Sneaky one

by Sunjee, May 16, 2025, 9:35 AM

Find minimum and maximum value of following function.
$f(x,y)=\frac{\sqrt{x^2+y^2}+\sqrt{(x-2)^2+(y-1)^2}}{\sqrt{x^2+(y-1)^2}+\sqrt{(x-2)^2+y^2}}$

inequalties

inequalities

IMO Solution mistake

by CHESSR1DER, May 16, 2025, 7:35 AM

I found a mistake in 4th solution at IMO 1982/1. It gives answer $660$ and $661$ . But right answer is only $660$ . Should it be reported somewhere in Aops?

IMO

power of a point

by BekzodMarupov, May 16, 2025, 5:41 AM

Epsilon 1.3. Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.

power of a point

geometry

4-vars inequality

by xytunghoanh, May 15, 2025, 2:10 PM

For $a,b,c,d \ge 0$ and $a\ge c$ , $b \ge d$ . Prove that
$a+b+c+d+ac+bd+8 \ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{cd}+\sqrt{da}+\sqrt{ac}+\sqrt{bd})$ .

This post has been edited 1 time. Last edited by xytunghoanh, Yesterday at 2:16 PM
Reason: add condition

inequalities

inequality

by mathematical-forest, May 15, 2025, 12:40 PM

For positive real intengers $x_{1} ,x_{2} ,\cdots,x_{n}$ , such that $\prod_{i=1}^{n} x_{i} =1$
proof:
$\sum_{i=1}^{n} \frac{1}{1+\sum _{j\ne i}x_{j} } \le 1$

inequalities

Ah, easy one

by irregular22104, May 14, 2025, 4:01 PM

In the number series $1,9,9,9,8,...,$ every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers $1234$ and $5678$ in this series?

combinatorics

Hard geometry

by Lukariman, May 14, 2025, 4:28 AM

Given circle (O) and chord AB with different diameters. The tangents of circle (O) at A and B intersect at point P. On the small arc AB, take point C so that triangle CAB is not isosceles. The lines CA and BP intersect at D, BC and AP intersect at E. Prove that the centers of the circles circumscribing triangles ACE, BCD and OPC are collinear.

Attachments:

This post has been edited 2 times. Last edited by Lukariman, May 14, 2025, 4:49 AM

geometry

Functional Equation!

by EthanWYX2009, Mar 29, 2025, 10:48 AM

Find all functions $f:\mathbb Z\to\mathbb Z$ such that $f$ is unbounded and
$2f(m)f(n)-f(n-m)-1$ is a perfect square for all integer $m,n.$

number theory

functional equation

algebra

IMO Shortlist 2014 G3

by hajimbrak, Jul 11, 2015, 8:49 AM

Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$ . The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$ . Let $\Gamma$ be the circle with diameter $BM$ . The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$ . Prove that $BR\parallel AC$ .
(Here we always assume that an angle bisector is a ray.)

Proposed by Sergey Berlov, Russia

This post has been edited 2 times. Last edited by hajimbrak, Jul 23, 2015, 10:35 AM
Reason: Added proposer

geometry

angle bisector

IMO Shortlist

Eight-point cicle

by sandu2508, May 4, 2010, 12:21 PM

Let $ABC$ be an acute triangle with orthocentre $H$ , and let $M$ be the midpoint of $AC$ . The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$ . Let $H_1$ be the reflection of $H$ in $AB$ . The orthogonal projections of $C_1$ onto the lines $AH_1$ , $AC$ and $BC$ are $P$ , $Q$ and $R$ , respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$ .
Prove that $M_1$ lies on the segment $BH_1$ .

geometry

geometric transformation

Submit

this blog is so orz
by ESAOPS, May 10, 2025, 6:51 PM
Can you give some thought to dropping a guide to STS? Just like how you presented your research (in your paper), what your essays were about, etc. Also cool blog!
by Shreyasharma, Mar 13, 2025, 7:03 PM
this is so good
by purpledonutdragon, Mar 4, 2025, 2:05 PM
orz usamts grader
by Lhaj3, Jan 23, 2025, 7:43 PM
Entertaining blog
by eduD_looC, Dec 31, 2024, 8:57 PM
wow really cool stuff
by kingu, Dec 4, 2024, 1:02 AM
Although I had a decent college essay, this isn't really my specialty so I don't really have anything useful to say that isn't already available online.
by greenturtle3141, Nov 3, 2024, 7:25 PM
Could you also make a blog post about college essay writing :skull:
by Shreyasharma, Nov 2, 2024, 9:04 PM
what gold
by peace09, Oct 15, 2024, 3:39 PM
oh lmao, i was confused because of the title initially. thanks! great read
by OlympusHero, Jul 20, 2024, 5:00 AM
It should be under August 2023
by greenturtle3141, Jul 11, 2024, 11:44 PM
does this blog still have the post about your math journey? for some reason i can't find it
by OlympusHero, Jul 10, 2024, 5:41 PM
imagine not tortoise math

no but seriously really interesting blog
by fruitmonster97, Apr 2, 2024, 12:39 AM
W blog man
by s12d34, Jan 24, 2024, 11:37 PM
very nice blog greenturtle it is very descriptive and fascinating to pay attention to
by StarLex1, Jan 3, 2024, 3:12 PM
Hi! Cool blog!
by mahaler, Nov 14, 2021, 3:08 PM
fascinating...
by pi271828, Nov 14, 2021, 12:00 AM
I believe that indeed, turtles come in all sorts of different colors. Even real ones! Some googling shows that indeed, red turtles seem to exist, though they're not that red.
by greenturtle3141, Nov 13, 2021, 9:21 PM
hello

since you have turtle knowledge, does your species of green turtles have a brethren that possess a color of red? what about blue? what about tawny mauve?
by pi271828, Nov 13, 2021, 9:18 PM
Nice Blog!
by Jwenslawski, Nov 13, 2021, 6:22 PM
Looking especially forward to how you improved on your AMC/AIME/USAMO so quickly
by StopSine, Nov 11, 2021, 2:01 AM
thanks the_turtle
by pog, Nov 10, 2021, 1:09 PM
another turtle blog?
by The_Turtle, Nov 9, 2021, 2:23 AM
Looking forward to future posts!
by Arr0w, Nov 2, 2021, 6:15 PM
just copy and paste every single one of your msm posts and thisll become the best blog ever hahahaha
by peace09, Nov 1, 2021, 9:33 PM
hello!
by tigerzhang, Nov 1, 2021, 9:09 PM
3rd shout! Yay greenturtle blog!
by violin21, Oct 25, 2021, 1:04 AM
hello $\text{[math]}$ $\text{[math]}$
by v4913, Oct 24, 2021, 5:33 PM
first shout
by mathlearner2357, Oct 24, 2021, 5:12 AM

68 shouts

Turtle Math

by Sunjee, May 16, 2025, 9:35 AM

by CHESSR1DER, May 16, 2025, 7:35 AM

by BekzodMarupov, May 16, 2025, 5:41 AM

by xytunghoanh, May 15, 2025, 2:10 PM

by mathematical-forest, May 15, 2025, 12:40 PM

by irregular22104, May 14, 2025, 4:01 PM

by Lukariman, May 14, 2025, 4:28 AM

by EthanWYX2009, Mar 29, 2025, 10:48 AM

by hajimbrak, Jul 11, 2015, 8:49 AM

by sandu2508, May 4, 2010, 12:21 PM