be six positive real numbers. 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!
G
Topic
First Poster
Last Poster
JBMO Combinatorics vibes
Sadigly 1
N
by Royal_mhyasd
Source: Azerbaijan Senior NMO 2018
Numbers 
are written on a board. 
and 
plays the following game: They take turns choosing a number from the board and deleting them. starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?
are written on a board. 
and 
plays the following game: They take turns choosing a number from the board and deleting them. starts first. They sum all the deleted numbers. If after a player's turn (after he deletes a number on the board) the sum of the deleted numbers can't be expressed as difference of two perfect squares,then he loses, if not, then the game continues as usual. Which player got a winning strategy?
1 reply
line JK of intersection points of 2 lines passes through the midpoint of BC
parmenides51 3
N
by cursed_tangent1434
Source: Rioplatense Olympiad 2018 level 3 p4
Let 
be an acute triangle with 
. be 
the circumcircle circumscribed to the triangle and 
the midpoint of the smallest arc 
of this circle. Let 
and 
points of the segments 
and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and 
be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
be an acute triangle with 
. be 
the circumcircle circumscribed to the triangle and 
the midpoint of the smallest arc 
of this circle. Let 
and 
points of the segments 
and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and 
be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
3 replies
1 viewing
Brilliant guessing game on triples
Assassino9931 2
N
by Mirjalol
Source: Al-Khwarizmi Junior International Olympiad 2025 P8
There are 
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to . In a move, Madina is allowed to choose any 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
Shubin Yakov, Russia
cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from 
to . In a move, Madina is allowed to choose any 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?Shubin Yakov, Russia
2 replies
ISI UGB 2025 P5
SomeonecoolLovesMaths 4
N
by Shiny_zubat
Source: ISI UGB 2025 P5
Let 
be nonzero real numbers such that 
. Assume that 
Show that for any odd integer 
, 
be nonzero real numbers such that 
. Assume that 
Show that for any odd integer 
, 
4 replies
ISI UGB 2025 P2
SomeonecoolLovesMaths 6
N
by quasar_lord
Source: ISI UGB 2025 P2
If the interior angles of a triangle satisfy the equality, 
prove that the triangle must have a right angle.
satisfy the equality, 
prove that the triangle must have a right angle.
6 replies
ISI UGB 2025 P6
SomeonecoolLovesMaths 3
N
by Shiny_zubat
Source: ISI UGB 2025 P6
Let 
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
3 replies
Shortest number theory you might've seen in your life
AlperenINAN 5
N
by Royal_mhyasd
Source: Turkey JBMO TST 2025 P4
Let 
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
5 replies
d+2 pts in R^d can partition
EthanWYX2009 0
Source: Radon's Theorem
Show that: any set of 
points in 
can be partitioned into two sets whose convex hulls intersect.
points in 
can be partitioned into two sets whose convex hulls intersect.
0 replies
hard inequality omg
tokitaohma 4
N
by arqady
1. Given 
and 
Prove that:
2. Given and 
Prove that:
and 
Prove that:
2. Given
and 
Prove that:
4 replies
Advanced topics in Inequalities
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.
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
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.
![$x+y+z \geq 3\sqrt[3]{xyz} \geq 3$](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.
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#6
Apr 16, 2025, 1:48 PM
Y by
You see NOTHING
This post has been edited 1 time. Last edited by giangtruong13, Apr 16, 2025, 1:55 PM
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#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 
.
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
are reals 
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Y by
Let 
This post has been edited 1 time. Last edited by sqing, Apr 16, 2025, 3:27 PM
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
.![\[
f''(x) = \frac{d^2}{dx^2} \ln(2x^2 + 3) = \frac{12}{(2x^2 + 3)^2} > 0 \quad \text{for all } x \in \mathbb{R}.
\]](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:
.
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.
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.
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.
Show that
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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.
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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:
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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)
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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!
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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)
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
#24
Apr 21, 2025, 8:22 PM
Y by
.
or 
,This post has been edited 1 time. Last edited by Primeniyazidayi, Apr 21, 2025, 8:34 PM
Z
K
Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
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
Z
K
Y
N Quick Reply
G
H
