MathDB

3

Part of 2016 China Team Selection Test

Problems(3)

Infinite set and primes

Source: China Team Selection Test 2016 Test 2 Day 1 Q3

3/20/2016
Let PP be a finite set of primes, AA an infinite set of positive integers, where every element of AA has a prime factor not in PP. Prove that there exist an infinite subset BB of AA, such that the sum of elements in any finite subset of BB has a prime factor not in PP.
number theory
Maximal Proper Subset Closed under Operations

Source: China Team Selection Test 2016 Day 1 Q3

3/15/2016
Let n≥2n \geq 2 be a natural. Define X={(a1,a2,⋯ ,an)∣ak∈{0,1,2,⋯ ,k},k=1,2,⋯ ,n}X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}. For any two elements s=(s1,s2,⋯ ,sn)∈X,t=(t1,t2,⋯ ,tn)∈Xs = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X, define s∨t=(max⁡{s1,t1},max⁡{s2,t2},⋯ ,max⁡{sn,tn})s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} ) s∧t=(min⁡{s1,t1},min⁡{s2,t2,},⋯ ,min⁡{sn,tn})s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\}) Find the largest possible size of a proper subset AA of XX such that for any s,t∈As,t \in A, one has s∨t∈A,s∧t∈As \vee t \in A, s \wedge t \in A.
combinatoricsnumber theory
Two points are reflections across diagonal

Source: China Team Selection Test 2016 Test 3 Day 1 Q3

3/25/2016
In cyclic quadrilateral ABCDABCD, AB>BCAB>BC, AD>DCAD>DC, I,JI,J are the incenters of △ABC\triangle ABC,△ADC\triangle ADC respectively. The circle with diameter ACAC meets segment IBIB at XX, and the extension of JDJD at YY. Prove that if the four points B,I,J,DB,I,J,D are concyclic, then X,YX,Y are the reflections of each other across ACAC.
geometrycyclic quadrilateralincenter