Circle and square

by Marrelia, Apr 10, 2025, 3:00 AM

Given a circle with center $O$, and square $ABCD$. Point $A$ and $B$ are on the circle, and $CD$ is tangent to the circle at point $E$. Let $M$ represent the midpoint of $AD$ and $F$ represent the intersection between $AD$ and circle. Prove that $MF = FD$.
Attachments:
geometry
geometric transformation
L

Hard number theory

by td12345, Apr 9, 2025, 11:32 PM

Let $q$ be a prime number. Define the set
\[ M_q = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2q^{2025} x} \in \mathbb{Q} \right\}. \]
Find the number of elements of \(M_2 \cup M_{2027}\).
number theory
L

Hardest Computational Problem?

by happypi31415, Apr 9, 2025, 11:06 PM

What do you guys think the hardest computational problem (for high school students) is?
L

Question abt directed angles

by idk12345678, Apr 9, 2025, 9:09 PM

If you have a diameter of a circle COA, and there is a point on the circle B, then how do you prove CBA is 90 degrees. Usually, i would use the inscribed angle theorem, but you cant divide directed angles by 2
directed angle
L

Random Question

by JerryZYang, Apr 9, 2025, 5:03 PM

Can anyone help me prove $\lim_{x\rightarrow\infty}(1+\dfrac{1}{x})^x=\sum_{n=0}^{\infty}\dfrac{1}{n!}$?
L

Inequalities

by sqing, Apr 9, 2025, 2:40 PM

Let $ a,b,c $ be real numbers so that $ a+2b+3c=2 $ and $ 2ab+6bc+3ca =1. $ Show that
$$-\frac{1}{6} \leq ab-bc+ ca\leq \frac{1}{2}$$$$\frac{5-\sqrt{61}}{9} \leq a-b+c\leq \frac{5+\sqrt{61}}{9} $$
This post has been edited 1 time. Last edited by sqing, Yesterday at 2:41 PM
algebra
inequalities
L

lcm(1,2,3,...,n)

by lgx57, Apr 9, 2025, 7:41 AM

Let $M=\operatorname{lcm}(1,2,3,\cdots,n)$.Estimate the range of $M$.
number theory
L

No bash for this inequality

by giangtruong13, Apr 8, 2025, 3:08 PM

Let $x,y,z$ be positive real number satisfy that: $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1$.Find the minimum: $$ \sum_{cyc} \frac{(xy)^2}{z(x^2+y^2)} $$
Inequality
inequalities
L

junior 3 and 4 var ineq (2019 Romanian NMO grade VII P1)

by parmenides51, Sep 4, 2024, 12:21 AM

a) Prove that for $x,y \ge 1$, holds $$x+y - \frac{1}{x}- \frac{1}{y} \ge 2\sqrt{xy} -\frac{2}{\sqrt{xy}}$$
b) Prove that for $a,b,c,d \ge 1$ with $abcd=16$ , holds $$a+b+c+d-\frac{1}{a}-\frac{1}{b}-\frac{1}{c}-\frac{1}{d}\ge 6$$
This post has been edited 2 times. Last edited by parmenides51, Oct 21, 2024, 1:15 PM
Reason: typo corrected
algebra
inequalities
L

A complicated fraction

by nsato, Mar 16, 2006, 3:54 PM

Compute
\[ \frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}. \]
L

It's not just good - it's revolutionary!

avatar

liberator
Shouts
Submit

  • whoa....

    by bachkieu, Jan 31, 2025, 1:40 AM

  • hello...

    by ethan2011, Jul 4, 2024, 5:13 PM

  • 2024 shout ftw

    by Shreyasharma, Feb 19, 2024, 10:28 PM

  • time flies

    by Asynchrone, Dec 13, 2023, 9:29 PM

  • first 2023 shout :D

    by gracemoon124, Aug 2, 2023, 4:58 AM

  • offline.................

    by 799786, Dec 27, 2021, 7:08 AM

  • YOU SHALL NOT PASS! - liberator

    by OlympusHero, Aug 16, 2021, 4:10 AM

  • Nice Blog!

    by geometry6, Jul 31, 2021, 1:39 PM

  • First shout out in 2021 :D

    by Aimingformygoal, May 31, 2021, 4:23 PM

  • indeed a pr0 blog :surf:

    by Kanep, Dec 3, 2020, 10:46 PM

  • pr0 blog !!

    by Hamroldt, Dec 2, 2020, 8:32 AM

  • niice bloog!

    by Eliot, Oct 1, 2020, 3:27 PM

  • nice blog :o

    by fukano_2, Aug 8, 2020, 7:49 AM

  • Nice blog :)

    by Feridimo, Mar 31, 2020, 9:29 AM

  • Very nice blog !

    by Kamran011, Oct 31, 2019, 5:48 PM

56 shouts
Tags
geometry
Reim s theorem
combinatorics
APMO
Miquel
power of a point
Inequality
9-point circle
Mannheim circles
number theory
Rioplatense Olympiad
Russian MO
algebra
Angle Chase
British TST
ceva
Extremal
fe
global
graph theory
homothety
IMO 2013
ISL
Isogonal conjugacy
Projective
Spiral Similarity
USAMO
Argentinian TST
Austria-Poland MO
Balkan MO
Bulgarian TST
complex numbers
Cylocevian conjugacy
Desargues
Determinants
Elmo
Euclid s algorithm
Excentral triangle
Greek TST
IMO 2011
IMO 2015
Iran MO
Iran TST
isogonality lemma
Japan MO
Menelaus
Pascal
polynomial
Ramsey Theory
Serbian MO
USA TST
Vietnam TST
About Owner
  • Posts: 95
  • Joined: May 28, 2014
Blog Stats
  • Blog created: Aug 13, 2014
  • Total entries: 46
  • Total visits: 37992
  • Total comments: 43
Search Blog
a