Inspired by Abelkonkurransen 2025

by sqing, Apr 13, 2025, 1:54 AM

Let $a,b,c$ be real numbers such that $\frac{a}{bc}+\frac{4b}{ca}+\frac{c}{ab}=24.$ Prove that
$\frac{1}{a}+\frac{1}{2b}+\frac{1}{ c}\geq -6$ $\frac{1}{a}+\frac{1}{2b}+\frac{1}{3c}\geq 1-\sqrt{73}$ $\frac{1}{a}+\frac{1}{4b}+\frac{1}{ c}\geq \frac{3}{2}(1-\sqrt{33} )$

inequalities proposed

algebra

Inspired by giangtruong13

by sqing, Apr 11, 2025, 2:57 AM

Let $a,b,c,d\geq 0 ,a-b+d=21$ and $a+3b+4c=101$ . Prove that
$61\leq a+b+2c+d\leq \frac{265}{3}$ $- \frac{2121}{2}\leq ab+bc-2cd+da\leq \frac{14045}{12}$ $\frac{519506-7471\sqrt{7471}}{27}\leq ab+bc-2cd+3da\leq 33620$

This post has been edited 2 times. Last edited by sqing, Friday at 3:13 AM

inequalities proposed

algebra

Inspired by Ruji2018252

by sqing, Apr 10, 2025, 2:00 AM

Let $a,b,c$ be reals such that $a^2+b^2+c^2-2a-4b-4c=7.$ Prove that
$-4\leq 2a+b+2c\leq 20$ $5-4\sqrt 3\leq a+b+c\leq 5+4\sqrt 3$ $11-4\sqrt {14}\leq a+2b+3c\leq 11+4\sqrt {14}$

inequalities proposed

algebra

inequalities

Sets With a Given Property

by oVlad, Apr 9, 2025, 2:32 PM

Determine the sets $S{}$ of positive integers satisfying the following two conditions:

For any positive integers $a, b, c{}$ , if $ab + bc + ca{}$ is in $S$ , then so are $a + b + c{}$ and $abc$ ; and
The set $S{}$ contains an integer $N \geqslant 160$ such that $N-2$ is not divisible by $4$ .

Bogdan Blaga, United Kingdom

combinatorics

Sets

number theory

Number Theory Chain!

by JetFire008, Apr 7, 2025, 7:14 AM

I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1

This post has been edited 1 time. Last edited by JetFire008, Apr 7, 2025, 7:14 AM

number theory

marathon

Inspired by KHOMNYO2

by sqing, Mar 28, 2025, 2:30 PM

Let $a,b>0$ and $a^2+b^2=\frac{5}{2}.$ Prove that $2a + 2b + \frac{1}{a} + \frac{1}{b} +\frac{ab}{\sqrt 2}\geq 5\sqrt 2$ $a + b +\frac{2}{a} + \frac{2}{b} + ab\geq \frac{5}{4} + \frac{13}{\sqrt 5}$ $a + b +\frac{2}{a} + \frac{2}{b} + \frac{ab}{\sqrt 2}\geq \frac{5}{4\sqrt 2} + \frac{13}{\sqrt 5}$

This post has been edited 1 time. Last edited by sqing, Mar 28, 2025, 2:37 PM

inequalities proposed

algebra

inequalities

2025 Caucasus MO Seniors P2

by BR1F1SZ, Mar 26, 2025, 12:39 AM

Let $ABC$ be a triangle, and let $B_1$ and $B_2$ be points on segment $AC$ symmetric with respect to the midpoint of $AC$ . Let $\gamma_A$ denote the circle passing through $B_1$ and tangent to line $AB$ at $A$ . Similarly, let $\gamma_C$ denote the circle passing through $B_1$ and tangent to line $BC$ at $C$ . Let the circles $\gamma_A$ and $\gamma_C$ intersect again at point $B'$ ( $B' \neq B_1$ ). Prove that $\angle ABB' = \angle CBB_2$ .

geometry

A cyclic problem

by KhuongTrang, Sep 5, 2024, 4:54 AM

Problem. Given non-negative real numbers $a,b,c: ab+bc+ca>0$ then $\frac{1}{a+kb}+\frac{1}{b+kc}+\frac{1}{c+ka}\le f(k)\cdot\frac{a+b+c}{ab+bc+ca}$ where $f(k)=\frac{(k^2-k+1)\left(2k^2+\sqrt{k^2-k+1}+2\sqrt{k^4-k^3+k^2}\right)}{\left(k^2+\sqrt{k^4-k^3+k^2}\right)\left(k^2-k+1+\sqrt{k^4-k^3+k^2}\right)}.$ Also, $k\ge k_{0}\approx 1.874799...$ and $k_{0}$ is largest real root of the equation $k^8 - 3 k^7 + 10 k^6 - 25 k^5 + 30 k^4 - 25 k^3 + 10 k^2 - 3 k + 1=0.$ k=2

inequalities

Problem 16

by Nguyenhuyen AG, May 3, 2010, 6:28 AM

Let $a,b,c >0$ such that $abc\ge1$ . Prove that
$\frac{1}{a^4+b^3+c^2}+\frac{1}{b^4+c^3+a^2}+\frac{1}{c^4+a^3+b^2}\le 1$

three variable inequality

algebra

Inequality

inequalities

USAMO 2003 Problem 4

by MithsApprentice, Sep 27, 2005, 8:01 PM

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$ , respectively. Lines $AB$ and $DE$ intersect at $F$ , while lines $BD$ and $CF$ intersect at $M$ . Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$ .

This post has been edited 1 time. Last edited by MithsApprentice, Sep 27, 2005, 10:00 PM

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
by owenccc, Sep 27, 2023, 12:59 AM
hullo :<
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hello $\text{[math]}$
by LeoLionTank, Feb 17, 2023, 9:02 PM
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!
by DCode10, Mar 10, 2021, 7:00 PM
Hi adityaguharoy! Nice blog!
by masadca, Feb 4, 2021, 9:19 PM
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
by Intellectuality123, Aug 12, 2016, 11:35 PM
Hello!!!!!
by MinionLA, Jul 18, 2016, 9:15 PM
hi everyone
by matthMaster, Jul 2, 2016, 1:36 AM
You have a great blog here; just add some more stuff and it will be even better!
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 sqing, Apr 13, 2025, 1:54 AM

by sqing, Apr 11, 2025, 2:57 AM

by sqing, Apr 10, 2025, 2:00 AM

by oVlad, Apr 9, 2025, 2:32 PM

by JetFire008, Apr 7, 2025, 7:14 AM

by sqing, Mar 28, 2025, 2:30 PM

by BR1F1SZ, Mar 26, 2025, 12:39 AM

by KhuongTrang, Sep 5, 2024, 4:54 AM

by Nguyenhuyen AG, May 3, 2010, 6:28 AM

by MithsApprentice, Sep 27, 2005, 8:01 PM