a combinatoric problem

by xyz123456, Apr 18, 2025, 12:31 PM

$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$ $prove\ that:\exists \ c_1{,}c_2>0{,}c_1n^{\frac{6}{5}}<f\left(n\right)<c_2n^{\frac{6}{5}}\ln n$

combinatorics

Geometry Problem

by Itoz, Apr 18, 2025, 11:49 AM

Given $\triangle ABC$ . Let the perpendicular line from $A$ to $BC$ meets $BC,\odot(ABC)$ at points $S,K$ , respectively, and the foot from $B$ to $AC$ is $L$ . $\odot (AKL)$ intersects line $AB$ at $T(\neq A)$ , $\odot(AST)$ intersects line $AC$ at $M(\neq A)$ , and lines $TM,CK$ intersect at $N$ .

Prove that $\odot(CNM)$ is tangent to $\odot (BST)$ .

Attachments:

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

geometry

Quadric function

by soryn, Apr 18, 2025, 2:47 AM

If f(x)=ax^2+bx+c, a,b,c integers, |a|>=3, and M îs the set of integers x for which f(x) is a prime number and f has exactly one integer solution,prove that M has at most three elements.

This post has been edited 1 time. Last edited by soryn, Today at 3:14 AM

function

number theory

Turbo's en route to visit each cell of the board

by Lukaluce, Apr 14, 2025, 11:01 AM

Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$ , the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey

This post has been edited 1 time. Last edited by Lukaluce, Apr 14, 2025, 11:54 AM

one cyclic formed by two cyclic

by CrazyInMath, Apr 13, 2025, 12:38 PM

Let $ABC$ be an acute triangle. Points $B, D, E$ , and $C$ lie on a line in this order and satisfy $BD = DE = EC$ . Let $M$ and $N$ be the midpoints of $AD$ and $AE$ , respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$ , respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.

geometry

EGMO 2025

Weird function?

by ItzsleepyXD, Apr 11, 2025, 12:43 PM

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ ,
$f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).$

function

algebra

functional equation

Inequality with a,b,c,d

by GeoMorocco, Apr 9, 2025, 1:35 PM

Let $a,b,c,d$ positive real numbers such that $a+b+c+d=3+\frac{1}{abcd}$ . Prove that :
$a^2+b^2+c^2+d^2+5abcd \geq 9$

inequalities

a combinatorial geometry problem

by xyz123456, Mar 3, 2025, 12:41 PM

$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$

combinatorics

combinatorial geometry

geometry

Circles tangent to AD and AB intersect on AC

by gghx, Aug 3, 2024, 2:33 AM

In an acute triangle $ABC$ , $AC>AB$ , $D$ is the point on $BC$ such that $AD=AB$ . Let $\omega_1$ be the circle through $C$ tangent to $AD$ at $D$ , and $\omega_2$ the circle through $C$ tangent to $AB$ at $B$ . Let $F(\ne C)$ be the second intersection of $\omega_1$ and $\omega_2$ . Prove that $F$ lies on $AC$ .

geometry

Shortlist 2017/G4

by fastlikearabbit, Jul 10, 2018, 11:02 AM

In triangle $ABC$ , let $\omega$ be the excircle opposite to $A$ . Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$ , and $AB$ , respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$ . Let $M$ be the midpoint of $AD$ . Prove that the circle $MPQ$ is tangent to $\omega$ .

This post has been edited 1 time. Last edited by Amir Hossein, Jul 16, 2018, 4:01 AM
Reason: No LIME

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hi guys $~~~$
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An Introduction to the Theory of Mathematics

by xyz123456, Apr 18, 2025, 12:31 PM

by Itoz, Apr 18, 2025, 11:49 AM

by soryn, Apr 18, 2025, 2:47 AM

by Lukaluce, Apr 14, 2025, 11:01 AM

by CrazyInMath, Apr 13, 2025, 12:38 PM

by ItzsleepyXD, Apr 11, 2025, 12:43 PM

by GeoMorocco, Apr 9, 2025, 1:35 PM

by xyz123456, Mar 3, 2025, 12:41 PM

by gghx, Aug 3, 2024, 2:33 AM

by fastlikearabbit, Jul 10, 2018, 11:02 AM