Final fool geometry

by giangtruong13, Apr 1, 2025, 4:10 PM

Let $ABC$ be a pointed triangle and altitudes $AD, BE, CF$ . Prove that: $S_{DEF}=(sin^2A*sin^2B+sin^2C-2)S_{ABC}$

geometry

Gut inequality

by giangtruong13, Apr 1, 2025, 3:54 PM

Let $a,b,c>0$ satisfy that $a+b+c=3$ . Find the minimum $\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$

This post has been edited 3 times. Last edited by giangtruong13, 2 hours ago

inequalities

Nut equation

by giangtruong13, Apr 1, 2025, 3:47 PM

Solve the quadratic equation: $[4(\sqrt{1+x})^3-3\sqrt{1+x^2}](4x^3+3x)=2$

algebra

quadratics

Can I find source of a geometry problem via Approach0?

by xytunghoanh, Apr 1, 2025, 1:59 PM

Can I find source of a geometry problem via Approach0 or AOPS search feature?
Thanks.

geometry

Olympiad problem - I can't solve it pls help

by kjhgyuio, Apr 1, 2025, 11:07 AM

It is given that x and y are positive integers such that x>y and
√x + √y=√2000
How many different possible values can x take?

This post has been edited 1 time. Last edited by kjhgyuio, Today at 11:22 AM
Reason: typo in problem

Maximum angle ratio

by miiirz30, Mar 31, 2025, 6:10 PM

Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$ , such that $CD = DB = 2AC.$ Find the ratio $\frac{CM}{MD}$ , where $M$ is a point on arc $AB$ , such that $\angle CMD$ is maximized.

Proposed by Andria Gvaramia, Georgia

Euler Olympiad

geometry

ratio

A geometry problem

by Lttgeometry, Mar 30, 2025, 4:02 PM

Given a non-isosceles triangle $ABC$ that is inscribed in $(O)$ . The incircle $(I)$ is tangent to $BC,CA,AB$ at $D,E,F$ respectively. A line through $A$ parallel to $BC$ intersects $(O)$ at $T$ , and $TD$ intersects $(O)$ again at $J$ . Let $N$ is the midpoint of $BC$ . $P,Q$ be the second intersection of $JE,JF$ with $(O)$ . $AI$ intersects $(O)$ again at $M$ . Prove that the line passing through $A$ perpendicular to $PQ$ bisects $MN$ .

geometry

FE over R

by IAmTheHazard, Jun 22, 2024, 3:40 PM

Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$ ,
$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$
Andrew Carratu

This post has been edited 1 time. Last edited by IAmTheHazard, Jun 22, 2024, 3:41 PM

algebra

ELMO Shortlist

functional equation

Find values of $a b+a c+b c$

by NJAX, May 31, 2024, 12:08 PM

Let $a, b, c$ be distinct real numbers such that $a+b+c=0$ and $a^{2}-b=b^{2}-c=c^{2}-a.$ Evaluate all the possible values of $a b+a c+b c$ .

Proposed by Nguyen Anh Vu, Vietnam

This post has been edited 1 time. Last edited by NJAX, May 31, 2024, 12:34 PM

algebra

Al-Khwarizmi IJMO

Interesting config

by TheUltimate123, Jun 26, 2023, 5:16 AM

Let $ABC$ be an acute scalene triangle with orthocenter $H$ . Line $BH$ intersects $\overline{AC}$ at $E$ and line $CH$ intersects $\overline{AB}$ at $F$ . Let $X$ be the foot of the perpendicular from $H$ to the line through $A$ parallel to $\overline{EF}$ . Point $B_1$ lies on line $XF$ such that $\overline{BB_1}$ is parallel to $\overline{AC}$ , and point $C_1$ lies on line $XE$ such that $\overline{CC_1}$ is parallel to $\overline{AB}$ . Prove that points $B$ , $C$ , $B_1$ , $C_1$ are concyclic.

Proposed by Luke Robitaille

Elmo

geometry

Submit

how do you have so many posts
by krithikrokcs, Jul 14, 2023, 6:20 PM
lol⠀⠀⠀⠀⠀
by math_explorer, Jan 20, 2021, 8:43 AM
woah ancient blog
by suvamkonar, Jan 20, 2021, 4:14 AM
https://artofproblemsolving.com/community/c47h361466
by math_explorer, Jun 10, 2020, 1:20 AM
when did the first greed control game start?
by piphi, May 30, 2020, 1:08 AM
ok..........
by asdf334, Sep 10, 2019, 3:48 PM
There is one existing way to obtain contributorship documented on this blog. See if you can find it.
by math_explorer, Sep 10, 2019, 2:03 PM
SO MANY VIEWS!!!
PLEASE CONTRIB

by asdf334, Sep 10, 2019, 1:58 PM
Hullo bye
by AnArtist, Jan 15, 2019, 8:59 AM
Hullo bye
by tastymath75025, Nov 22, 2018, 9:08 PM
Hullo bye
by Kayak, Jul 22, 2018, 1:29 PM
It's sad; the blog is still active but not really ;-;
by GeneralCobra19, Sep 21, 2017, 1:09 AM
dope css
by zxcv1337, Mar 27, 2017, 4:44 AM
nice blog ^_^
by chezbgone, Mar 28, 2016, 5:18 AM
shouts make blogs happier
by briantix, Mar 18, 2016, 9:58 PM
I like your CSS. Can you send me your shoutbox block? I'll credit you.
by bluecarneal, Feb 21, 2011, 8:24 PM
Hey contrib?
by Dsquared, Dec 16, 2010, 3:14 PM
$\sum_{n=0}^{\infty}2^n=-1$
by AwesomeToad, Dec 13, 2010, 2:12 PM
Delete the shouts...

If you wish something more mathematical, here's some digits.
1679350278936041982752
Enjoy. (no offense)
by chaotic_iak, Dec 3, 2010, 1:00 PM
Now I feel guilty my shoutbox is so nonmathematical.
by math_explorer, Dec 3, 2010, 12:36 PM
Congratulations on winning Achievement Unlocked: AoPS Style! I'm looking to your participation in Achievement Unlocked 2: Reunlock the Achievements!
by chaotic_iak, Dec 3, 2010, 12:32 PM
El_ectric won game 11
by halowarfare17, Nov 17, 2010, 2:36 PM
Did you win reaper or did electric?
by dragon96, Nov 11, 2010, 2:14 AM
Very nice blog
by Amir Hossein, Oct 28, 2010, 2:58 PM
Changed your avatar?
by asf, Oct 15, 2010, 3:47 AM
Gosh, post count over 9000.
by math_explorer, Sep 20, 2010, 1:44 PM
Nice blog I read pretty much whenever you make a new post, but I don't comment.
by AIME15, Sep 19, 2010, 2:38 PM
I see, that part of the CSS needs a lot of tweaking.
by math_explorer, Sep 12, 2010, 9:20 AM
Hello. Shout?
by asf, Sep 11, 2010, 5:18 PM

91 shouts

math_exploration

by giangtruong13, Apr 1, 2025, 4:10 PM

by giangtruong13, Apr 1, 2025, 3:54 PM

by giangtruong13, Apr 1, 2025, 3:47 PM

by xytunghoanh, Apr 1, 2025, 1:59 PM

by kjhgyuio, Apr 1, 2025, 11:07 AM

by miiirz30, Mar 31, 2025, 6:10 PM

by Lttgeometry, Mar 30, 2025, 4:02 PM

by IAmTheHazard, Jun 22, 2024, 3:40 PM

by NJAX, May 31, 2024, 12:08 PM

by TheUltimate123, Jun 26, 2023, 5:16 AM