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All even perfect numbers >6 equal x^3 + y^3 + z^3
TUAN2k8 4
N
by Lufin
Source: 2025 VIASM summer challenge P5
Let 
be an even positive integer greater than 6 such that 
is equal to the sum of all distinct positive divisors of . Prove that there exist distinct integers 
satisfying:
![\[
x^3 + y^3 + z^3 = n.
\]](//latex.artofproblemsolving.com/a/e/a/aea654a9c463a6bd34ea81ffb156fa480b17951e.png)
be an even positive integer greater than 6 such that 
is equal to the sum of all distinct positive divisors of . Prove that there exist distinct integers 
satisfying:![\[
x^3 + y^3 + z^3 = n.
\]](http://latex.artofproblemsolving.com/a/e/a/aea654a9c463a6bd34ea81ffb156fa480b17951e.png)
4 replies
1 viewing
Sword in row
Eeightqx 0
Source: 2025 China South East Mathematical Olympiad Grade10 P7
We put some swords which can only face to up, down, left and right into a 
grid with only 1 sword in a row and 1 sword in a column to form a " star sword in row". A square is said controlled if a sword is put in it or it is pointed by some swords. Try to work out the maximum number of the squares which is controlled in a star sword in row and when it gets its maximum, there are how many situations.
Below is an example of a
star sword in row, swords are represented by arrows and the controlled squares are colored grey. Here 8 squares are controlled.
grid with only 1 sword in a row and 1 sword in a column to form a " star sword in row". A square is said controlled if a sword is put in it or it is pointed by some swords. Try to work out the maximum number of the squares which is controlled in a star sword in row and when it gets its maximum, there are how many situations.Below is an example of a
star sword in row, swords are represented by arrows and the controlled squares are colored grey. Here 8 squares are controlled.
0 replies
Payable numbers
mathscrazy 15
N
by Cats_on_a_computer
Source: INMO 2025/6
Let 
be a positive integer. Anu has an infinite collection of notes with exactly 
copies of a note worth 
rupees, for every integer 
. A positive integer is called payable if Anu can pay exactly 
rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers.
Proposed by Shantanu Nene
be a positive integer. Anu has an infinite collection of notes with exactly 
copies of a note worth 
rupees, for every integer 
. A positive integer is called payable if Anu can pay exactly 
rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers.Proposed by Shantanu Nene
15 replies
Training turkeys to get together
plagueis 10
N
by SimplisticFormulas
Source: Mexico National Olympiad 2020 P3
Let 
be an integer. Two players, Ana and Beto, play the following game. Ana tags the vertices of a regular - gon with the numbers from 
to , in any order she wants. Every vertex must be tagged with a different number. Then, we place a turkey in each of the vertices.
These turkeys are trained for the following. If Beto whistles, each turkey moves to the adjacent vertex with greater tag. If Beto claps, each turkey moves to the adjacent vertex with lower tag.
Beto wins if, after some number of whistles and claps, he gets to move all the turkeys to the same vertex. Ana wins if she can tag the vertices so that Beto can't do this. For each, determine which player has a winning strategy.
Proposed by Victor and Isaías de la Fuente
be an integer. Two players, Ana and Beto, play the following game. Ana tags the vertices of a regular - gon with the numbers from 
to , in any order she wants. Every vertex must be tagged with a different number. Then, we place a turkey in each of the vertices.These turkeys are trained for the following. If Beto whistles, each turkey moves to the adjacent vertex with greater tag. If Beto claps, each turkey moves to the adjacent vertex with lower tag.
Beto wins if, after some number of whistles and claps, he gets to move all the turkeys to the same vertex. Ana wins if she can tag the vertices so that Beto can't do this. For each
, determine which player has a winning strategy.Proposed by Victor and Isaías de la Fuente
10 replies
Positive reals FE
VicKmath7 8
N
by thaiquan2008
Source: Bulgaria NMO 2024, Problem 3
Find all functions 
, such that 
for any positive reals 
.
, such that 
for any positive reals 
.
8 replies
![\[
\int_0^{2\pi} \frac{4\sin x + 8\cos x}{5\sin x + 4\cos x} \, dx
\]](http://latex.artofproblemsolving.com/b/1/e/b1e9d7cdf2a396b53d804a7c1c70ae898e7313e6.png)
![\[
\sin x = t \Rightarrow \cos x \, dx = dt
\]](http://latex.artofproblemsolving.com/4/7/c/47c2a268200339e1bad48c77d8dde23a7c513469.png)
![\[
x = 0 \Rightarrow \sin 0 = 0,\quad x = 2\pi \Rightarrow \sin 2\pi = 0
\]](http://latex.artofproblemsolving.com/e/6/9/e6982ff44fca9896150146d993936b02323dae43.png)
So the bounds in 
become:![\[
t = 0 \text{ to } t = 0
\]](http://latex.artofproblemsolving.com/3/1/d/31d5f0a15353165b1ceb26d59887a28d15bf5f45.png)
Which may misleadingly suggest:![\[
\int_0^0 \ldots \, dt = 0
\]](http://latex.artofproblemsolving.com/7/3/f/73fbf03ee05a5deed199fe20e7014476bbab1aa6.png)
But this is incorrect, where did i go wrong?
I am [not] a parallelogram
peppapig_ 20
N
by AN1729
Source: ISL 2024/G4
Let 
be a quadrilateral with 
parallel to 
and 
. Lines 
and 
intersect at a point 
. Point 
distinct from 
lies on the circumcircle of triangle 
such that 
. Point 
distinct from 
lies on the circumcircle of triangle 
such that 
. Lines 
and 
intersect at 
.
Prove that
is parallel to .
Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine
be a quadrilateral with 
parallel to 
and 
. Lines 
and 
intersect at a point 
. Point 
distinct from 
lies on the circumcircle of triangle 
such that 
. Point 
distinct from 
lies on the circumcircle of triangle 
such that 
. Lines 
and 
intersect at 
.Prove that
is parallel to .Fedir Yudin, Mykhailo Shtandenko, Anton Trygub, Ukraine
20 replies
2 var inquality
Iveela 20
N
by TigerOnion
Source: Izho 2025 P1
Let be positive reals such that 
. Prove that ![\[(a-b)^2 + a + b \geq 2\]](//latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
be positive reals such that 
. Prove that ![\[(a-b)^2 + a + b \geq 2\]](http://latex.artofproblemsolving.com/c/f/a/cfa01a5b54a72afe6746f547390e65dbbb82ea64.png)
20 replies
Four concyclic points
jayme 2
N
by whwlqkd
Source: own?
Dear Mathlinkers,
1. ABC an A-isoceles triangle
2. (O) the circumcircle
3. D a point on the segment AC
4. E le second point d’intersection de (O) avec (BD)
5. Te the tangent to (O) at E
6. Z the point of intersection of the parallel to BC through D and Te
7. Y the second point of intersection of CZ and (O).
Question : E, D, Y et Z are cocyclic.
Sincerely
Jean-Louis
1. ABC an A-isoceles triangle
2. (O) the circumcircle
3. D a point on the segment AC
4. E le second point d’intersection de (O) avec (BD)
5. Te the tangent to (O) at E
6. Z the point of intersection of the parallel to BC through D and Te
7. Y the second point of intersection of CZ and (O).
Question : E, D, Y et Z are cocyclic.
Sincerely
Jean-Louis
2 replies
van der Waerden Theorem
steven_zhang123 0
Source: 2025 CSMO Grade 11 P8
Do there exist sets 
and 
such that 
, 
, and for every positive integer 
, there exists an integer 
such that neither nor contains an arithmetic progression of common difference and length 
? Prove your answer.
and 
such that 
, 
, and for every positive integer 
, there exists an integer 
such that neither nor contains an arithmetic progression of common difference and length 
? Prove your answer.
0 replies
