Prove that
Y by DrMath, jh235, TheCrafter, Eugenis, mathleticguyyy, somebodyyouusedtoknow, Adventure10, Mango247, NicoN9, cubres, ehuseyinyigit
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
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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
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
find question
mathematical-forest 5
N
by Jupiterballs
Are there any contest questions that seem simple but are actually difficult? :-D
5 replies
Own made functional equation
Primeniyazidayi 10
N
by Phat_23000245
Source: own(probably)
Find all functions 
such that 
for all 
such that 
for all 
10 replies
Tough inequality
TUAN2k8 4
N
by Phat_23000245
Source: Own
Let 
be an even integer and let 
be real numbers satisfying 
.
Prove that
be an even integer and let 
be real numbers satisfying 
.Prove that
4 replies
Guess period of function
a1267ab 9
N
by HamstPan38825
Source: USA TST 2025
Let 
be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
. He tells Ana that 
is nonconstant, 
, and 
for all integers 
. Ana's goal is to determine the value of . She writes down integers 
. After seeing this list, Banana writes down 
in order. Ana wins if she can determine the value of from this information. Find the smallest value of for which Ana has a winning strategy.
Anthony Wang
be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
. He tells Ana that 
is nonconstant, 
, and 
for all integers 
. Ana's goal is to determine the value of . She writes down integers 
. After seeing this list, Banana writes down 
in order. Ana wins if she can determine the value of from this information. Find the smallest value of for which Ana has a winning strategy.Anthony Wang
9 replies
Inequality with abc=1
tenplusten 11
N
by sqing
Source: JBMO 2011 Shortlist A7
Let be positive reals such that 
.Prove the inequality 
11 replies
Central sequences
EeEeRUT 13
N
by v_Enhance
Source: EGMO 2025 P2
An infinite increasing sequence 
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
Show that there exists an infinite sequence
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .Show that there exists an infinite sequence
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
13 replies
Advanced topics in Inequalities
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Y by Adventure10
Will edit later maybe
4.1:
5.1:
[/hide]
4.1:
by AM-GM so it suffices to show
but clearly 
giving the result.5.1:
so then
so it suffices to show
or 
.![$xy+yz+zx \ge 3 \sqrt[3]{x^2y^2z^2} \ge 3$](http://latex.artofproblemsolving.com/9/1/1/911e8dbf8e67fc68c37c72b890036797739666cb.png)
and
holds because 
h[/hide]
This post has been edited 2 times. Last edited by tastymath75025, Mar 7, 2015, 5:31 AM
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![$x+y+z \geq 3\sqrt[3]{xyz} \geq 3$](http://latex.artofproblemsolving.com/3/1/4/314f429df203762a8392644a33d2f0d55cdd43d9.png)
and ![$xy+yz+xz \geq 3\sqrt[3]{x^2y^2z^2} \geq 3$](http://latex.artofproblemsolving.com/d/b/3/db32383ff1c260e0c691e722f49e7e02e0af9499.png)
.
,
.
and symmetrically others.
.
,
due to Cauchy,problem is over.
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Y by Strangett
Problem 5.1. (Tuan Le)
Let be positive real numbers such that 
Prove the inequality
Problem4.2. (T.Q.Anh)
Let
and 
.Prove that
Problem 3.1. (Titu Andresscu)
Let
and 
Prove that
https://artofproblemsolving.com/community/c6h1838005p12333824
Let
be positive real numbers such that 
Prove the inequality
Problem4.2. (T.Q.Anh)
Let
and 
.Prove that
Problem 3.1. (Titu Andresscu)
Let
and 
Prove that
https://artofproblemsolving.com/community/c6h1838005p12333824This post has been edited 2 times. Last edited by sqing, Apr 16, 2025, 2:33 PM
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#6
Apr 16, 2025, 1:48 PM
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You see NOTHING
This post has been edited 1 time. Last edited by giangtruong13, Apr 16, 2025, 1:55 PM
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#7
Apr 16, 2025, 2:54 PM
Y by
The direction of 3.2 is false.Anyway,here is an approach from "Problems from the book,Titu Andreescu and Gabriel Dospinescu"
and 
.
Proof:Apply Titu's Lemma and it succifies to show that 
or 
Unfortunaly,there is just a note which says "This is not an easy inequality.We will use the proposed problem 6 form the chepter "Two useful Substitutions" to reduce the above inequality to the form 
which follows from 
.
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are reals 
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Y by
Let 
This post has been edited 1 time. Last edited by sqing, Apr 16, 2025, 3:27 PM
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.![\[
f''(x) = \frac{d^2}{dx^2} \ln(2x^2 + 3) = \frac{12}{(2x^2 + 3)^2} > 0 \quad \text{for all } x \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/e/5/d/e5d9804fd04315833f902731f3cd4baaf563c7eb.png)
Hence, 
is strictly convex.![\[
f(a) + f(b) + f(c) \ge 3f\left( \frac{a + b + c}{3} \right) = 3f(1).
\]](http://latex.artofproblemsolving.com/9/0/3/903988a564a1e33e476d7a575cf090c3b13247eb.png)
Now,![\[
f(1) = \ln(2 \cdot 1^2 + 3) = \ln 5 \Rightarrow f(a) + f(b) + f(c) \ge 3 \ln 5 = \ln 125.
\]](http://latex.artofproblemsolving.com/3/1/9/3192bb0efde4cf7d3ffd7c2b5e0a3359e34d1783.png)
Exponentiating both sides gives:
.
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Y by
Novmath wrote:
Strangett wrote:
Problem 3.4. Given that 
, prove that
![\[
(2a^2 + 3)(2b^2 + 3)(2c^2 + 3) \ge 125.
\]](//latex.artofproblemsolving.com/c/1/e/c1e561355e2a6d57be0f874a515ad8c4db9bb352.png)
, prove that![\[
(2a^2 + 3)(2b^2 + 3)(2c^2 + 3) \ge 125.
\]](http://latex.artofproblemsolving.com/c/1/e/c1e561355e2a6d57be0f874a515ad8c4db9bb352.png)
Nice solution ,can you explain how did you think of taking f(x)= something with ln. When do we take f(x) as ln(something) ?
Just in order to use the sum property of the natural logarithm (to apply Jensen's inequality to sums of function values). Also when taking the derivatives it is pretty obvious we are going to get something with its denominator squared and it is surely a convex function.
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Show that
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Y by
Strangett wrote:
Just in order to use the sum property of the natural logarithm (to apply Jensen's inequality to sums of function values). Also when taking the derivatives it is pretty obvious we are going to get something with its denominator squared and it is surely a convex function.
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Y by
sqing wrote:
Novmath wrote:
va2010 wrote:
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
The problem 2.1 (USAMO 2001 P3) is wrong. The condition was
Let
and satisfy 
Show that![\[ 0 \le ab + bc + ca - abc \leq 2. \]](http://latex.artofproblemsolving.com/2/0/9/209b3fcb56c01c335bbcb8d5fb781e28e49a0798.png)
Among a, b, c there must exist 2 numbers, say a and b, such that
Then:
On the other hand, from
we have:
Thus:
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Y by
sqing wrote:
Can you share solution of this one:
Problem: Prove that for all real numbers
, the following inequality holds:![\[
4(1 + a^2)(1 + b^2)(1 + c^2) \geq 3(a + b + c)^2
\]](http://latex.artofproblemsolving.com/d/a/7/da7212d7aab5c44dd78b993b3afa5dd938d50c6f.png)
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Y by
Novmath wrote:
sqing wrote:
Can you share solution of this one:
Problem: Prove that for all real numbers
, the following inequality holds:https://artofproblemsolving.com/community/c6h1228946p6197808
https://artofproblemsolving.com/community/c6h2239135p17158474
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Y by
sqing wrote:
Novmath wrote:
sqing wrote:
Can you share solution of this one:
Problem: Prove that for all real numbers
, the following inequality holds:https://artofproblemsolving.com/community/c6h1228946p6197808
https://artofproblemsolving.com/community/c6h2239135p17158474
Thank you very much
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Y by
sqing wrote:
Novmath wrote:
sqing wrote:
Can you share solution of this one:
Problem: Prove that for all real numbers
, the following inequality holds:https://artofproblemsolving.com/community/c6h1228946p6197808
https://artofproblemsolving.com/community/c6h2239135p17158474
Bro where can I learn separation trick in inequalities ,do you have some source which explaines it detailed, if you send it to me it will be appreciated!
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and![\[
x = a + \frac{1}{b}, \quad y = b + \frac{1}{c}, \quad z = c + \frac{1}{a},
\]](http://latex.artofproblemsolving.com/e/f/e/efee2962579a7343e83a3e1b8e9f03af08b832bb.png)
then![\[
xy + yz + zx \geq 2 + x + y + z.
\]](http://latex.artofproblemsolving.com/7/e/7/7e7499966a274bdbf64234b7eef6a5093afa148f.png)
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#24
Apr 21, 2025, 8:22 PM
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or 
,This post has been edited 1 time. Last edited by Primeniyazidayi, Apr 21, 2025, 8:34 PM
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Y by
sqing wrote:
Novmath wrote:
sqing wrote:
Can you share solution of this one:
Problem: Prove that for all real numbers
, the following inequality holds:https://artofproblemsolving.com/community/c6h1228946p6197808
https://artofproblemsolving.com/community/c6h2239135p17158474
Please share the solution of Vasc one that I shared
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