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Fake number theory
Davdav1232 4
N
by NO_SQUARES
Source: Israel TST p1
For a positive integer 
, does there exist positive integer solutions to the following system of equations?
![\[
\begin{cases}
a^n - 2b^n = 1, \\
b^n - 2c^n = 1.
\end{cases}
\]](//latex.artofproblemsolving.com/3/f/5/3f51406329aed6e3a915205c0bce3965ef547527.png)
, does there exist positive integer solutions to the following system of equations?![\[
\begin{cases}
a^n - 2b^n = 1, \\
b^n - 2c^n = 1.
\end{cases}
\]](http://latex.artofproblemsolving.com/3/f/5/3f51406329aed6e3a915205c0bce3965ef547527.png)
4 replies
Interesting inequalities
sqing 1
N
by sqing
Source: Own
Let 
Prove that
Where 
Let
Prove that
Let 
Prove that
Let 
Prove that
Prove that
Where 
Let
Prove that
Let 
Prove that
Let 
Prove that
1 reply
1 viewing
Prime sums of pairs
Assassino9931 1
N
by Nuran2010
Source: Al-Khwarizmi Junior International Olympiad 2025 P5
Sevara writes in red 
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be prime?
Marin Hristov, Bulgaria
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be prime?Marin Hristov, Bulgaria
1 reply
Curious inequality
produit 0
Positive real numbers x_1, x_2, . . . x_n satisfy x_1 + x_2 + . . . + x_n = 1.
Prove that
1/(1 −√x_1)+1/(1 −√x_2)+ . . . +1/(1 −√x_n)⩾ n + 4.
Prove that
1/(1 −√x_1)+1/(1 −√x_2)+ . . . +1/(1 −√x_n)⩾ n + 4.
0 replies
the number of fractions in lowest terms in (0,1) s.t. fractional part >=1/2
tom-nowy 0
Source: Own
Let 
be a positive integer. Find the number of reduced fractions 
for which 
, where 
denotes the fractional part of 
. Express your answer in terms of .
be a positive integer. Find the number of reduced fractions 
for which 
, where 
denotes the fractional part of 
. Express your answer in terms of .
0 replies
Hard geometry
Lukariman 0
Given triangle ABC, a line d intersects the sides AB, AC and the line BC at D, E, F respectively.
(a) Prove that the circles circumscribing triangles ADE, BDF and CEF pass through a point P and P belongs to the circumcircle of triangle ABC.
(b) Prove that the centers of the circles circumscribing triangles ADE, BDF, CEF and ABC are all on the circle.
(c) Let
,
, 
be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle 
belongs to d.
(d) Prove that the orthocenters of triangles ADE, ABC, BDF, CEF are collinear.
(a) Prove that the circles circumscribing triangles ADE, BDF and CEF pass through a point P and P belongs to the circumcircle of triangle ABC.
(b) Prove that the centers of the circles circumscribing triangles ADE, BDF, CEF and ABC are all on the circle.
(c) Let
,
, 
be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle 
belongs to d.(d) Prove that the orthocenters of triangles ADE, ABC, BDF, CEF are collinear.
0 replies
CGMO5: Carlos Shine's Fact 5
v_Enhance 61
N
by Adywastaken
Source: 2012 China Girl's Mathematical Olympiad
As shown in the figure below, the in-circle of 
is tangent to sides 
and 
at 
and 
respectively, and 
is the circumcenter of 
. Prove that 
.
IMAGE
is tangent to sides 
and 
at 
and 
respectively, and 
is the circumcenter of 
. Prove that 
.IMAGE
61 replies
Grid combi with T-tetrominos
Davdav1232 1
N
by NO_SQUARES
Source: Israel TST 8 2025 p1
Let 
denote the maximum number of 
-tetrominoes that can be placed on an 
board such that each -tetromino covers at least one cell that is not covered by any other -tetromino.
Find the smallest real number
such that
![\[
f(N) \leq cN^2
\]](//latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
denote the maximum number of 
-tetrominoes that can be placed on an 
board such that each -tetromino covers at least one cell that is not covered by any other -tetromino.Find the smallest real number
such that![\[
f(N) \leq cN^2
\]](http://latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
1 reply
pqr/uvw convert
Nguyenhuyen_AG 10
N
by Nguyenhuyen_AG
Source: https://github.com/nguyenhuyenag/pqr_convert
Hi everyone,
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
I hope you’ll find it helpful!
Download: pqr_convert
Screenshot:
IMAGE
IMAGE
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.I hope you’ll find it helpful!
Download: pqr_convert
Screenshot:
IMAGE
IMAGE
10 replies
interesting functional
Pomegranat 2
N
by Pomegranat
Source: I don't know sorry
Find all functions 
such that for all positive real numbers 
and 
, the following equation holds:
![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](//latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
such that for all positive real numbers 
and 
, the following equation holds:![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
2 replies
