Easy Combinatorics

by MuradSafarli, Apr 29, 2025, 6:40 PM

A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$ . Is this possible to make the number $(20^{25}+2024)$ on the board?

combinatorics

Geometry with orthocenter config

by thdnder, Apr 29, 2025, 5:26 PM

Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$ . Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$ . Prove that the lines $DG$ , $EF$ , and $A$ -median of $\triangle ABC$ are concurrent.

prefix sum QRs

by optimusprime154, Apr 29, 2025, 5:23 PM

oops.....

This post has been edited 3 times. Last edited by optimusprime154, 2 hours ago

Balkan Mathematics Olympiad

number theory

Inequality with 3 variables and a special condition

by Nuran2010, Apr 29, 2025, 5:06 PM

For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$ .
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$ .

Determine the equality case.

inequalities

n = a*b , numbers of the form a^b

by falantrng, Aug 24, 2023, 7:43 PM

The teacher calculates and writes on the board all the numbers $a^b$ that satisfy the condition $n = a\times b$ for the natural number $n.$ Here $a$ and $b$ are natural numbers. Is there a natural number $n$ such that each of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.

number theory

AZE SENIOR NATIONAL MO

find all functions

by DNCT1, Oct 10, 2020, 6:43 AM

Find all functions $f:\mathbb{R^+}\rightarrow \mathbb{R^+}$ such that
$f(2f(x)+2y)=f(2x+y)+y\quad\forall x,y,\in\mathbb{R^+}$

function

Sum of products is n mod 2

by y-is-the-best-_, Sep 22, 2020, 11:35 PM

Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that
$\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.$

Reflection of D moves on a line

by Ankoganit, Nov 11, 2017, 3:45 PM

Triangle $ABC$ has orthocenter $H$ . Let $D$ be a point distinct from the vertices on the circumcircle of $ABC$ . Suppose that circle $BHD$ meets $AB$ at $P\ne B$ , and circle $CHD$ meets $AC$ at $Q\ne C$ . Prove that as $D$ moves on the circumcircle, the reflection of $D$ across line $PQ$ also moves on a fixed circle.

Michael Ren

geometry

geometric transformation

reflection

Functional equation on the set of reals

by abeker, Aug 25, 2017, 2:35 PM

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
$f(x^2 + f(x)f(y)) = xf(x + y)$ for all real numbers $x$ and $y$ .

algebra

functional equation

USAMO 2003 Problem 1

by MithsApprentice, Sep 27, 2005, 7:45 PM

Prove that for every positive integer $n$ there exists an $n$ -digit number divisible by $5^n$ all of whose digits are odd.

This post has been edited 1 time. Last edited by MithsApprentice, Sep 27, 2005, 10:00 PM

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 MuradSafarli, Apr 29, 2025, 6:40 PM

by thdnder, Apr 29, 2025, 5:26 PM

by optimusprime154, Apr 29, 2025, 5:23 PM

by Nuran2010, Apr 29, 2025, 5:06 PM

by falantrng, Aug 24, 2023, 7:43 PM

by DNCT1, Oct 10, 2020, 6:43 AM

by y-is-the-best-_, Sep 22, 2020, 11:35 PM

by Ankoganit, Nov 11, 2017, 3:45 PM

by abeker, Aug 25, 2017, 2:35 PM

by MithsApprentice, Sep 27, 2005, 7:45 PM