Inequality on non-nagative numbers

by TUAN2k8, May 18, 2025, 12:42 PM

Let $a,b,c$ be non-nagative real numbers such that $a+b+c=3$ .
Prove that $ab+bc+ca-abc \leq \frac{9}{4}$ .

algebra

inequalities

Mathematical Olympiad Finals 2013

by parkjungmin, May 18, 2025, 11:20 AM

Mathematical Olympiad Finals 2013

Attachments:

Difficult combinatorics problem

by shactal, May 18, 2025, 10:40 AM

Can someone help me with this problem? Let $n\in \mathbb N^*$ . We call a distribution the act of distributing the integers from $1$
to $n^2$ represented by tokens to players $A_1$ to $A_n$ so that they all have the same number of tokens in their urns.
We say that $A_i$ beats $A_j$ when, when $A_i$ and $A_j$ each draw a token from their urn, $A_i$ has a strictly greater chance of drawing a larger number than $A_j$ . We then denote $A_i>A_j$ . A distribution is said to be chicken-fox-viper when $A_1>A_2>\ldots>A_n>A_1$ What is $R(n)$
, the number of chicken-fox-viper distributions?

combinatorics

Incircle in an isoscoles triangle

by Sadigly, May 16, 2025, 9:21 PM

Let $ABC$ be an isosceles triangle with $AB=AC$ , and let $I$ be its incenter. Incircle touches sides $BC,CA,AB$ at $D,E,F$ , respectively. Foot of altitudes from $E,F$ to $BC$ are $X,Y$ , respectively. Rays $XI,YI$ intersect $(ABC)$ at $P,Q$ , respectively. Prove that $(PQD)$ touches incircle at $D$ .

geometry

incenter

Prove that the triangle is isosceles.

by TUAN2k8, May 16, 2025, 9:55 AM

Given acute triangle $ABC$ with two altitudes $CF$ and $BE$ .Let $D$ be the point on the line $CF$ such that $DB \perp BC$ .The lines $AD$ and $EF$ intersect at point $X$ , and $Y$ is the point on segment $BX$ such that $CY \perp BY$ .Suppose that $CF$ bisects $BE$ .Prove that triangle $ACY$ is isosceles.

geometry

Triangle

A very beautiful geo problem

by TheMathBob, Mar 29, 2023, 2:16 PM

Given an acute triangle $ABC$ with their incenter $I$ . Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$ . Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$ . It is given that $\angle AIX = \angle XYA = 120^\circ.$ Prove that $YI$ is the angle bisector of $XYA$ .

This post has been edited 1 time. Last edited by TheMathBob, Mar 29, 2023, 2:24 PM

geometry

incenter

angle bisector

Cubic and Quadratic

by mathisreal, Oct 26, 2020, 7:59 PM

Find all triples of positive integers $(a,b,c)$ such that the following equations are both true:
I- $a^2+b^2=c^2$
II- $a^3+b^3+1=(c-1)^3$

This post has been edited 1 time. Last edited by mathisreal, Oct 26, 2020, 7:59 PM

number theory

algebra

n^k + mn^l + 1 divides n^(k+1) - 1

by cjquines0, Jul 19, 2017, 4:40 PM

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$ . Prove that

$m = 1$ and $l = 2k$ ; or
$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$ .

This post has been edited 1 time. Last edited by cjquines0, Jul 19, 2017, 5:09 PM

number theory

IMO Shortlist

Circle is tangent to circumcircle and incircle

by ABCDE, Jun 24, 2016, 2:05 PM

Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$ , let its incircle be tangent to sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$ , respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$ . Finally, let $\gamma$ be the circumcircle of $\triangle AST$ .

(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$ .

(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$ .

James Lin

This post has been edited 1 time. Last edited by ABCDE, Jun 24, 2016, 2:07 PM

Locus of Mobile points on Circle and Square

by Kunihiko_Chikaya, Feb 28, 2012, 2:58 AM

In the $xyz$ -plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$ -plane.

(1) Let $P(0,\ 0,\ 2)$ . When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$ ,
find the equation of the locus of the point $R$ .

(2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$ . When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$ , draw the domain swept by the point $R$ on the $xy$ -plane, then find the area.

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 TUAN2k8, May 18, 2025, 12:42 PM

by parkjungmin, May 18, 2025, 11:20 AM

by shactal, May 18, 2025, 10:40 AM

by Sadigly, May 16, 2025, 9:21 PM

by TUAN2k8, May 16, 2025, 9:55 AM

by TheMathBob, Mar 29, 2023, 2:16 PM

by mathisreal, Oct 26, 2020, 7:59 PM

by cjquines0, Jul 19, 2017, 4:40 PM

by ABCDE, Jun 24, 2016, 2:05 PM

by Kunihiko_Chikaya, Feb 28, 2012, 2:58 AM