Incredible combinatorics problem

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0 replies

jlacosta

May 1, 2025

0 replies

A final attempt to make a combinatorics problem

JARP091 0

9 minutes ago

Source: At the time of posting the problem I do not know the source if any

Let $N$ be a positive integer and consider the set $S = \{1, 2, \ldots, N\}$ .

Two players alternate moves. On each turn, the current player must select a nonempty subset $T \subseteq S$ of numbers not previously chosen such that for every distinct $x, y \in T$ , neither $x$ divides $y$ nor $y$ divides $x$ .

After selecting $T$ , all multiples of every element in $T$ , including those in $T$ itself, are removed from $S$ .

The game continues with the reduced set $S$ until no moves are possible.
Determine, for each $N$ , which player has a winning strategy if any

Note: It might be wrong or maybe too easy.

0 replies

JARP091

9 minutes ago

0 replies

Serbian selection contest for the IMO 2025 - P2

OgnjenTesic 10

N 27 minutes ago by Mahdi_Mashayekhi

Source: Serbian selection contest for the IMO 2025

Let $ABC$ be an acute triangle. Let $A'$ be the reflection of point $A$ over the line $BC$ . Let $O$ and $H$ be the circumcenter and the orthocenter of triangle $ABC$ , respectively, and let $E$ be the midpoint of segment $OH$ . Let $D$ and $L$ be the points where the reflection of line $AA'$ with respect to line $OA'$ intersects the circumcircle of triangle $ABC$ , where point $D$ lies on the arc $BC$ not containing $A$ . If $M$ is a point on the line $BC$ such that $OM \perp AD$ , prove that $\angle MAD = \angle EAL$ .

Proposed by Strahinja Gvozdić

10 replies

OgnjenTesic

Thursday at 4:02 PM

Mahdi_Mashayekhi

27 minutes ago

find all Polynomials

andria 9

N 30 minutes ago by A.H.H

Source: Iranian third round 2015 algebra problem 5

Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$ :
$p(5x)^2-3=p(5x^2+1)$ such that:
$a) p(0)\neq 0$
$b) p(0)=0$

9 replies

andria

Sep 8, 2015

A.H.H

30 minutes ago

AT // BC wanted

parmenides51 105

N 31 minutes ago by Adywastaken

Source: IMO 2019 SL G1

Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$ , meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$ . The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$ . Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$ .

(Nigeria)

105 replies