G
Topic
First Poster
Last Poster
Inspired by a cool result
DoThinh2001 1
N
by arqady
Source: Old?
Let three real numbers 
, no two of which are 
. Prove that:
, no two of which are 
. Prove that:
1 reply
Crossing ٍٍChords
matinyousefi 1
N
by Trenod
Source: Iranian Combinatorics Olympiad 2020 P3
points and some chords between them is given.
In every step we can take two chords 
with a common point other than 
and erase exactly one of and draw 
let 
be the minimum of chords after some steps. Find the maximum of over all initial positions.
In every step we can take two chords with a common point other than and erase both of and draw let be the minimum of chords after some steps. Find the maximum of over all initial positions.Proposed by Afrouz Jabalameli, Abolfazl Asadi
1 reply
Nice NT with powers of two
oVlad 7
N
by SimplisticFormulas
Source: Romania TST 2024 Day 1 P3
Let 
be a positive integer and let 
and 
be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer 
such that at least one of the numbers 
and 
is divisible by 
Cătălin Liviu Gherghe
be a positive integer and let 
and 
be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer 
such that at least one of the numbers 
and 
is divisible by 
Cătălin Liviu Gherghe
7 replies
Inequality in triangle
Nguyenhuyen_AG 0
Let 
be the lengths of the sides of a triangle. Prove that
![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](//latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
be the lengths of the sides of a triangle. Prove that![\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]](http://latex.artofproblemsolving.com/7/4/b/74b7aa5621bc52977d9acd448d74293dc27a8e00.png)
0 replies
is a positive integer. Prove that
JBMO Shortlist 2023 G7
Orestis_Lignos 7
N
by tilya_TASh
Source: JBMO Shortlist 2023, G7
Let 
and 
be arbitrary points on the sides 
and 
of triangle 
, respectively. The circumcircle of 
meets for the second time the circumcircle of 
at point 
. Line 
meets line 
at point 
, while line 
meets line 
at point 
. Prove that lines 
and 
pass through the same point.
and 
be arbitrary points on the sides 
and 
of triangle 
, respectively. The circumcircle of 
meets for the second time the circumcircle of 
at point 
. Line 
meets line 
at point 
, while line 
meets line 
at point 
. Prove that lines 
and 
pass through the same point.
7 replies
Reflected point lies on radical axis
Mahdi_Mashayekhi 5
N
by Mahdi_Mashayekhi
Source: Iran 2025 second round P4
Given is an acute and scalene triangle with circumcenter 
. 
and 
intersect the altitude from 
to at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
with circumcenter 
. 
and 
intersect the altitude from 
to at points 
and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
5 replies
Find the value
sqing 18
N
by Yiyj
Source: 2024 China Fujian High School Mathematics Competition
Let 
and 
Find the value of 
and 
Find the value of 
18 replies
Number Theory
fasttrust_12-mn 14
N
by Namisgood
Source: Pan African Mathematics Olympiad P1
Find all positive intgers 
and 
such that 
and 
is a prime number
and 
such that 
and 
is a prime number
14 replies
