My Unsolved Problem

by ZeltaQN2008, May 22, 2025, 3:07 PM

Let triangle $ABC$ be inscribed in circle $(O)$ . Let $(I_a)$ be the $A$ -excircle of triangle $ABC$ , which is tangent to $BC$ , the extension of $AB$ , and the extension of $AC$ . Let $BE$ and $CF$ be the angle bisectors of triangle $ABC$ . Let $EF$ intersect $(O)$ at two points $S$ and $T$ .

a) Prove that circle $(O)$ bisects the segments $I_aT$ and $I_aS$ .
b) Prove that $S$ and $T$ are the points of tangency of the common external tangents of circles $(O)$ and $(I_a)$ .

geometry

geometry unssolve

Functional equations

by mathematical-forest, May 22, 2025, 12:43 PM

Find all funtion $f:C\to C$ , s.t. $\forall x \in C$
$xf(x)=\overline{x} f(\overline{x})$

function

functional equation

algebra

Moving stones on an infinite row

by miiirz30, May 22, 2025, 10:53 AM

We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations:

1. Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between.

2. Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right.

3. Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells.

The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player.

Proposed by Luka Tsulaia, Georgia

This post has been edited 1 time. Last edited by miiirz30, 5 hours ago

Cauchy and multiplicative function over a field extension

by miiirz30, May 22, 2025, 10:40 AM

Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$ ,
$f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y),$ where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$ .

Proposed by Stijn Cambie, Belgium

Euler Olympiad

functional equation

Cauchy functional equation

multiplicative function

algebra

functional inequality with equality

by miiirz30, May 22, 2025, 10:32 AM

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following two conditions hold:

1. For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$ , We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$ .

2. For all real numbers $x$ and $y$ , there exist real numbers $a$ and $b$ , such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$ .

Proposed by Zaza Melikidze, Georgia

This post has been edited 1 time. Last edited by miiirz30, 5 hours ago

Interesting inequalities

by sqing, May 21, 2025, 1:25 PM

Let $a,b,c,d\geq 0 , a+b+c+d \leq 4.$ Prove that
$a(bc+bd+cd) \leq \frac{256}{81}$ $ab(a+2c+2d ) \leq \frac{256}{27}$ $ab(a+3c+3d ) \leq \frac{32}{3}$ $ab(c+d ) \leq \frac{64}{27}$

This post has been edited 1 time. Last edited by sqing, Yesterday at 2:06 PM

inequalities proposed

inequalities

A sharp one with 3 var

by mihaig, May 13, 2025, 7:20 PM

Let $a,b,c\geq0$ satisfying
$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$ Prove
$ab+bc+ca+abc\geq4.$

inequalities

Hard Functional Equation in the Complex Numbers

by yaybanana, Apr 9, 2025, 3:29 PM

Find all functions $f:\mathbb {C}\rightarrow \mathbb {C}$ , s.t :

$f(xf(y)) + f(x^2+y) = f(x+y)x + f(f(y))$

for all $x,y \in \mathbb{C}$

Circles, Tangents, Variable point on BC

by SerdarBozdag, Mar 19, 2022, 5:38 PM

In triangle $ABC$ , $D$ is an arbitrary point on $BC$ . $(ADC), (ADB)$ cuts $AB$ , $AC$ at $F$ and $E$ respectively. Tangents at $B$ and $C$ intersect at $X$ . $Z=EF \cap BX$ and $Y=EF \cap CX$ . $P$ is a point on $(ABC)$ such that $AP, YZ, BC$ are concurrent. Prove that $P$ lies on $(XYZ)$

Proposed by SerdarBozdag and k12byda5h

geometry

DAMO

DJMO DAMO

22 light bulbs

by dangerousliri, Mar 6, 2022, 2:33 PM

$22$ light bulbs are given. Each light bulb is connected to exactly one switch, but a switch can be connected to one or more light bulbs. Find the least number of switches we should have such that we can turn on whatever number of light bulbs.

combinatorics

Submit

Take a look at The British Flag Theorem post. I've included a working Python program.
by aoum, May 17, 2025, 1:05 AM
Check out the Pascal's Law post. I included a cartoon from the xkcd serial webcomic.
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If you leave a comment on one of my posts—especially older ones—I might not see it right away.
by aoum, May 2, 2025, 11:55 PM
100 posts!
by aoum, Apr 21, 2025, 9:11 PM
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1,000 views!
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Are you asking to contribute or to be notified whenever a post is published?
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nice blog! love the dedication c:
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by Catcumber, Apr 4, 2025, 11:16 PM
The first few posts for April are out!
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Sure! I understand that it would be quite a bit to take in.
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That's nice to know. I'm also learning new, interesting things on here myself, too.
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Reading the fun facts and all from this blog's material makes me feel so at ease when using formulas. like, I finally understand the backstory of it and all that even teachers don't teach
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Thanks! There are many interesting things about math out there, and I hope to share them with you all. I'll be posting more of these!
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Yes, I'll be doing problems of the day every day.
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I think it would also be cool if you did a problem of the day every day, as I see from today's problem.
by jocaleby1, Mar 5, 2025, 1:13 AM
Do you guys like my "lectures" or would you like something else?
by aoum, Mar 4, 2025, 10:37 PM
Yeah, keep on making these "lectures"
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Thanks! Glad to hear that!
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Anyone wants to contribute to my blog? Shout or give me a friend request!
by aoum, Mar 2, 2025, 3:22 PM
Nice blog! Contrib?
by jocaleby1, Mar 1, 2025, 6:43 PM

61 shouts

Pi in the Sky

by ZeltaQN2008, May 22, 2025, 3:07 PM

by mathematical-forest, May 22, 2025, 12:43 PM

by miiirz30, May 22, 2025, 10:53 AM

by miiirz30, May 22, 2025, 10:40 AM

by miiirz30, May 22, 2025, 10:32 AM

by sqing, May 21, 2025, 1:25 PM

by mihaig, May 13, 2025, 7:20 PM

by yaybanana, Apr 9, 2025, 3:29 PM

by SerdarBozdag, Mar 19, 2022, 5:38 PM

by dangerousliri, Mar 6, 2022, 2:33 PM