square root problem

by kjhgyuio, May 3, 2025, 4:48 AM

........
Attachments:
square root
algebra
Real
L

Comics and triangles in perspective

by srirampanchapakesan, May 3, 2025, 4:20 AM

Let a conic intersect the sides BC, CA, AB of triangle ABC at A1,A2,B1,B2,C1,C2.

T1 is the triangle formed by A1B2, B1C2, and C1A2.

T2 is the triangle formed by A2B1, B2C1 and C2A1.

Prove that the triangles ABC, T1 and T2 are pair-wise in perspective.

Also prove that all three centers of perspective coincide.
geometry
L

inequalities

by Cobedangiu, May 3, 2025, 4:06 AM

$a,b,c>0$ and $\sum ab=\dfrac{1}{3}$. Prove that:
$\sum \dfrac{1}{a^2-bc+1}\le 3$
inequalities
L

3-var inequality

by sqing, May 3, 2025, 3:56 AM

Let $ a,b,c\geq 0 ,a+b+c =4. $ Prove that
$$a +ab^2 +ab^2c \leq\frac{33}{4}+2\sqrt 2$$$$a +ab^2 +abc \leq \frac{2(100+13\sqrt {13})}{27}$$$$a +a^2b + a b^2c^3\leq \frac{2(82+19\sqrt {19})}{27}$$
inequalities proposed
inequalities
L

A nice and easy gem off of StackExchange

by NamelyOrange, May 2, 2025, 8:13 PM

Define $S$ as the set of all numbers of the form $2^i5^j$ for some nonnegative $i$ and $j$. Find (with proof) all pairs $(m,n)$ such that $m,n\in S$ and $m-n=1$.


Rephrased: Solve $2^a5^b-2^c5^d=1$ over $(\mathbb{N}_0)^4$, and prove that your solution(s) is/are the only one(s).
This post has been edited 2 times. Last edited by NamelyOrange, Today at 12:56 AM
number theory
Exponents
easy
L

Construct

by Pomegranat, Apr 30, 2025, 7:22 AM

Let \( p \) be a prime number. Prove that there exists a natural number \( n \) such that
\[ p \mid m^n - n. \]
This post has been edited 1 time. Last edited by Pomegranat, Thursday at 1:58 PM
number theory
L

Almost Squarefree Integers

by oVlad, Apr 12, 2025, 9:35 AM

A positive integer $n\geqslant 3$ is almost squarefree if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.
number theory
TST
L

this geo is scarier than the omega variant

by AwesomeYRY, Dec 13, 2021, 5:00 PM

Triangles $ABC$ and $DEF$ share circumcircle $\Omega$ and incircle $\omega$ so that points $A,F,B,D,C,$ and $E$ occur in this order along $\Omega$. Let $\Delta_A$ be the triangle formed by lines $AB,AC,$ and $EF,$ and define triangles $\Delta_B, \Delta_C, \ldots, \Delta_F$ similarly. Furthermore, let $\Omega_A$ and $\omega_A$ be the circumcircle and incircle of triangle $\Delta_A$, respectively, and define circles $\Omega_B, \omega_B, \ldots, \Omega_F, \omega_F$ similarly.

(a) Prove that the two common external tangents to circles $\Omega_A$ and $\Omega_D$ and the two common external tangents to $\omega_A$ and $\omega_D$ are either concurrent or pairwise parallel.

(b) Suppose that these four lines meet at point $T_A$, and define points $T_B$ and $T_C$ similarly. Prove that points $T_A,T_B$, and $T_C$ are collinear.

Nikolai Beluhov
This post has been edited 1 time. Last edited by AwesomeYRY, Dec 13, 2021, 10:32 PM
USA TSTST
geometry
L

minimum of \sqrt{\frac{a}{b(3a+2)}}+\sqrt{\frac{b}{a(3b+2)}}

by parmenides51, Jul 25, 2018, 2:56 PM

Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}} $
This post has been edited 1 time. Last edited by parmenides51, Apr 20, 2019, 4:58 PM
Reason: typo in the last fraction, see post #4 for details
algebra
minimum value
JBMO
L

Integer a_k such that b - a^n_k is divisible by k

by orl, Jul 13, 2008, 1:41 PM

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$.

Author: Dan Brown, Canada
This post has been edited 3 times. Last edited by v_Enhance, Jan 18, 2016, 2:13 AM
Reason: Fix obsolete TeX
modular arithmetic
number theory
Divisibility
IMO Shortlist
Hi
L

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

avatar

googology101
Shouts
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
Tags
numbers
Misc
Notations
aleph-null
bentley
calvin
chained arrow
choogol
factors
gaz
gazillion
gongol
gongolplex
googol
hyperlicious
illion
kaboodol
shoogol
thoogol
wakamoogol
wakapoogol
wakoogol
welcome
zhoogol
About Owner
  • Posts: 21
  • Joined: Jan 5, 2009
Blog Stats
  • Blog created: Jan 11, 2009
  • Total entries: 35
  • Total visits: 24137
  • Total comments: 37
Search Blog
a