Prove that
and 
Prove that
Let 
Prove that
Y by
Without using a calculator, show that 
, and for a positive integer 
, prove that
.
, and for a positive integer 
, prove that
.G
Topic
First Poster
Last Poster
IMO ShortList 2001, number theory problem 4
orl 43
N
by Zany9998
Source: IMO ShortList 2001, number theory problem 4
Let 
be a prime number. Prove that there exists an integer 
with 
such that neither 
nor 
is divisible by 
.
be a prime number. Prove that there exists an integer 
with 
such that neither 
nor 
is divisible by 
.
43 replies
Number theory - Iran
soroush.MG 32
N
by Nobitasolvesproblems1979
Source: Iran MO 2017 - 2nd Round - P1
a) Prove that there doesn't exist sequence 
such that: 
b) Let
be an odd prime number. Prove that there exist sequence such that: 
such that: 
b) Let
be an odd prime number. Prove that there exist sequence such that: 
32 replies
Line through incenter tangent to a circle
Kayak 32
N
by L13832
Source: Indian TST D1 P1
In an acute angled triangle 
with 
, let 
denote the incenter and 
the midpoint of side 
. The line through 
perpendicular to 
intersects the tangent from to the incircle (different from line ) at a point 
> Show that is tangent to the circumcircle of triangle 
.
Proposed by Tejaswi Navilarekallu
with 
, let 
denote the incenter and 
the midpoint of side 
. The line through 
perpendicular to 
intersects the tangent from to the incircle (different from line ) at a point 
> Show that is tangent to the circumcircle of triangle 
.Proposed by Tejaswi Navilarekallu
32 replies
D1015 : A strange EF for polynomials
Dattier 3
N
by Dattier
Source: les dattes à Dattier
Find all ![$P \in \mathbb R[x,y]$](//latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](//latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
, 
is an homoemorphisme too.
![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
, 
is an homoemorphisme too.
3 replies
Turkey EGMO TST 2017 P6
nimueh 4
N
by Nobitasolvesproblems1979
Source: Turkey EGMO TST 2017 P6
Find all pairs of prime numbers 
, such that 
and 
are both integers.
, such that 
and 
are both integers.
4 replies
An inequality
JK1603JK 4
N
by Quantum-Phantom
Source: unknown
Let a,b,c>=0: ab+bc+ca=3 then maximize P=\frac{a^2b+b^2c+c^2a+9}{a+b+c}+\frac{abc}{2}.
4 replies
Inspired by Abelkonkurransen 2025
sqing 1
N
by kiyoras_2001
Source: Own
Let 
be real numbers such that 
Prove that 
Let be real numbers such that 
Prove that 
be real numbers such that 
Prove that 
Let be real numbers such that 
Prove that 
1 reply
Inequalities
sqing 11
N
by DAVROS
Let 
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
11 replies
Geometry challenging question
srnjbr 0
Given a triangle ABC. A1, B1 and C1 are the points of contact of the inner circumcircle of the triangle with the sides BC, AC and AB respectively. The point of contact of AA1 with B1C1 and the circumcircle are called L and Q respectively. M is the midpoint of B1C1. The point of intersection of lines BC and B1C1 is called T. P is the foot of the perpendicular drawn to AT from point L. Show that points A1, M, Q and P lie on a circle.
0 replies
Inequality with integers and indices
G
H
G
H
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#2
Mar 17, 2025, 1:09 PM
Y by
In order to solve the first problem, I have the following idea: raise both sides to the 6th power, and you're done! Right?
In the general case, consider the function f(x)=lnx/x, whose derivative is negative for x>e, so the function is decreasing. Let x=n and x=n+1, and compare them! I think you understand, if not, write and I'll answer!
In the general case, consider the function f(x)=lnx/x, whose derivative is negative for x>e, so the function is decreasing. Let x=n and x=n+1, and compare them! I think you understand, if not, write and I'll answer!
This post has been edited 3 times. Last edited by invisibleman, Mar 18, 2025, 11:36 AM
Reason: typo
Reason: typo
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by
The inequality is equivalent to 
. Taking natural logarithm on both sides, this is equivalent to 
Dividing, this is equivalent to 
. Since 
is increasing (obviously linear functions increase more quickly than logarithms, or using the derivative like @above said), we are done by definition.
i hope this is right!
well someone taught me the solution hehe
. Taking natural logarithm on both sides, this is equivalent to 
Dividing, this is equivalent to 
. Since 
is increasing (obviously linear functions increase more quickly than logarithms, or using the derivative like @above said), we are done by definition.i hope this is right!
well someone taught me the solution hehe
This post has been edited 2 times. Last edited by krish6_9, Mar 18, 2025, 12:31 AM
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#6
Mar 18, 2025, 1:43 AM
Y by
Bro! Let's show n^(n+1) > (n+1)^n , or (1+1/n)^n < n. Note that (1+1/n)^n < e because this limits upwards towards e. So for n >= 3, this is obviously true as e<3. Bruh! Now that's rizz!
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#7
Mar 18, 2025, 7:32 AM
Y by
f(x)=\frac{\ln x}{x}\Rightarrow f'(x)=\frac{1-\ln x}{{{x}^{2}}}<0,\forall x>e\Rightarrow f(n)>f(n+1),\forall n\ge 2
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Z
K
Y
N Quick Reply
G
H
