Combo problem

by soryn, Apr 22, 2025, 6:33 AM

The school A has m1 boys and m2 girls, and ,the school B has n1 boys and n2 girls. Each school is represented by one team formed by p students,boys and girls. If f(k) is the number of cases for which,the twice schools has,togheter k girls, fund f(k) and the valute of k, for which f(k) is maximum.

combinatorics

Apple sharing in Iran

by mojyla222, Apr 20, 2025, 4:17 AM

Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).

Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.

combinatorics

Iran 2nd Round

infinite grid

Iran second round 2025-q1

by mohsen, Apr 19, 2025, 10:21 AM

Find all positive integers n>2 such that sum of n and any of its prime divisors is a perfect square.

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

Divisibility on 101 integers

by BR1F1SZ, Aug 9, 2024, 12:31 AM

There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$ , with $1 \leqslant i \leqslant 101$ , $a_i+1$ is a multiple of $a_{i+1}$ . Determine the greatest possible value of the largest of the $101$ numbers.

This post has been edited 2 times. Last edited by BR1F1SZ, Jan 27, 2025, 5:01 PM

number theory

Some nice summations

by amitwa.exe, May 24, 2024, 8:52 AM

Problem 1: $\Omega=\left(\sum_{0\le i\le j\le k}^{\infty} \frac{1}{3^i\cdot4^j\cdot5^k}\right)\left(\mathop{{\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}}}_{i\neq j\neq k}\frac{1}{3^i\cdot3^j\cdot3^k}\right)=?$

This post has been edited 4 times. Last edited by amitwa.exe, Aug 6, 2024, 5:43 AM

A Familiar Point

by v4913, Apr 16, 2023, 10:00 PM

Let $ABC$ be a triangle with circumcircle $\Omega$ . Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$ ). Let $I$ be the incenter of $ABC$ . Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$ , and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$ . Show that the line $IN_a$ , and the lines through the intersections of $\omega_b$ and $\omega_c$ , meet on $\Omega$ .

Looking for the smallest ghost

by Justpassingby, Jan 17, 2022, 9:52 AM

Let $p$ be an odd prime number. Let $S=a_1,a_2,\dots$ be the sequence defined as follows: $a_1=1,a_2=2,\dots,a_{p-1}=p-1$ , and for $n\ge p$ , $a_n$ is the smallest integer greater than $a_{n-1}$ such that in $a_1,a_2,\dots,a_n$ there are no arithmetic progressions of length $p$ . We say that a positive integer is a ghost if it doesn’t appear in $S$ .
What is the smallest ghost that is not a multiple of $p$ ?

Proposed by Guerrero

This post has been edited 2 times. Last edited by Justpassingby, Jan 20, 2022, 1:29 AM
Reason: Added proposer

Iran Team Selection Test 2016

by MRF2017, Jul 15, 2016, 7:46 PM

Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$ .Points $X,Y$ lie on $AB,AC$ ,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$ .Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$ .