MathDB

3

Part of 2017 China Team Selection Test

Problems(5)

2017 China TSTST 1 Day 1 Problem 3

Source: 2017 China TSTST 1 Day 1 Problem 3

3/7/2017
Suppose S={1,2,3,...,2017}S=\{1,2,3,...,2017\},for every subset AA of SS,define a real number f(A)≥0f(A)\geq 0 such that: (1)(1) For any A,B⊂SA,B\subset S,f(A∪B)+f(A∩B)≤f(A)+f(B)f(A\cup B)+f(A\cap B)\leq f(A)+f(B); (2)(2) For any A⊂B⊂SA\subset B\subset S, f(A)≤f(B)f(A)\leq f(B); (3)(3) For any k,j∈Sk,j\in S,f({1,2,…,k+1})≥f({1,2,…,k}∪{j});f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\}); (4)(4) For the empty set ∅\varnothing, f(∅)=0f(\varnothing)=0. Confirm that for any three-element subset TT of SS,the inequality f(T)≤2719f({1,2,3})f(T)\leq \frac{27}{19}f(\{1,2,3\}) holds.
set theorySubsetscombinatoricsinequalitiesalgebra
Coaxal Circles

Source: China TSTST Test 2 Day 1 Q3

3/13/2017
Let ABCDABCD be a quadrilateral and let ll be a line. Let ll intersect the lines AB,CD,BC,DA,AC,BDAB,CD,BC,DA,AC,BD at points X,X′,Y,Y′,Z,Z′X,X',Y,Y',Z,Z' respectively. Given that these six points on ll are in the order X,Y,Z,X′,Y′,Z′X,Y,Z,X',Y',Z', show that the circles with diameter XX′,YY′,ZZ′XX',YY',ZZ' are coaxal.
geometrycirclescoaxal
Finding a chain within subsets

Source: China TSTST 3 Day 1 Problem 3

3/17/2017
Let XX be a set of 100100 elements. Find the smallest possible nn satisfying the following condition: Given a sequence of nn subsets of XX, A1,A2,…,AnA_1,A_2,\ldots,A_n, there exists 1≤i<j<k≤n1 \leq i < j < k \leq n such that Ai⊆Aj⊆Ak or Ai⊇Aj⊇Ak.A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.
combinatoricsSetsChina TST
a problem of ordered array from China TST

Source: 2017 China TST 4 Problem 3

3/22/2017
Find the numbers of ordered array (x1,...,x100)(x_1,...,x_{100}) that satisfies the following conditions: (ii)x1,...,x100∈{1,2,..,2017}x_1,...,x_{100}\in\{1,2,..,2017\}; (iiii)2017∣x1+...+x1002017|x_1+...+x_{100}; (iiiiii)2017∣x12+...+x10022017|x_1^2+...+x_{100}^2.
combinatoricsset theoryQuadratic ResiduesChina TST
2017 blue points and 58 red points

Source: 2017 China TST 5 P3

4/8/2017
For a rational point (x,y), if xy is an integer that divided by 2 but not 3, color (x,y) red, if xy is an integer that divided by 3 but not 2, color (x,y) blue. Determine whether there is a line segment in the plane such that it contains exactly 2017 blue points and 58 red points.
combinatorics