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Random Points = Problem
kingu   5
N 6 minutes ago by happypi31415
Source: Chinese Geometry Handout
Let $ABC$ be a triangle. Let $\omega$ be a circle passing through $B$ intersecting $AB$ at $D$ and $BC$ at $F$. Let $G$ be the intersection of $AF$ and $\omega$. Further, let $M$ and $N$ be the intersections of $FD$ and $DG$ with the tangent to $(ABC)$ at $A$. Now, let $L$ be the second intersection of $MC$ and $(ABC)$. Then, prove that $M$ , $L$ , $D$ , $E$ and $N$ are concyclic.
5 replies
kingu
Apr 27, 2024
happypi31415
6 minutes ago
Very odd geo
Royal_mhyasd   2
N an hour ago by Royal_mhyasd
Source: own (i think)
Let $\triangle ABC$ be an acute triangle with $AC>AB>BC$ and let $H$ be its orthocenter. Let $P$ be a point on the perpendicular bisector of $AH$ such that $\angle APH=2(\angle ABC - \angle ACB)$ and $P$ and $C$ are on different sides of $AB$, $Q$ a point on the perpendicular bisector of $BH$ such that $\angle BQH = 2(\angle ACB-\angle BAC)$ and $R$ a point on the perpendicular bisector of $CH$ such that $\angle CRH=2(\angle ABC - \angle BAC)$ and $Q,R$ lie on the opposite side of $BC$ w.r.t $A$. Prove that $P,Q$ and $R$ are collinear.
2 replies
Royal_mhyasd
Yesterday at 6:10 PM
Royal_mhyasd
an hour ago
Polynomial Application Sequences and GCDs
pieater314159   46
N an hour ago by cursed_tangent1434
Source: ELMO 2019 Problem 1, 2019 ELMO Shortlist N1
Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$.

Proposed by Milan Haiman and Carl Schildkraut
46 replies
1 viewing
pieater314159
Jun 19, 2019
cursed_tangent1434
an hour ago
c^a + a = 2^b
Havu   10
N an hour ago by Havu
Find $a, b, c\in\mathbb{Z}^+$ such that $a,b,c$ coprime, $a + b = 2c$ and $c^a + a = 2^b$.
10 replies
1 viewing
Havu
May 10, 2025
Havu
an hour ago
Own made functional equation
JARP091   0
2 hours ago
Source: Own (Maybe?)
\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]
0 replies
JARP091
2 hours ago
0 replies
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   16
N 2 hours ago by JARP091
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
16 replies
OgnjenTesic
May 22, 2025
JARP091
2 hours ago
equal segments on radiuses
danepale   8
N 2 hours ago by zuat.e
Source: Croatia TST 2016
Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.
8 replies
danepale
Apr 25, 2016
zuat.e
2 hours ago
Inequality
SunnyEvan   8
N 2 hours ago by arqady
Let $a$, $b$, $c$ be non-negative real numbers, no two of which are zero. Prove that :
$$ \sum \frac{3ab-2bc+3ca}{3b^2+bc+3c^2} \geq \frac{12}{7}$$
8 replies
SunnyEvan
Apr 1, 2025
arqady
2 hours ago
Inequality conjecture
RainbowNeos   2
N 2 hours ago by RainbowNeos
Show (or deny) that there exists an absolute constant $C>0$ that, for all $n$ and $n$ positive real numbers $x_i ,1\leq i \leq n$, there is
\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]
2 replies
RainbowNeos
May 29, 2025
RainbowNeos
2 hours ago
2- player game on a strip of n squares with two game pieces
parmenides51   2
N 2 hours ago by Gggvds1
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 3
Alice and Bob play a game on a strip of $n \ge  3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of $ n = 7$ squares.
IMAGE
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.

For which $n$ can Bob ensure a win no matter how Alice plays?
For which $n$ can Alice ensure a win no matter how Bob plays?

(Karl Czakler)
2 replies
parmenides51
Mar 26, 2024
Gggvds1
2 hours ago
Incenters and Circles
rkm0959   6
N 2 hours ago by happypi31415
Source: Korean National Junior Olympiad Problem 1
In a triangle $\triangle ABC$ with incenter $I$,
Let $D$ = $AI$ $\cap$ $BC$
$E$ = incenter of $\triangle ABD$
$F$ = incenter of $\triangle ACD$
$P$ = intersection of $\odot BCE$ and $\overline {ED}$
$Q$ = intersection of $\odot BCF$ and $\overline {FD}$
$M$ = midpoint of $\overline {BC}$

Prove that $D, M, P, Q$ concycle
6 replies
rkm0959
Nov 2, 2014
happypi31415
2 hours ago
Reflected point lies on radical axis
Mahdi_Mashayekhi   6
N 2 hours ago by khanhnx
Source: Iran 2025 second round P4
Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.
6 replies
Mahdi_Mashayekhi
Apr 19, 2025
khanhnx
2 hours ago
Question on Balkan SL
Fmimch   4
N 3 hours ago by BreezeCrowd
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
4 replies
Fmimch
Apr 30, 2025
BreezeCrowd
3 hours ago
Question on Balkan SL
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Fmimch
4 posts
#1
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Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
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Fmimch
4 posts
#2
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bump...please help, thank you
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Assassino9931
1382 posts
#3
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I have posted almost all problems (some users posted the other ones), someone needs to request a collection.
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GreekIdiot
277 posts
#4 • 3 Y
Y by NO_SQUARES, cursed_tangent1434, BreezeCrowd
https://drive.google.com/file/d/15S5byn0y4RLfceYbc-8XA9BItE3rz49H/view?usp=sharing
This post has been edited 1 time. Last edited by GreekIdiot, Apr 30, 2025, 12:05 PM
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BreezeCrowd
5 posts
#5
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Do you have 2022 and 2023 Shortlist?
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