help me~~

by Imshyso, May 31, 2025, 8:10 AM

Given triangle ABC inscribed in (O) with orthocenter H. Let K be the midpoint of AH. Take E,F on AC, AB so that BKE=CKF=90. Prove that E,O,F are collinear.

geometry

Inspired by a cool result

by DoThinh2001, May 31, 2025, 12:08 AM

Let three real numbers $a,b,c\geq 0$ , no two of which are $0$ . Prove that:
$\sqrt{\frac{a^2+bc}{b^2+c^2}}+\sqrt{\frac{b^2+ca}{c^2+a^2}}+\sqrt{\frac{c^2+ab}{a^2+b^2}}\geq 2+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}.$
Inspiration

inequalities

algebra

find question

by mathematical-forest, May 29, 2025, 10:19 AM

Are there any contest questions that seem simple but are actually difficult? :-D

combinatorics

D,E,F are collinear.

by TUAN2k8, May 28, 2025, 1:07 AM

Help me with this:

Attachments:

geometry

orthocenter

Basic ideas in junior diophantine equations

by Maths_VC, May 27, 2025, 7:54 PM

Determine all positive integers $a, b$ and $c$ such that
$2$ $\cdot$ $10^a + 5^b = 2025^c$

number theory

Diophantine equation

Nice NT with powers of two

by oVlad, Jul 31, 2024, 1:38 PM

Let $n{}$ be a positive integer and let $a{}$ and $b{}$ be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer $k{}$ such that at least one of the numbers $a^k-b$ and $b^k-a$ is divisible by $2^n.$

Cătălin Liviu Gherghe

number theory

Divisibility

Center lies on circumcircle of other

by Philomath_314, Jan 21, 2024, 11:43 AM

In triangle $ABC$ with $CA=CB$ , point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB=90^{\circ}$ . The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$ . Prove that the center of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$ .

Proposed by Prithwijit De

This post has been edited 6 times. Last edited by Philomath_314, Jan 21, 2024, 4:51 PM

geometry

Angle Chasing

INMO

Find values for a, b ,c

by Ferum_2710, Apr 30, 2023, 8:00 AM

Let $M \geq 1$ be a real number. Determine all natural numbers $n$ for which there exist distinct natural numbers $a$ , $b$ , $c > M$ , such that
$n = (a,b) \cdot (b,c) + (b,c) \cdot (c,a) + (c,a) \cdot (a,b)$
(where $(x,y)$ denotes the greatest common divisor of natural numbers $x$ and $y$ ).

number theory

Combinatorial identity

by MehdiGolafshan, Jan 16, 2023, 4:21 PM

Let $n$ is a positive integer. Prove that
$\sum_{k=0}^{n-1}\frac{1}{k+1}\binom{n-1}{k} = \frac{2^n-1}{n}.$

combinatorics

counting

double counting

Combinatorial Identity

Crossing ٍٍChords

by matinyousefi, Apr 24, 2020, 8:22 AM

$1399$ points and some chords between them is given.
$a)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase exactly one of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions.
$b)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase both of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions.

Proposed by Afrouz Jabalameli, Abolfazl Asadi

This post has been edited 1 time. Last edited by matinyousefi, Apr 26, 2020, 8:05 PM

combinatorics

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 Imshyso, May 31, 2025, 8:10 AM

by DoThinh2001, May 31, 2025, 12:08 AM

by mathematical-forest, May 29, 2025, 10:19 AM

by TUAN2k8, May 28, 2025, 1:07 AM

by Maths_VC, May 27, 2025, 7:54 PM

by oVlad, Jul 31, 2024, 1:38 PM

by Philomath_314, Jan 21, 2024, 11:43 AM

by Ferum_2710, Apr 30, 2023, 8:00 AM

by MehdiGolafshan, Jan 16, 2023, 4:21 PM

by matinyousefi, Apr 24, 2020, 8:22 AM