BOMC-2012 Test Day 2

[Tahmid Hasan]

I shouldn't have written that, please delete my commnt if it creates any confusions.

[nayel]

Here is an example of partition: a triangle is partitioned into $6$ triangles by its medians.

For the rest of the questions: I am not going to answer them. Do what you think is right. The problem statements are clear enough.

[Phlembac Adib Hasan]

নায়েল ভাই, প্রবলেম 3 নিয়ে একটা প্রশ্ন। যেকোন সুষম বহুভুজেই তো যদি আমরা কিছুই না করি তবে সেটাকেও একটা পার্টিশন হিসেবে ভাবা যায় যেখানে মাত্র একটা বহুভুজই তৈরি হয়েছে।এই কেসটাও কি হিসাবের মাঝে আসবে ?

[nafistiham]

I think all the problems are now clear enough.The topic can be locked to stop further discussion.

[*Mahi*]

Problem 1:
$a_0,a_1,\ldots$ and $b_0,b_1,\ldots$ are arithmetic progressions of integers such that $a_1-a_0$ and $b_1-b_0$ share no common factor greater than $1$. Prove that for any integer $n$ there exist $i,j$ such that $a_i-b_j=n$.

Problem 2:
Find all finite sets $S$ of nonnegative integers with the property that for any $m,n\in S$ we have $|m-n+1|\in S$.

Problem 3:
Find all ordered pairs $(n,m)$ of positive integers $\ge 3$ such that a regular $n$-gon can be partitioned into some congruent regular $m$-gons.

Discussion threads:

Problem 1: http://www.matholympiad.org.bd/forum/vi ... =14&t=2020

Problem 2: http://www.matholympiad.org.bd/forum/vi ... =14&t=2021

Problem 3: http://www.matholympiad.org.bd/forum/vi ... =14&t=2022