Cycle in a graph with a minimal number of chords

by GeorgeRP, May 14, 2025, 7:51 AM

In King Arthur's court every knight is friends with at least $d>2$ other knights where friendship is mutual. Prove that King Arthur can place his knights around a round table in such a way that every knight is friends with the $2$ people adjacent to him and between them there are at least $\frac{d^2}{10}$ friendships of knights that are not adjacent to each other.

Interesting inequality of sequence

by GeorgeRP, May 14, 2025, 7:47 AM

Let $d\geq 2$ be an integer and $a_0,a_1,\ldots$ is a sequence of real numbers for which $a_0=a_1=\cdots=a_d=1$ and:
$$a_{k+1}\geq a_k-\frac{a_{k-d}}{4d}, \forall_{k\geq d}$$Prove that all elements of the sequence are positive.
This post has been edited 1 time. Last edited by GeorgeRP, 2 hours ago

Parallel lines in incircle configuration

by GeorgeRP, May 14, 2025, 7:46 AM

Let $I$ be the incenter of triangle $\triangle ABC$. Let $H_A$, $H_B$, and $H_C$ be the orthocenters of triangles $\triangle BCI$, $\triangle ACI$, and $\triangle ABI$, respectively. Prove that the lines through $H_A$, $H_B$, and $H_C$, parallel to $AI$, $BI$, and $CI$, respectively, are concurrent.
This post has been edited 1 time. Last edited by GeorgeRP, 2 hours ago

Number theory

by EeEeRUT, May 14, 2025, 6:49 AM

Let $n$ be a positive integer. Show that there exist a polynomial $P(x)$ with integer coefficient that satisfy the following
  • Degree of $P(x)$ is at most $2^n - n -1$
  • $|P(k)| = (k-1)!(2^n-k)!$ for each $k \in \{1,2,3,\dots,2^n\}$
This post has been edited 2 times. Last edited by EeEeRUT, 3 hours ago

inequality

by xytunghoanh, May 14, 2025, 4:12 AM

For $a,b,c\ge 0$. Let $a+b+c=3$.
Prove or disprove
\[\sum ab +\sum ab^2 \le 6\]
This post has been edited 2 times. Last edited by xytunghoanh, 5 hours ago

Inspired by old results

by sqing, May 14, 2025, 3:58 AM

Trigonometric Product

by Henryfamz, May 13, 2025, 4:52 PM

Interesting inequalities

by sqing, May 13, 2025, 11:48 AM

ISI UGB 2025 P2

by SomeonecoolLovesMaths, May 11, 2025, 11:16 AM

If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^2 A + \sin ^2 B + \sin^2  C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$$prove that the triangle must have a right angle.
This post has been edited 1 time. Last edited by SomeonecoolLovesMaths, May 11, 2025, 12:00 PM

BMO Shortlist 2021 G2

by Lukaluce, May 8, 2022, 5:06 PM

Let $I$ and $O$ be the incenter and the circumcenter of a triangle $ABC$, respectively, and let
$s_a$ be the exterior bisector of angle $\angle BAC$. The line through $I$ perpendicular to $IO$ meets the
lines $BC$ and $s_a$ at points $P$ and $Q$, respectively. Prove that $IQ = 2IP$.

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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