Four variables and n variables

by Nguyenhuyen_AG, Jul 26, 2025, 10:43 AM

(1) Let $a,\,b,\,c,\,d$ be non-negative real numbers, such that
$a+b+c+d = a^2+b^2+c^2+d^2.$ Prove that
$ab+bc+ca+da+db+dc \geqslant a^2b^2+b^2c^2+c^2a^2+d^2a^2+d^2b^2+d^2c^2.$ (2) Let $a_1,a_2,\ldots,n_n \, (n \geqslant 1)$ be non-negative real numbers, such that
$\sum_{i=1}^{n} a_i = \sum_{i=1}^{n} a_i^2.$ Prove that
$\left( \sum_{i=1}^{n} a_i \right)^2 - \sum_{i=1}^{n} a_i^2 \geqslant \left( \sum_{i=1}^{n} a_i^2 \right)^2 - \sum_{i=1}^{n} a_i^4 .$

inequalities

inequalities proposed

Inequality

by SunnyEvan, Jul 26, 2025, 9:00 AM

Let $a,b,c \in R .$ Prove that:
$7(a^6+b^6+c^6)+ 2\prod_{cyc}(a^2+7bc) \geq 345a^2b^2c^2$ $7(a^6+b^6+c^6) \geq 2\prod_{cyc}(a^2-7bc)+341a^2b^2c^2$ When does the equality holds ?

This post has been edited 1 time. Last edited by SunnyEvan, 3 hours ago

Inequality

inequalities

2-var inequality

by sqing, Jul 26, 2025, 3:12 AM

Let $a,b \geq 0 , \frac {a+1} {a^2+ab+1}+ \frac {b+1} {b^2+ab+1} \leq 1 .$ Prove that
$a+b\geq 1+\sqrt 3$ Let $a,b \geq 0, \frac {a+1} {a^2+ab+1}+ \frac {b+1} {b^2+ab+1} \geq \frac {4} {3} .$ Prove that
$a+b\leq \frac {3+\sqrt{17}} {2}$ Let $a,b \geq 0, \frac {a} {a^2+ab+1}+ \frac {b} {b^2+ab+1} \geq \frac {2} {5} .$ Prove that
$a+b\leq \frac {5-\sqrt{17}} {2}$

This post has been edited 3 times. Last edited by sqing, Today at 3:33 AM

inequalities proposed

inequalities

Peru IMO TST 2023

by diegoca1, Jul 25, 2025, 7:22 PM

Let $x, y, z$ be non-negative real numbers such that $x + y + z \leq 1$ . Prove the inequality
$6xyz \leq x(1 - x) + y(1 - y) + z(1 - z),$ and determine when equality holds.

inequalities

D1053 : Set of Dirichlet

by Dattier, Jul 22, 2025, 1:23 PM

We say a set $D$ have the Dirichlet propriety, if $\forall (a,b) \in (\mathbb N^*)^2,\gcd(a,b)=1, a<b$ , $\text{card}(\{ n \in\mathbb N, n \mod b=a \} \cap D)=+\infty$ .

Let $D=\{d_1,....,d_n,...\}$ with $\forall i \in \mathbb N^*,d_{i+1}>d_{i}$ subset of $\mathbb N$ and have the Dirichlet propriety.

1) Is it true that $\lim \dfrac{d_{n+1}}{d_n}=1$ ?

2) Is it true that $\liminf \dfrac{d_{n+1}}{d_n}=1$ ?

number theory

Amazing rook and chessboard question

by egxa, Dec 17, 2024, 8:09 AM

Let $m,n\ge2$ be positive integers. On an $m\times n$ chessboard, some unit squares are occupied by rooks such that each rook attacked by odd number of other rooks. Determine the maximum number of rooks that can be placed on the chessboard.

combinatorics

Symmetric inequality

by nexu, Feb 12, 2023, 6:54 AM

Let $x,y,z \ge 0$ . Prove that:
$\sum_{\mathrm{cyc}}{\left( y-z \right) ^2\left( 7x^2-y^2-z^2 \right) ^2}\ge 112\left( x-y \right) ^2\left( y-z \right) ^2\left( z-x \right) ^2.$

inequalities

Iran geometry

by Dadgarnia, Mar 11, 2020, 2:07 PM

Given a triangle $ABC$ with circumcircle $\Gamma$ . Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$ , $I$ is incenter and $K$ is the intersection of $AI$ and $EF$ . Suppose that $T$ be the midpoint of arc $BAC$ . Circle $\Gamma$ intersects the $A$ -median and circumcircle of $AEF$ for the second time at $X$ and $S$ . Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$ . Prove that quadrilateral $TJIX$ is cyclic.

Proposed by Alireza Dadgarnia and Amir Parsa Hosseini

This post has been edited 1 time. Last edited by Dadgarnia, Mar 12, 2020, 10:38 AM

A sequence must be bounded

by math90, Jul 10, 2018, 11:08 AM

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$ .

This post has been edited 2 times. Last edited by math90, Jul 11, 2018, 2:47 PM
Reason: ISL=IMO Shortlist

function

algebra

IMO Shortlist

Tangent intersect intersect tangent intersect

by cjquines0, May 26, 2017, 11:12 AM

Let two circles $C_1$ and $C_2$ intersect in points $A$ and $B$ . The tangent to $C_1$ at $A$ intersects $C_2$ in $P$ and the line $PB$ intersects $C_1$ for the second time in $Q$ (suppose that $Q$ is outside $C_2$ ). The tangent to $C_2$ from $Q$ intersects $C_1$ and $C_2$ in $C$ and $D$ , respectively. (The points $A$ and $D$ lie on different sides of the line $PQ$ .) Show that $AD$ is the bisector of $\angle CAP$ .

Proposed by Iman Maghsoudi

geometry

Submit

i really gotta find a better platform
by Welp..., Mar 8, 2023, 1:15 PM
wassup $\text{[math]}$
by hop_skip_jump, Jun 5, 2022, 2:10 PM
yeah you got it
by Welp..., Jan 2, 2022, 7:05 AM
no NT november
don't die december
by fidgetboss_4000, Jan 1, 2022, 9:52 PM
failed NNN AND DDD
by Welp..., Dec 2, 2021, 8:12 PM
howdy pardner
by madhavramaprasad, May 12, 2021, 9:00 PM
e e e e hewo
by floof421, Oct 28, 2020, 3:42 PM
hi ???? ? ??
by Welp..., Sep 7, 2020, 3:42 AM
henlooooooooooo
by Emathmaster, Sep 6, 2020, 2:44 PM
henlooooooooooo
by mathlete0825, Aug 12, 2020, 2:09 PM
aged most terribly
by Welp..., Jul 28, 2020, 3:30 AM
@below rip did not age well
by rzlng, Jul 24, 2020, 6:25 AM
Welp...在2020APChinese得了很好的成绩！
Welp...在2020APCaSh得了很好的成绩！
Welp...在2020APCalcBC得了很好的成绩！
Welp...是我们的superstar！
by I-_-I, May 20, 2020, 11:13 PM
mine's a fork of AIME12345's anyways so go ahead
by Welp..., May 16, 2020, 5:59 PM
Can I use but modify your css??
by duwipr, May 16, 2020, 10:55 AM
what the actual hell

blank accusations, smartninja was banned on the public blog for a great reason

he's such a troll. i troll on occasion to be light hearted but this is clearly malicious
by Welp..., Jan 11, 2020, 12:44 AM
smartninja2000 wrote:

Welp, at least i dont try to cheat on mts like you and other people in your area.

Is this true o.O
by Radio2, Jan 7, 2020, 8:47 AM
that too @below
by Welp..., Jan 3, 2020, 6:32 AM
yet another reason to make blogs private is if you ban someone, they can just log out or create a multi and still access your blog
by ccx09, Dec 28, 2019, 2:45 AM
because banning people makes them angrier than just privatizing the blog and denying readers
by Welp..., Dec 7, 2019, 1:18 AM
Hi so why is this blog private?!
by Radio2, Dec 5, 2019, 1:40 PM
hi and sure!
by Welp..., Nov 8, 2019, 4:21 AM
Hi contrib pls?
by Stormersyle, Nov 8, 2019, 4:03 AM
it's just aime12345's css
by Welp..., Nov 3, 2019, 5:21 PM
what is this css lol
by rzlng, Nov 3, 2019, 1:45 AM
bruh $\text{[math]}$ $\text{[math]}$
by Welp..., Nov 3, 2019, 12:38 AM
breh $~~$
by pretzel, Nov 3, 2019, 12:37 AM
that's really nice i care so much
by Welp..., Nov 2, 2019, 6:35 PM
First shout.
by hannahptl, Nov 2, 2019, 6:06 PM

66 shouts

relatively consistent post

by Nguyenhuyen_AG, Jul 26, 2025, 10:43 AM

by SunnyEvan, Jul 26, 2025, 9:00 AM

by sqing, Jul 26, 2025, 3:12 AM

by diegoca1, Jul 25, 2025, 7:22 PM

by Dattier, Jul 22, 2025, 1:23 PM

by egxa, Dec 17, 2024, 8:09 AM

by nexu, Feb 12, 2023, 6:54 AM

by Dadgarnia, Mar 11, 2020, 2:07 PM

by math90, Jul 10, 2018, 11:08 AM

by cjquines0, May 26, 2017, 11:12 AM