MathDB

6

Part of 2018 China Team Selection Test

Problems(4)

2018 China TST 1 Day 2 Q3

Source: China Wuhan . Dec 31, 2017

1/2/2018
Let A1A_1, A2A_2, ⋯\cdots, AmA_m be mm subsets of a set of size nn. Prove that ∑i=1m∑j=1m∣Ai∣⋅∣Ai∩Aj∣≥1mn(∑i=1m∣Ai∣)3. \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.
combinatoricsinequalitiesChina TST
Iterated functional equation with divisibility condition

Source: 2018 China TST 2 Day 2 Q3

1/9/2018
Let M,a,b,rM,a,b,r be non-negative integers with a,r≥2a,r\ge 2, and suppose there exists a function f:Z→Zf:\mathbb{Z}\rightarrow\mathbb{Z} satisfying the following conditions: (1) For all n∈Zn\in \mathbb{Z}, f(r)(n)=an+bf^{(r)}(n)=an+b where f(r)f^{(r)} denotes the composition of rr copies of ff (2) For all n≥Mn\ge M, f(n)≥0f(n)\ge 0 (3) For all n>m>Mn>m>M, n−m∣f(n)−f(m)n-m|f(n)-f(m) Show that aa is a perfect rr-th power.
functionDivisibilityIterationalgebrafunctional equation
2018 China TST 3 Day 2 Q3

Source: Mar 21, 2018

3/27/2018
Find all pairs of positive integers (x,y)(x, y) such that (xy+1)(xy+x+2)(xy+1)(xy+x+2) be a perfect square .
number theoryalgebraChina TSTPell equations
The set of 3-Tuple Subscript

Source: 2018 China TST 4 Day 2 Problem 6

3/27/2018
Suppose ai,bi,ci,i=1,2,⋯ ,na_i, b_i, c_i, i=1,2,\cdots ,n, are 3n3n real numbers in the interval [0,1].\left [ 0,1 \right ]. Define S={(i,j,k)∣ ai+bj+ck<1},    T={(i,j,k)∣ ai+bj+ck>2}.S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}. Now we know that ∣S∣≥2018, ∣T∣≥2018.\left | S \right |\ge 2018,\, \left | T \right |\ge 2018. Try to find the minimal possible value of nn.
combinatoricsalgebraic combinatoricsalgebracombinatorics unsolved