MathDB

2

Part of 2014 Danube Mathematical Competition

Problems(2)

at least 806 palindrome words to stick for a palindrome of 2014 letters

Source: Danube 2014 junior p2

7/22/2019
We call word a sequence of letters l1l2...ln‾,n≥1\overline {l_1l_2...l_n}, n\ge 1 . A word l1l2...ln‾,n≥1\overline {l_1l_2...l_n}, n\ge 1 is called palindrome if lk=ln−k+1l_k=l_{n-k+1} , for any k,1≤k≤nk, 1 \le k \le n. Consider a word X=l1l2...l2014‾X=\overline {l_1l_2...l_{2014}} in which lk∈{A,B} l_k\in\{A,B\} , for any k,1≤k≤2014k, 1\le k \le 2014. Prove that there are at least 806806 palindrome words to ''stick" together to get word XX.
palindromescombinatorics
set with \lfloor x\rfloor =\lfloor y\rfloor , for any x,y

Source: Danube 2014 p2

7/22/2019
Let SS be a set of positive integers such that ⌊x⌋=⌊y⌋\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor for all x,y∈Sx, y \in S. Show that the products xyxy, where x,y∈Sx, y \in S, are pairwise distinct.
floor functionsetalgebraProduct