile 
unit positions to the right along the number line. What is the largest natural number 
on which a token can be placed? (Express as a function of 
.)
0 replies
Inequality with rational function
MathMystic33 0
Source: Macedonian Mathematical Olympiad 2025 Problem 2
Let 
be an integer, 
a real number, and 
be positive real numbers such that 
. Prove that:
![\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]](//latex.artofproblemsolving.com/0/8/d/08de0c74a4e36b50a64d17875d3fd93eeb5b52de.png)
When does equality hold?
be an integer, 
a real number, and 
be positive real numbers such that 
. Prove that:![\[
\frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n.
\]](http://latex.artofproblemsolving.com/0/8/d/08de0c74a4e36b50a64d17875d3fd93eeb5b52de.png)
When does equality hold?
0 replies
Circumcircle of MUV tangent to two circles at once
MathMystic33 0
Source: Macedonian Mathematical Olympiad 2025 Problem 1
Given is an acute triangle 
with 
. Let 
be the midpoint of side 
, and let 
and 
be points on segments 
and 
, respectively, such that 
. Let 
be the circumcircle of 
, and 
the circumcircle of 
. The common tangent 
to and , which lies closer to point 
, touches and at points 
and 
, respectively. Let the line 
intersect again at 
, and the line 
intersect again at 
. Prove that the circumcircle of triangle 
is tangent to both and .
with 
. Let 
be the midpoint of side 
, and let 
and 
be points on segments 
and 
, respectively, such that 
. Let 
be the circumcircle of 
, and 
the circumcircle of 
. The common tangent 
to and , which lies closer to point 
, touches and at points 
and 
, respectively. Let the line 
intersect again at 
, and the line 
intersect again at 
. Prove that the circumcircle of triangle 
is tangent to both and .0 replies
USAMO 2003 Problem 5
MithsApprentice 93
N
by endless_abyss
Let 
, 
, 
be positive real numbers. Prove that
![\[ \dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.
\]](//latex.artofproblemsolving.com/0/5/e/05e90cc0c9d60f8943608b690f75d20bcd4f6220.png)
, 
, 
be positive real numbers. Prove that![\[ \dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.
\]](http://latex.artofproblemsolving.com/0/5/e/05e90cc0c9d60f8943608b690f75d20bcd4f6220.png)
93 replies
