A final attempt to make a combinatorics problem

by JARP091, May 24, 2025, 5:27 AM

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.

This post has been edited 1 time. Last edited by JARP091, 2 hours ago

combinatorics

combinatorics proposed

combinatorics unsolved

FE with conditions on $x,y$

by Adywastaken, May 23, 2025, 6:18 PM

Find all functions $f:\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ such that $\forall y>x>0$ ,
$f(x^2+f(y))=f(xf(x))+y$

This post has been edited 1 time. Last edited by Adywastaken, an hour ago

function

algebra

positive

Simple Geometry

by AbdulWaheed, May 23, 2025, 5:15 AM

Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle $\Omega$ . Let $X$ be the midpoint of the arc $\overarc{BC}$ not containing $A$ and define $Y, Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ of $ABC$ .

Angle Chase

geometry

Serbian selection contest for the IMO 2025 - P6

by OgnjenTesic, May 22, 2025, 4:07 PM

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

number theory

Serbian selection contest for the IMO 2025 - P2

by OgnjenTesic, May 22, 2025, 4:02 PM

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

This post has been edited 1 time. Last edited by OgnjenTesic, Thursday at 4:24 PM
Reason: Last sentence

geometry

2021 EGMO P4: Reflection of A over EF lies on BC

by anser, Apr 13, 2021, 12:03 PM

Let $ABC$ be a triangle with incenter $I$ and let $D$ be an arbitrary point on the side $BC$ . Let the line through $D$ perpendicular to $BI$ intersect $CI$ at $E$ . Let the line through $D$ perpendicular to $CI$ intersect $BI$ at $F$ . Prove that the reflection of $A$ across the line $EF$ lies on the line $BC$ .

This post has been edited 1 time. Last edited by v_Enhance, Apr 18, 2021, 2:48 AM
Reason: "lines on the line" -> "lies on the line"

Two lines concur on (ABC)

by amar_04, Mar 2, 2021, 3:58 AM

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$ , point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$ . The excircle opposite to $A$ touches $\overline{BC}$ at point $E$ . Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$ . Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$ .

Sharygin Geometry Olympiad

anant mudgal geo

AT // BC wanted

by parmenides51, Sep 22, 2020, 11:26 PM

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)

This post has been edited 1 time. Last edited by parmenides51, Sep 22, 2020, 11:27 PM

geometry

IMO Shortlist

find all Polynomials

by andria, Sep 8, 2015, 1:13 PM

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:
$a) p(0)\neq 0$
$b) p(0)=0$

Less than or equal to 30°

by orl, Nov 11, 2005, 6:27 PM

Let $\,ABC\,$ be a triangle and $\,P\,$ an interior point of $\,ABC\,$ . Show that at least one of the angles $\,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $30^{\circ }$ .

This post has been edited 1 time. Last edited by orl, Aug 15, 2008, 1:39 PM

Submit

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PLEASE CONTRIB

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by asf, Sep 11, 2010, 5:18 PM

91 shouts

math_exploration

by JARP091, May 24, 2025, 5:27 AM

by Adywastaken, May 23, 2025, 6:18 PM

by AbdulWaheed, May 23, 2025, 5:15 AM

by OgnjenTesic, May 22, 2025, 4:07 PM

by OgnjenTesic, May 22, 2025, 4:02 PM

by anser, Apr 13, 2021, 12:03 PM

by amar_04, Mar 2, 2021, 3:58 AM

by parmenides51, Sep 22, 2020, 11:26 PM

by andria, Sep 8, 2015, 1:13 PM

by orl, Nov 11, 2005, 6:27 PM