Function equation

by luci1337, Apr 17, 2025, 3:01 PM

find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number

functional equation

algebra

inequalities

by pennypc123456789, Apr 17, 2025, 1:53 PM

If $a,b,c$ are positive real numbers, then
$\frac{a + b}{a + 7b + c} + \dfrac{b + c}{b + 7c + a}+\dfrac{c + a}{c + 7a + b} \geq \dfrac{2}{3}$
we can generalize this problem

inequalities

Using Humpty point on Trapezoid ??

by FireBreathers, Apr 17, 2025, 12:21 PM

Given a trapezoid $ABCD$ with $AD//BC$ . Let point $H$ be orthocenter $ABD$ and $M$ midpoint $AD$ . It is also known that $HC$ perpendicular to $BM$ . Let $X$ be a point on the segment $AB$ such that $XH=BH$ and point $Y$ be the intersection of $CX$ and $BD$ . Prove that $AXYD$ concyclic

geometry

trapezoid

11^n+2^n+6=m^3

by jungle_wang, Aug 1, 2024, 1:37 PM

Find all positive integers $(m,n)$ such that
$11^n+2^n+6=m^3$

This post has been edited 1 time. Last edited by jungle_wang, Aug 4, 2024, 2:20 AM

number theory

Permutations of Integers from 1 to n

by Twoisntawholenumber, Jul 20, 2021, 9:01 PM

Let $n$ be a positive integer. Find the number of permutations $a_1$ , $a_2$ , $\dots a_n$ of the
sequence $1$ , $2$ , $\dots$ , $n$ satisfying
$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$ .

Proposed by United Kingdom

This post has been edited 1 time. Last edited by Twoisntawholenumber, Jul 20, 2021, 9:36 PM
Reason: Added the country that proposed the problem

combinatorics

recursion

The Bank of Bath

by TelMarin, Jul 17, 2019, 12:18 PM

The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$ , then he turns over the $k$ th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$ , which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$ , let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$ . Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$ .

Proposed by David Altizio, USA

This post has been edited 2 times. Last edited by djmathman, Jul 17, 2019, 12:26 PM

Coaxal Circles

by fattypiggy123, Mar 13, 2017, 1:07 AM

Let $ABCD$ be a quadrilateral and let $l$ be a line. Let $l$ intersect the lines $AB,CD,BC,DA,AC,BD$ at points $X,X',Y,Y',Z,Z'$ respectively. Given that these six points on $l$ are in the order $X,Y,Z,X',Y',Z'$ , show that the circles with diameter $XX',YY',ZZ'$ are coaxal.

This post has been edited 1 time. Last edited by fattypiggy123, Mar 13, 2017, 1:34 AM
Reason: Typo

geometry

circles

coaxal

Inequality from my inequality training.

by Orkhan-Ashraf_2002, Aug 21, 2016, 4:38 PM

Let $a,b,c$ non-negative real numbers,but $ab+bc+ca\not=$ 0.Prove that
$1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}$

inequalities

IMO 2014 Problem 2

by v_Enhance, Jul 8, 2014, 12:16 PM

Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.

Counting friends in two ways

by joybangla, May 11, 2014, 12:51 PM

Suppose a class contains $100$ students. Let, for $1\le i\le 100$ , the $i^{\text{th}}$ student have $a_i$ many friends. For $0\le j\le 99$ let us define $c_j$ to be the number of students who have strictly more than $j$ friends. Show that $\begin{align*} & \sum_{i=1}^{100}a_i=\sum_{j=0}^{99}c_j \end{align*}$

This post has been edited 1 time. Last edited by joybangla, May 11, 2014, 3:56 PM

combinatorics proposed

combinatorics

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 luci1337, Apr 17, 2025, 3:01 PM

by pennypc123456789, Apr 17, 2025, 1:53 PM

by FireBreathers, Apr 17, 2025, 12:21 PM

by jungle_wang, Aug 1, 2024, 1:37 PM

by Twoisntawholenumber, Jul 20, 2021, 9:01 PM

by TelMarin, Jul 17, 2019, 12:18 PM

by fattypiggy123, Mar 13, 2017, 1:07 AM

by Orkhan-Ashraf_2002, Aug 21, 2016, 4:38 PM

by v_Enhance, Jul 8, 2014, 12:16 PM

by joybangla, May 11, 2014, 12:51 PM