Geometry hard problem.

by noneofyou34, May 20, 2025, 9:50 AM

In a circle of radius R, three chords of length R are given. Their ends are joined with segments to
obtain a hexagon inscribed in the circle. Show that the midpoints of the new chords are the vertices of
an equilateral triang

Hard Inequality

by danilorj, May 20, 2025, 5:17 AM

Let $a, b, c > 0$ with $a + b + c = 1$. Prove that:
\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]with equality if and only if $a = b = c = \frac{1}{3}$.
This post has been edited 1 time. Last edited by danilorj, Today at 5:18 AM
Reason: ..

Problem 3

by blug, May 19, 2025, 4:47 PM

In a triangle $ABC$, $\angle ACB=60^{\circ}$. Points $D, E$ lie on segments $BC, AC$ respectively. Points $K, L$ are such that $ADK$ and $BEL$ are equlateral, $A$ and $L$ lie on opposite sides of $BE$, $B$ and $K$ lie on the opposite siedes of $AD$. Prove that
$$AE+BD=KL.$$

Inspired by SXJX (12)2022 Q1167

by sqing, May 19, 2025, 4:01 AM

Computing functions

by BBNoDollar, May 18, 2025, 5:25 PM

Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )

Find the minimum

by sqing, May 17, 2025, 9:12 AM

easy geo

by ErTeeEs06, Apr 26, 2025, 11:13 AM

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.
This post has been edited 1 time. Last edited by ErTeeEs06, Apr 26, 2025, 11:15 AM
Reason: latex

Where are the Circles?

by luminescent, Apr 9, 2022, 10:00 PM

Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.
This post has been edited 1 time. Last edited by luminescent, Apr 9, 2022, 10:07 PM
Reason: change source format to match other egmo problems

Number of Solutions is 2

by Miku3D, Jun 9, 2021, 6:29 AM

Prove that for each real number $r>2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^2=r\lfloor x \rfloor$.

IMO 2016 Shortlist, N6

by dangerousliri, Jul 19, 2017, 4:31 PM

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.

Proposed by Dorlir Ahmeti, Albania

This blog is about large numbers. There. Have I bored you yet?

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googology101
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  • 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

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