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Combinatorics from EGMO 2018
BarishNamazov   27
N an hour ago by HamstPan38825
Source: EGMO 2018 P3
The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$. After the competition, they queue in front of the restaurant according to the following rules.
[list]
[*]The Jury chooses the initial order of the contestants in the queue.
[*]Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.
[list]
[*]If contestant $C_i$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.
[*]If contestant $C_i$ has fewer than $i$ other contestants in front of her, the restaurant opens and process ends.
[/list]
[/list]
[list=a]
[*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices.
[*]Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
[/list]
27 replies
BarishNamazov
Apr 11, 2018
HamstPan38825
an hour ago
Do you have any idea why they all call their problems' characters "Mykhailo"???
mshtand1   1
N an hour ago by sarjinius
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 10.7
In a row, $1000$ numbers \(2\) and $2000$ numbers \(-1\) are written in some order.
Mykhailo counted the number of groups of adjacent numbers, consisting of at least two numbers, whose sum equals \(0\).
(a) Find the smallest possible value of this number.
(b) Find the largest possible value of this number.

Proposed by Anton Trygub
1 reply
+1 w
mshtand1
Mar 14, 2025
sarjinius
an hour ago
Polynomial divisible by x^2+1
Miquel-point   2
N 2 hours ago by lksb
Source: Romanian IMO TST 1981, P1 Day 1
Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]is divisible by $X^2+1$.

Mircea Becheanu
2 replies
Miquel-point
Apr 6, 2025
lksb
2 hours ago
D1030 : An inequalitie
Dattier   1
N 2 hours ago by lbh_qys
Source: les dattes à Dattier
Let $0<a<b<c<d$ reals, and $n \in \mathbb N^*$.

Is it true that $a^n(b-a)+b^n(c-b)+c^n(d-c) \leq \dfrac {d^{n+1}}{n+1}$ ?
1 reply
Dattier
Yesterday at 7:17 PM
lbh_qys
2 hours ago
IGO 2021 P1
SPHS1234   14
N 3 hours ago by LeYohan
Source: igo 2021 intermediate p1
Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point
$E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that
$BE \perp HD$.

Proposed by Tran Quang Hung - Vietnam
14 replies
SPHS1234
Dec 30, 2021
LeYohan
3 hours ago
Nationalist Combo
blacksheep2003   16
N 3 hours ago by Martin2001
Source: USEMO 2019 Problem 5
Let $\mathcal{P}$ be a regular polygon, and let $\mathcal{V}$ be its set of vertices. Each point in $\mathcal{V}$ is colored red, white, or blue. A subset of $\mathcal{V}$ is patriotic if it contains an equal number of points of each color, and a side of $\mathcal{P}$ is dazzling if its endpoints are of different colors.

Suppose that $\mathcal{V}$ is patriotic and the number of dazzling edges of $\mathcal{P}$ is even. Prove that there exists a line, not passing through any point in $\mathcal{V}$, dividing $\mathcal{V}$ into two nonempty patriotic subsets.

Ankan Bhattacharya
16 replies
blacksheep2003
May 24, 2020
Martin2001
3 hours ago
subsets of {1,2,...,mn}
N.T.TUAN   10
N 3 hours ago by de-Kirschbaum
Source: USA TST 2005, Problem 1
Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that
(a) each element of $T$ is an $m$-element subset of $S_{m}$;
(b) each pair of elements of $T$ shares at most one common element;
and
(c) each element of $S_{m}$ is contained in exactly two elements of $T$.

Determine the maximum possible value of $m$ in terms of $n$.
10 replies
N.T.TUAN
May 14, 2007
de-Kirschbaum
3 hours ago
Sum and product of digits
Sadigly   4
N 3 hours ago by jasperE3
Source: Azerbaijan NMO 2018
For a positive integer $n$, define $f(n)=n+P(n)$ and $g(n)=n\cdot S(n)$, where $P(n)$ and $S(n)$ denote the product and sum of the digits of $n$, respectively. Find all solutions to $f(n)=g(n)$
4 replies
Sadigly
Sunday at 9:19 PM
jasperE3
3 hours ago
Geometry
smartvong   0
3 hours ago
Source: UM Mathematical Olympiad 2024
Let $P$ be a point inside a triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2, C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
0 replies
smartvong
3 hours ago
0 replies
angles in triangle
AndrewTom   34
N 4 hours ago by happypi31415
Source: BrMO 2012/13 Round 2
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
34 replies
AndrewTom
Feb 1, 2013
happypi31415
4 hours ago
a