inequalities

by Ducksohappi, May 16, 2025, 12:57 PM

let a,b,c be non-negative numbers such that ab+bc+ca>0. Prove:
$\sum_{cyc} \frac{b+c}{2a^2+bc}\ge \frac{6}{a+b+c}$
P/s: I have analysed: $S_a=\frac{b^2+c^2+3bc-ab-ac}{(2b^2+ac)(2c^2+2ab)}$ , similarly to $S_b, S_c$ , by SOS

This post has been edited 1 time. Last edited by Ducksohappi, an hour ago

inequalities

Interesting inequalities

by sqing, May 16, 2025, 4:34 AM

Let $a,b,c \geq 0$ and $ab+bc+ca- abc =3.$ Show that
$a+k(b+c)\geq 2\sqrt{3 k}$ Where $k\geq 1.$
Let $a,b,c \geq 0$ and $2(ab+bc+ca)- abc =31.$ Show that
$a+k(b+c)\geq \sqrt{62k}$ Where $k\geq 1.$

inequalities proposed

inequalities

Simple but hard

by Lukariman, May 16, 2025, 2:47 AM

Given triangle ABC. Outside the triangle, construct rectangles ACDE and BCFG with equal areas. Let M be the midpoint of DF. Prove that CM passes through the center of the circle circumscribing triangle ABC.

Attachments:

This post has been edited 2 times. Last edited by Lukariman, 5 hours ago

geometry

Good Permutations in Modulo n

by swynca, Apr 27, 2025, 2:03 PM

An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$ , such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$ ;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$ .
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

This post has been edited 2 times. Last edited by swynca, Apr 27, 2025, 4:15 PM

BMO

number theory

sqrt(2)<=|1+z|+|1+z^2|<=4

by SuiePaprude, Jan 23, 2025, 10:53 PM

let z be a complex number with |z|=1 show that sqrt2 <=|1+z|+|1+z^2|<=4

complex numbers

complex inequality

concyclic wanted, diameter related

by parmenides51, May 5, 2024, 1:26 AM

As shown in the figure, $AB$ is the diameter of circle $\odot O$ , and chords $AC$ and $BD$ intersect at point $E$ , $EF\perp AB$ intersects at point $F$ , and $FC$ intersects $BD$ at point $G$ . Point $M$ lies on $AB$ such that $MD=MG$ . Prove that points $F$ , $M$ , $D$ , $G$ lies on a circle.

https://cdn.artofproblemsolving.com/attachments/2/3/614ef5b9e8c8b16a29b8b960290ef9d7297529.jpg

This post has been edited 1 time. Last edited by parmenides51, May 5, 2024, 1:27 AM

geometry

Concyclic

Grid combo with tilings

by a_507_bc, Apr 23, 2023, 3:40 PM

A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$ . What is the least number of dominoes that are entirely inside some square $2 \times 2$ ?

combinatorics

<ACB=90^o if AD = BD , <ACD = 3 <BAC, AM=//MD, CM//AB,

by parmenides51, Oct 7, 2022, 8:15 PM

In the convex quadrilateral $ABCD$ , $AD = BD$ and $\angle ACD = 3 \angle BAC$ . Let $M$ be the midpoint of side $AD$ . If the lines $CM$ and $AB$ are parallel, prove that the angle $\angle ACB$ is right.

geometry

right angle

equal angles

bulgarian concurrency, parallelograms and midpoints related

by parmenides51, May 28, 2019, 2:10 PM

In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$ , respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$ . Prove that the lines $AQ, BP$ and $LT$ intersect in a point.

Concurrency

by Omid Hatami, May 20, 2008, 9:29 AM

Suppose that $I$ is incenter of triangle $ABC$ and $l'$ is a line tangent to the incircle. Let $l$ be another line such that intersects $AB,AC,BC$ respectively at $C',B',A'$ . We draw a tangent from $A'$ to the incircle other than $BC$ , and this line intersects with $l'$ at $A_1$ . $B_1,C_1$ are similarly defined. Prove that $AA_1,BB_1,CC_1$ are concurrent.

Submit

11 year bump
by john0512, Dec 28, 2020, 3:03 AM
Sure.
by googology101, Feb 12, 2009, 12:38 AM
me want contrib
by gauss1181, Feb 11, 2009, 5:49 PM
Sorry, I ran out of time on the last shout...
by googology101, Feb 8, 2009, 11:19 PM
and contrib?
by james4l, Feb 8, 2009, 4:42 AM
I guess...
by googology101, Feb 7, 2009, 11:49 PM
contrib please
by james4l, Feb 7, 2009, 10:18 PM
Yeah, I noticed that too. I'll change it later, but there's a weird glitch for you. The top and left properties should be automatically set to 0 if position is set to absolute or fixed. Go Firefox!
by googology101, Jan 21, 2009, 10:03 PM
i viewed this from ie and it looks like position fixed also ruins the left aligned so put left:0. nice submit buttons
by Sephiroth, Jan 21, 2009, 5:32 AM
Use filter:alpha(opacity=80) to make it semitransparent in IE. And make sure you use -moz-opacity instead of opacity to make sure that older versions of Firefox will cooperate. And by the way, you're free to copy the code. The whole thing is at http://www.artofproblemsolving.com/Forum/templates/blogs/Hyperion/style/c1366.css
by googology101, Jan 19, 2009, 9:50 PM
add filter: alpha(opacity=80) and opacity:0.8
by Sephiroth, Jan 19, 2009, 3:51 AM
Could I use that code in my blog?
by dysfunctionalequations, Jan 19, 2009, 1:51 AM
Watch out! position:fixed ruins the box's 100% width. Here's the code I used:

#navigation_box {
background-color: #005500 !important;
border-bottom-style:solid !important;
-moz-opacity:0.80;
color:white !important;
position:fixed !important;
width:100%;
padding-right:10px;
}

Don't overdo the opacity. 0.80 or 0.90 will do. Sorry, IE and Safari users, you aren't seeing it.
by googology101, Jan 19, 2009, 1:26 AM
what do you mean by floating menus? like how he made navigation_box fixed? he used "#navigation_box{ position: fixed;}" i like how you made it opacity
by Sephiroth, Jan 18, 2009, 4:45 AM
Floating menus? How do you do that?
by dysfunctionalequations, Jan 18, 2009, 1:25 AM
Yup. I'm trying to post daily.
by googology101, Jan 14, 2009, 12:05 AM
3rd post!
by 007math, Jan 13, 2009, 11:57 PM
Do you mean to this blog?
by googology101, Jan 12, 2009, 9:30 PM
Can i be a contributor?
by Wickedestjr, Jan 12, 2009, 5:49 PM

19 shouts

A googol is a tiny dot

by Ducksohappi, May 16, 2025, 12:57 PM

by sqing, May 16, 2025, 4:34 AM

by Lukariman, May 16, 2025, 2:47 AM

by swynca, Apr 27, 2025, 2:03 PM

by SuiePaprude, Jan 23, 2025, 10:53 PM

by parmenides51, May 5, 2024, 1:26 AM

by a_507_bc, Apr 23, 2023, 3:40 PM

by parmenides51, Oct 7, 2022, 8:15 PM

by parmenides51, May 28, 2019, 2:10 PM

by Omid Hatami, May 20, 2008, 9:29 AM