Question on Balkan SL

by Fmimch, Apr 30, 2025, 12:13 AM

Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!

algebra

combinatorics

geometry

number theory

IMO Shortlist

hard inequalities

by pennypc123456789, Apr 30, 2025, 12:12 AM

Given $x,y,z$ be the positive real number. Prove that

$\frac{2xy}{\sqrt{2xy(x^2+y^2)}} + \frac{2yz}{\sqrt{2yz(y^2+z^2)}} + \frac{2xz}{\sqrt{2xz(x^2+z^2)}} \le \frac{2(x^2+y^2+z^2) + xy+yz+xz}{x^2+y^2+z^2}$

inequalities

Functional Geometry

by GreekIdiot, Apr 27, 2025, 1:08 PM

Let $f: \pi \to \mathbb R$ be a function from the Euclidean plane to the real numbers such that $f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$ for any acute triangle $\Delta ABC$ with circumcenter $O$ , centroid $G$ and orthocenter $H$ . Prove that $f$ is constant.

This post has been edited 1 time. Last edited by GreekIdiot, Apr 27, 2025, 1:08 PM

geometry

function

circumcircle

2^x+3^x = yx^2

by truongphatt2668, Apr 22, 2025, 3:38 PM

Prove that the following equation has infinite integer solutions:
$2^x+3^x = yx^2$

number theory

Easy Geometry Problem in Taiwan TST

by chengbilly, Mar 6, 2025, 5:05 AM

Suppose $I$ and $I_A$ are the incenter and the $A$ -excenter of triangle $ABC$ , respectively.
Let $M$ be the midpoint of arc $BAC$ on the circumcircle, and $D$ be the foot of the
perpendicular from $I_A$ to $BC$ . The line $MI$ intersects the circumcircle again at $T$ . For
any point $X$ on the circumcircle of triangle $ABC$ , let $XT$ intersect $BC$ at $Y$ . Prove
that $A, D, X, Y$ are concyclic.

geometry

Easy Combinatorial Game Problem in Taiwan TST

by chengbilly, Mar 5, 2025, 5:05 AM

Alice and Bob are playing game on an $n \times n$ grid. Alice goes first, and they take turns drawing a black point from the coordinate set
$\{(i, j) \mid i, j \in \mathbb{N}, 1 \leq i, j \leq n\}$ There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers $n$ such that Alice has a winning strategy.

Proposed by chengbilly

This post has been edited 1 time. Last edited by chengbilly, Mar 5, 2025, 5:09 AM

Cute R+ fe

by Aryan-23, Jan 27, 2024, 2:57 PM

Find all functions $f\colon \mathbb R^+ \mapsto \mathbb R^+$ , such that for all positive reals $x,y$ , the following is true:

$xf(1+xf(y))= f\left(f(x) + \frac 1y\right)$
Kazi Aryan Amin

This post has been edited 1 time. Last edited by Aryan-23, Jan 27, 2024, 2:58 PM

functional equation

algebra

Interesting Function

by Kei0923, Jan 9, 2024, 7:36 AM

Function $f:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}$ satisfies
$f(m+n)^2=f(m|f(n)|)+f(n^2)$ for any non-negative integers $m$ and $n$ . Determine the number of possible sets of integers $\{f(0), f(1), \dots, f(2024)\}$ .

This post has been edited 1 time. Last edited by Kei0923, Jan 9, 2024, 8:41 AM

algebra

Overlapping game

by Kei0923, Feb 11, 2023, 11:03 AM

On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.

This post has been edited 1 time. Last edited by Kei0923, Feb 11, 2023, 11:22 AM

combinatorics

An easy ineq; ISI BS 2011, P1

by Sayan, Mar 31, 2013, 4:15 AM

Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$ . Then show that
$\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}$

This post has been edited 1 time. Last edited by Sayan, Mar 31, 2013, 5:32 AM

calculus

inequalities

rearrangement inequality

n-variable inequality

Submit

I still exist as well.
by G.G.Otto, Aug 11, 2023, 2:44 AM
hello I'm still here lol
by player01, Aug 6, 2022, 6:24 PM
[REVIVAL] I will start posting more calculator relating posts very soon. Even though school has been busy, I have been programming my calculators a decent amount, so I have a lot to share...
by sonone, Feb 18, 2022, 10:29 PM
wow its been like 2.5 years since geo class
by pieMax2713, Feb 4, 2022, 8:38 PM
@violin21, I've been very busy with school lately and haven't been able to add another lesson. I will when i get a free moment
by sonone, Aug 19, 2021, 12:45 AM
ORZ CODER
by samrocksnature, Aug 9, 2021, 9:57 PM
Could you make more Asymptote lessons on your "How to do Asymptote" blog?
by violin21, Aug 9, 2021, 7:26 PM
You can take it, just C&P the CSS into your CSS area
by sonone, Apr 17, 2021, 10:08 PM
how can we take the CSS if we have permission to not take it?
by GoogleNebula, Apr 17, 2021, 5:22 PM
That is awesome!
by sonone, Apr 15, 2021, 10:09 PM
I modified your dodecahedron and got:
[asy]
import three;
import solids;
size(300);
currentprojection=orthographic(0,1.3,1.2);
light(0,5,10);

real phi=(sqrt(6)+1)/3;
real g=(phi-1)/2;
real s=1/2;
real a=sqrt(1-phi*phi/4-g*g)+phi/2;

triple[] d;
d[0]=(phi
by Andrew2019, Mar 26, 2021, 12:15 AM
Not too many, just changing the color here and there. I really like your CSS!
by sonone, Feb 2, 2021, 10:35 AM
Nice!

I see you're making changes to the CSS.
by G.G.Otto, Feb 1, 2021, 9:26 PM
I'm learning Java now!
by sonone, Feb 1, 2021, 5:56 PM
And I took part of it from CaptainFlint and then added a ton of modifications.
by G.G.Otto, Dec 1, 2020, 8:56 AM
fixed $\text{[math]}$
by sonone, Apr 25, 2020, 8:43 PM
Change in progress...
by sonone, Apr 25, 2020, 8:42 PM
When it says "At least 11 users clicked that button" it's incorrect since one person could have just posted 11 times. It should say "At most 11 users click that button".
by G.G.Otto, Apr 25, 2020, 8:26 PM
see this PM
by sonone, Apr 25, 2020, 7:03 PM
I can't get the cursor to work.
by G.G.Otto, Apr 25, 2020, 6:38 PM
If the cursor is confusing, where you are actually pointing in a front of the ship
by sonone, Apr 25, 2020, 11:04 AM
It's ok $\text{[math]}$
by sonone, Apr 23, 2020, 5:30 PM
Sorry I haven't been posting. I've got a Research Paper (for Roman History) due in a week.
by G.G.Otto, Apr 22, 2020, 11:15 PM
You should download it, as it's really useful when learning python. You can find out more here
by G.G.Otto, Apr 19, 2020, 10:04 PM
No. I don't have IDLE.
by sonone, Apr 18, 2020, 7:49 PM
Do you have IDLE installed on your computer? If so, I can give you a couple really cool codes that made that don't work in the AoPS widgets.
by G.G.Otto, Apr 14, 2020, 6:47 AM
I don't know how to change it....
by G.G.Otto, Apr 5, 2020, 5:58 PM
Do you know how the change the into thing above my avatar a different color? It's too hard to see.
by sonone, Apr 5, 2020, 2:08 PM
WELCOME TO MY BLOG!
by sonone, Mar 31, 2020, 4:56 PM