6
Part of 2015 China Team Selection Test
Problems(2)
Ping Pong and directed graph
Source: 2015 China Tst 1 Day 2 Q3
3/14/2015
There are some players in a Ping Pong tournament, where every players play with each other at most once. Given:
\$$1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$)
\$$2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$).
\\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
combinatoricscombinatorics proposedChina TST2015China
Squarefree Numbers
Source: 2015 China TST 3 Problem 3
3/19/2015
Prove that there exist infinitely many integers such that is squarefree.
number theoryAnalytic Number Theorysquarefree