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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Help me this problem. Thank you
illybest 3
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by jasperE3
Find f: R->R such that
f( xy + f(z) ) = (( xf(y) + yf(x) )/2) + z
f( xy + f(z) ) = (( xf(y) + yf(x) )/2) + z
3 replies
line JK of intersection points of 2 lines passes through the midpoint of BC
parmenides51 4
N
by reni_wee
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
4 replies
AGM Problem(Turkey JBMO TST 2025)
HeshTarg 3
N
by ehuseyinyigit
Source: Turkey JBMO TST Problem 6
Given that 
are real numbers, find the smallest possible value of the expression:
are real numbers, find the smallest possible value of the expression:
3 replies
Shortest number theory you might've seen in your life
AlperenINAN 8
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by HeshTarg
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.
8 replies
Squeezing Between Perfect Squares and Modular Arithmetic(JBMO TST Turkey 2025)
HeshTarg 3
N
by BrilliantScorpion85
Source: Turkey JBMO TST Problem 4
If and are primes and is a perfect square, prove that 
is a perfect square.
and are primes and is a perfect square, prove that 
is a perfect square.
3 replies
A bit too easy for P2(Turkey 2025 JBMO TST)
HeshTarg 0
Source: Turkey 2025 JBMO TST P2
Let 
be a positive integer. Aslı and Zehra are playing a game on an 
chessboard. Initially, 
stones are placed on the squares of the board. In each move, Aslı chooses a row or a column; Zehra chooses a row or a column. The number of stones in each square of the chosen row or column must change such that the difference between the number of stones in a square with the most stones and a square with the fewest stones in that same row or column is at most 1. For which values of can Aslı guarantee that after a finite number of moves, all squares on the board will have an equal number of stones, regardless of the initial distribution?
be a positive integer. Aslı and Zehra are playing a game on an 
chessboard. Initially, 
stones are placed on the squares of the board. In each move, Aslı chooses a row or a column; Zehra chooses a row or a column. The number of stones in each square of the chosen row or column must change such that the difference between the number of stones in a square with the most stones and a square with the fewest stones in that same row or column is at most 1. For which values of can Aslı guarantee that after a finite number of moves, all squares on the board will have an equal number of stones, regardless of the initial distribution?
0 replies
D1030 : An inequalitie
Dattier 0
Source: les dattes à Dattier
Let 
reals, and 
.
Is it true that
?
reals, and 
.Is it true that
?
0 replies
Long and wacky inequality
Royal_mhyasd 0
Source: Me
Let 
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
0 replies
Vietnamese national Olympiad 2007, problem 4
hien 16
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by de-Kirschbaum
Given a regular 2007-gon. Find the minimal number 
such that: Among every vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.
such that: Among every vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.
16 replies
2n^2+4n-1 and 3n+4 have common powers
bin_sherlo 4
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by CM1910
Source: Türkiye 2025 JBMO TST P5
Find all positive integers such that a positive integer power of 
equals to a positive integer power of 
.
such that a positive integer power of 
equals to a positive integer power of 
.
4 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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,
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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
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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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