Bigger Cyclic Sets Exist?

by FireBreathers, May 2, 2025, 10:17 AM

Define the set of numbers $a_1, . . . , a_m$ is $bigger$ than the set of numbers $b_1, . . . , b_n$ if among all inequalities of the form $a_i > b_j$ the number of true inequalities is at least $2$ times greater than the number of false ones. Prove that there do not exist three sets $X, Y, Z$ such that $X$ is $bigger$ than $Y$ , $Y$ is $bigger$ than $Z$ , $Z$ is $bigger$ than $X$ .

combinatorics

graph

directed

Consecutive sum of integers sum up to 2020

by NicoN9, May 2, 2025, 6:09 AM

Let $a$ and $b$ be positive integers. Suppose that the sum of integers between $a$ and $b$ , including $a$ and $b$ , are equal to $2020$ .
All among those pairs $(a, b)$ , find the pair such that $a$ achieves the minimum.

algebra

AIME

4-var inequality

by RainbowNeos, May 1, 2025, 9:31 AM

Given $a,b,c,d>0$ , show that
$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.$

inequalities

inequalities unsolved

inequalities proposed

4 variables with quadrilateral sides 2

by mihaig, Apr 29, 2025, 8:47 PM

Let $a,b,c,d\geq0$ satisfying
$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$ Prove
$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$

inequalities

geometry

D1024 : Can you do that?

by Dattier, Apr 29, 2025, 5:11 PM

Let $x_{n+1}=x_n^2+1$ and $x_0=1$ .

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$ ?

This post has been edited 1 time. Last edited by Dattier, Apr 29, 2025, 5:11 PM

Computer solutions

number theory

Inequality with 3 variables and a special condition

by Nuran2010, Apr 29, 2025, 5:06 PM

For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$ .
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$ .

Determine the equality case.

inequalities

inequalities proposed

4 lines concurrent

by Zavyk09, Apr 9, 2025, 11:51 AM

Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$ . $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$ .

IMO 2023 P2

by 799786, Jul 8, 2023, 4:47 AM

Let $ABC$ be an acute-angled triangle with $AB < AC$ . Let $\Omega$ be the circumcircle of $ABC$ . Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$ . The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$ . The line through $D$ parallel to $BC$ meets line $BE$ at $L$ . Denote the circumcircle of triangle $BDL$ by $\omega$ . Let $\omega$ meet $\Omega$ again at $P \neq B$ . Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$ .

This post has been edited 5 times. Last edited by Amir Hossein, Aug 8, 2023, 5:43 PM

IMO

IMO 2023

geometry

Geometry

by VicKmath7, Dec 26, 2019, 7:36 PM

Let $ABC$ be a triangle with circumcircle $\omega$ . Let $l_B$ and $l_C$ be two lines through the points $B$ and $C$ , respectively, such that $l_B \parallel l_C$ . The second intersections of $l_B$ and $l_C$ with $\omega$ are $D$ and $E$ , respectively. Assume that $D$ and $E$ are on the same side of $BC$ as $A$ . Let $DA$ intersect $l_C$ at $F$ and let $EA$ intersect $l_B$ at $G$ . If $O$ , $O_1$ and $O_2$ are circumcenters of the triangles $ABC$ , $ADG$ and $AEF$ , respectively, and $P$ is the circumcenter of the triangle $OO_1O_2$ , prove that $l_B \parallel OP \parallel l_C$ .

Proposed by Stefan Lozanovski, Macedonia

This post has been edited 8 times. Last edited by VicKmath7, Dec 22, 2022, 4:10 PM

geometry

Generalized mirror problem

by Taha1381, May 2, 2019, 9:14 AM

We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.)

geometry

rectangle

combinatorics

Convex and concave functions in Real numbers -- Basic 1

by adityaguharoy, Mar 1, 2018, 1:44 PM

Convex functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$ . Then we say that $f$ is a convex function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$\boxed{f(tx_1 + (1-t)x_2) \le t \cdot f(x_1) + (1-t) \cdot f(x_2)}$ And we say that $f$ is strictly convex on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$ .

Concave functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$ . Then we say that $f$ is a concave function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$\boxed{f(tx_1 + (1-t)x_2) \ge t \cdot f(x_1) + (1-t) \cdot f(x_2)}$ And we say that $f$ is strictly concave on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$ .

Quick exercises

Let us celebrate

This post has been edited 2 times. Last edited by adityaguharoy, Mar 4, 2018, 7:25 AM

Submit

hi guys $~~~$
by Yiyj1, Apr 9, 2025, 7:25 AM
You will be remembered
by giangtruong13, Feb 26, 2025, 3:38 PM
2025 shout!
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2024 shout
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helooooooooooo
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hullo :<
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hello $\text{[math]}$
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Halo thear
by HoRI_DA_GRe8, Oct 18, 2022, 7:44 AM
hi!!
just found this and I can't wait to read more!
so happy to have found this blog!
by Morrigan_Black, Jan 28, 2022, 1:05 PM
still waiting for the mathlinks camp lol
by CinarArslan, Jan 9, 2022, 1:55 PM
hello
by CyclicISLscelesTrapezoid, Nov 29, 2021, 6:36 PM
Right below the shout box it says how many it has.
by pith0n, May 11, 2021, 5:08 AM
Oh really the blog has 100 posts! I never counted the number of posts here. If I get some free time I will create a new page on my wordpress website and there I will post all the contents of this blog. So, make sure that you check the wordpress site.
by adityaguharoy, Mar 21, 2021, 2:58 PM
Nice blog!
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Hi adityaguharoy! Nice blog!
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Nice blog!
by Mathisfun04, Dec 2, 2016, 12:00 AM
Nice blog!!!
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really nice CSS
by StarFrost7, Aug 16, 2016, 3:37 AM
hi!
Post some more!
Also nice css
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Hello!!!!!
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hi everyone
by matthMaster, Jul 2, 2016, 1:36 AM
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by bobert1, Jun 30, 2016, 9:29 PM
yeah sure we must and will discuss real numbers.
by adityaguharoy, Jun 20, 2016, 4:11 AM
Why not real numbers
by skipiano, Jun 17, 2016, 2:35 PM
very interesting topic !! ?? !!
by adityaguharoy, Jun 12, 2016, 6:52 AM
Hello!!!!!!! Very interesting topic, here!
by MinionLA, Jun 9, 2016, 11:43 PM
Should it not start by the discussion of Set Theory.
by alexanderoldspartan, Jun 9, 2016, 11:03 AM
Darn I missed it
by skipiano, Jun 9, 2016, 4:52 AM
first shout >
by doitsudoitsu, Apr 26, 2016, 8:03 PM

118 shouts

An Introduction to the Theory of Mathematics

by FireBreathers, May 2, 2025, 10:17 AM

by NicoN9, May 2, 2025, 6:09 AM

by RainbowNeos, May 1, 2025, 9:31 AM

by mihaig, Apr 29, 2025, 8:47 PM

by Dattier, Apr 29, 2025, 5:11 PM

by Nuran2010, Apr 29, 2025, 5:06 PM

by Zavyk09, Apr 9, 2025, 11:51 AM

by 799786, Jul 8, 2023, 4:47 AM

by VicKmath7, Dec 26, 2019, 7:36 PM

by Taha1381, May 2, 2019, 9:14 AM

by adityaguharoy, Mar 1, 2018, 1:44 PM