satisfying
Prove
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
Y by
Let 
and 
Prove that
Where 
and 
Prove that
Where 
This post has been edited 1 time. Last edited by sqing, Apr 16, 2025, 5:16 AM
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Inequality with 3 variables and a special condition
Nuran2010 2
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by Assassino9931
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers 
we have 
.
Prove that:
.
Determine the equality case.
we have 
.Prove that:
.Determine the equality case.
2 replies
Good divisors and special numbers.
Nuran2010 1
N
by Assassino9931
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least 
beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
1 reply
Find points with sames integer distances as given
nAalniaOMliO 2
N
by nAalniaOMliO
Source: Belarus TST 2024
Points 
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
N. Sheshko, D. Zmiaikou
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
N. Sheshko, D. Zmiaikou
2 replies
Geometry tangent circles
Stefan4024 68
N
by zuat.e
Source: EGMO 2016 Day 2 Problem 4
Two circles 
and 
, of equal radius intersect at different points 
and 
. Consider a circle 
externally tangent to at 
and internally tangent to at point 
. Prove that lines 
and 
intersect at a point lying on .
and 
, of equal radius intersect at different points 
and 
. Consider a circle 
externally tangent to at 
and internally tangent to at point 
. Prove that lines 
and 
intersect at a point lying on .
68 replies
My Unsolved Problem
MinhDucDangCHL2000 2
N
by hukilau17
Source: 2024 HSGS Olympiad
Let triangle 
be inscribed in the circle 
. A line through point 
intersects 
and 
at points 
and 
, respectively. Let 
be the reflection of across the midpoint of , and 
be the reflection of across the midpoint of . Prove that:
a) the reflection of the orthocenter
of triangle across line 
lies on the circle .
b) the orthocenters of triangles
and 
coincide.
Im looking for a solution used complex bashing :(
be inscribed in the circle 
. A line through point 
intersects 
and 
at points 
and 
, respectively. Let 
be the reflection of across the midpoint of , and 
be the reflection of across the midpoint of . Prove that:a) the reflection of the orthocenter
of triangle across line 
lies on the circle .b) the orthocenters of triangles
and 
coincide.Im looking for a solution used complex bashing :(
2 replies
Easy Combinatorics
MuradSafarli 2
N
by Sadigly
A student firstly wrote 
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
. Is this possible to make the number 
on the board?
on the board. For each procces, the stutent deletes the number x and replaces it with either 
or 
or 
. Is this possible to make the number 
on the board?
2 replies
D1025 : Can you do that?
Dattier 0
Source: les dattes à Dattier
Let 
and 
.
Can you calculate
?
and 
.Can you calculate
?
0 replies
Perpendicularity
April 32
N
by zuat.e
Source: CGMO 2007 P5
Point 
lies inside triangle such that 
and 
. Point is the midpoint of segment 
. Point lies on segment with 
. Prove that 
.
lies inside triangle such that 
and 
. Point is the midpoint of segment 
. Point lies on segment with 
. Prove that 
.
32 replies
The number of integers
Fang-jh 16
N
by ihategeo_1969
Source: ChInese TST 2009 P3
Prove that for any odd prime number 
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
the number of positive integer 
satisfying 
is less than or equal to 
where 
is a constant independent of 
16 replies
Inspired by Omerking
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Source: Own
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![\[
4ka + 4b + 4kc = (4k-1)(a+c) + (a+c+4b) \geq 2\sqrt{(4k-1)(a+c)(a+c+4b)} \geq 2\sqrt{(4k-1)\cdot 4(ab+bc+ca)} \geq 4\sqrt{\frac{4k-1}{3}}.
\]](http://latex.artofproblemsolving.com/a/5/0/a50d93eb84d5ec5aae8800fe70e29af6a9d9f52a.png)
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