Ant wanna come to A

by Rohit-2006, Apr 15, 2025, 6:47 PM

An insect starts from $A$ and in $10$ steps and has to reach $A$ again. But in between one of the s steps and can't go $A$ . Find probability. For example $ABCDCDEDEA$ is valid but $ABCDEDEDEA$ is not valid.

Attachments:

This post has been edited 2 times. Last edited by Rohit-2006, 3 hours ago

combinatorics

quadratic with at least 1 roots

by giangtruong13, Apr 15, 2025, 3:56 PM

Find $m$ to satisfy that the equation $x^2+mx-1=0$ has at least 1 roots $\leq -2$

quadratics

number theory

Divisibility NT FE

by CHESSR1DER, Apr 14, 2025, 7:07 PM

Find all functions $f$ $N \rightarrow N$ such for any $a,b$ :
$(a+b)|a^{f(b)} + b^{f(a)}$ .

This post has been edited 3 times. Last edited by CHESSR1DER, 3 hours ago

number theory

functional equation

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, Yesterday at 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

Prove that x1=x2=....=x2025

by Rohit-2006, Apr 9, 2025, 5:22 AM

The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$ Where { $\bar{x_1},\cdots,\bar{x_{2025}}$ } is a permutation of $x_1,x_2,\cdots,x_{2025}$ . Prove that $x_1=x_2=\cdots=x_{2025}$

This post has been edited 1 time. Last edited by Rohit-2006, Apr 9, 2025, 5:23 AM

algebra

Number Theory Chain!

by JetFire008, Apr 7, 2025, 7:14 AM

I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1

This post has been edited 1 time. Last edited by JetFire008, Apr 7, 2025, 7:14 AM

number theory

marathon

Squares in an Octagon

by kred9, Apr 5, 2025, 11:50 PM

A regular octagon and all of its diagonals are drawn. Find, with proof, the number of squares that appear in the resulting diagram. (The side of each square must lie along one of the edges or diagonals of the octagon.)

geometry

all solutions of (p,n)

by Sayan, Feb 14, 2012, 3:56 PM

Determine all solutions $(p,n)$ of the equation
$n^3=p^2-p-1$
where $p$ is a prime number and $n$ is an integer

number theory proposed

number theory

IMO 2011 Problem 4

by Amir Hossein, Jul 19, 2011, 11:54 AM

Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1, \cdots, 2^{n-1}$ . We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.

Proposed by Morteza Saghafian, Iran