FE with a lot of terms

by MrHeccMcHecc, Apr 6, 2025, 1:44 PM

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ $f(x)f(y)+f(x+y)=xf(y)+yf(x)+f(xy)+x+y+1$

function

functional equation

algebra

Set Combo <-> Grid Combo

by Mathdreams, Apr 6, 2025, 1:36 PM

Consider an $n \times n$ grid, where $n$ is a composite integer.

The $n^2$ unit squares are divided up into $a$ disjoint sets of $b$ unit squares arbitrarily such that $ab = n^2$ . Denote this family of sets as $S$ .

The $n^2$ unit squares are again divided up into $c$ disjoint sets of $d$ unit squares arbitrarily such that $cd = n^2$ . Denote this family of sets as $T$ .

Is it necessarily possible to choose $\min(a,c)$ unit squares such that no two unit squares are in the same set of $S$ or the same set of $T$ ?

(Shining Sun, USA)

combinatorics

Two Functional Inequalities

by Mathdreams, Apr 6, 2025, 1:34 PM

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x) \le x^3$ and $f(x + y) \le f(x) + f(y) + 3xy(x + y)$ for any real numbers $x$ and $y$ .

(Miroslav Marinov, Bulgaria)

functional inequalities

algebra

Two Orthocenters and an Invariant Point

by Mathdreams, Apr 6, 2025, 1:30 PM

Let $\triangle{ABC}$ be a triangle, and let $P$ be an arbitrary point on line $AO$ , where $O$ is the circumcenter of $\triangle{ABC}$ . Define $H_1$ and $H_2$ as the orthocenters of triangles $\triangle{APB}$ and $\triangle{APC}$ . Prove that $H_1H_2$ passes through a fixed point which is independent of the choice of $P$ .

(Kritesh Dhakal, Nepal)

geometry

Inspired by 2012 Romania and 2021 BH

by sqing, Apr 6, 2025, 1:28 PM

Let $a, b, c, d\geq 0 , bc + d + a = 5, cd + a + b = 2$ and $da + b + c = 6.$ Prove that
$3\leq ab + c + d\leq 2\sqrt{13}-1$ $5\leq a+ b+ c +d \leq\frac{1}{2}(11+\sqrt{13})$ $\sqrt{13}+1 \leq a b +bc+ c d+d a \leq 6$

This post has been edited 2 times. Last edited by sqing, 24 minutes ago

inequalities proposed

algebra

Sum of Squares of Digits is Periodic

by Mathdreams, Apr 6, 2025, 1:28 PM

For any positive integer $n$ , let $f(n)$ denote the sum of squares of digits of $n$ . Prove that the sequence $f(n), f(f(n)), f(f(f(n))), \cdots$ is eventually periodic.

(Kritesh Dhakal, Nepal)

number theory

Digits

Periodic sequence

3 var inquality

by sqing, Apr 6, 2025, 1:11 PM

Let $a,b,c>0$ and $\dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}\leq 12.$ Prove that $a^2+b^2+c^2\geq 1$

inequalities proposed

inequalities

Common tangent to diameter circles

by Stuttgarden, Mar 31, 2025, 1:06 PM

The cyclic quadrilateral $ABCD$ , inscribed in the circle $\Gamma$ , satisfies $AB=BC$ and $CD=DA$ , and $E$ is the intersection point of the diagonals $AC$ and $BD$ . The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$ . Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$ .

geometry

Spain

inequality

by pennypc123456789, Mar 24, 2025, 11:17 AM

Let $x, y$ be positive real numbers satisfying $x + y = 2$ . Prove that

$3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.$

This post has been edited 1 time. Last edited by pennypc123456789, Mar 24, 2025, 11:22 AM

inequalities

inequalities proposed

Ratios in a right triangle

by PNT, Jun 9, 2023, 10:58 PM

Let $ABC$ be a right triangle in $A$ with $AB<AC$ . Let $M$ be the midpoint of $AB$ and $D$ a point on $AC$ such that $DC=DB$ . Let $X=(BDC)\cap MD$ .
Compute in terms of $AB,BC$ and $AC$ the ratio $\frac{BX}{DX}$ .

geometry

right triangle

ratio

Submit

how do you have so many posts
by krithikrokcs, Jul 14, 2023, 6:20 PM
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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
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by math_explorer, Sep 10, 2019, 2:03 PM
SO MANY VIEWS!!!
PLEASE CONTRIB

by asdf334, Sep 10, 2019, 1:58 PM
Hullo bye
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Hullo bye
by Kayak, Jul 22, 2018, 1:29 PM
It's sad; the blog is still active but not really ;-;
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dope css
by zxcv1337, Mar 27, 2017, 4:44 AM
nice blog ^_^
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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...

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1679350278936041982752
Enjoy. (no offense)
by chaotic_iak, Dec 3, 2010, 1:00 PM
Now I feel guilty my shoutbox is so nonmathematical.
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by chaotic_iak, Dec 3, 2010, 12:32 PM
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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 MrHeccMcHecc, Apr 6, 2025, 1:44 PM

by Mathdreams, Apr 6, 2025, 1:36 PM

by Mathdreams, Apr 6, 2025, 1:34 PM

by Mathdreams, Apr 6, 2025, 1:30 PM

by sqing, Apr 6, 2025, 1:28 PM

by Mathdreams, Apr 6, 2025, 1:28 PM

by sqing, Apr 6, 2025, 1:11 PM

by Stuttgarden, Mar 31, 2025, 1:06 PM

by pennypc123456789, Mar 24, 2025, 11:17 AM

by PNT, Jun 9, 2023, 10:58 PM