MathDB

1

Part of 2021 Romania Team Selection Test

Problems(3)

Romania TST 2021 Day 1 P1

Source:

5/15/2021
Let k>1k>1 be a positive integer. A set SS{} is called good if there exists a colouring of the positive integers with kk{} colours, such that no element from SS{} can be written as the sum of two distinct positive integers having the same colour. Find the greatest positive integer tt{} (in terms of kk{}) for which the set S={a+1,a+2,…,a+t}S=\{a+1,a+2,\ldots,a+t\}is good, for any positive integer aa{}.
combinatoricsSetsRomanian TSTTST
Divisibility

Source: Turkey TST 2014 Day 2 Problem 4

3/12/2014
Find all pairs (m,n)(m,n) of positive odd integers, such that n∣3m+1n \mid 3m+1 and m∣n2+3m \mid n^2+3.
modular arithmeticnumber theory proposednumber theory
Romania TST 2021 Day 3 P1

Source:

6/14/2021
Consider a fixed triangle ABCABC such that AB=AC.AB=AC. Let MM be the midpoint of BC.BC. Let PP be a variable point inside △ABC,\triangle ABC, such that ∠PBC=∠PCA.\angle PBC=\angle PCA. Prove that the sum of the measures of ∠BPM\angle BPM and ∠APC\angle APC is constant.
geometryconstantromaniaRomanian TSTTST