MathDB

4

Part of 2016 China Team Selection Test

Problems(3)

Sequence with Primitive Prime Factor

Source: China 2016 TST Day 2 Q4

3/16/2016
Let c,d≥2c,d \geq 2 be naturals. Let {an}\{a_n\} be the sequence satisfying a1=c,an+1=and+ca_1 = c, a_{n+1} = a_n^d + c for n=1,2,⋯n = 1,2,\cdots. Prove that for any n≥2n \geq 2, there exists a prime number pp such that p∣anp|a_n and p∤aip \not | a_i for i=1,2,⋯n−1i = 1,2,\cdots n-1.
number theorySequenceprime numbers
Product of f(m) multiple of odd integers

Source: China Team Selection Test 2016 Test 2 Day 2 Q4

3/21/2016
Set positive integer m=2k⋅tm=2^k\cdot t, where kk is a non-negative integer, tt is an odd number, and let f(m)=t1−kf(m)=t^{1-k}. Prove that for any positive integer nn and for any positive odd number a≤na\le n, ∏m=1nf(m)\prod_{m=1}^n f(m) is a multiple of aa.
number theoryfloor functionHi
Congruency in sum of digits base q

Source: China Team Selection Test 2016 Test 3 Day 2 Q4

3/26/2016
Let a,b,b′,c,m,qa,b,b',c,m,q be positive integers, where m>1,q>1,∣b−b′∣≥am>1,q>1,|b-b'|\ge a. It is given that there exist a positive integer MM such that Sq(an+b)≡Sq(an+b′)+c(modm)S_q(an+b)\equiv S_q(an+b')+c\pmod{m}
holds for all integers n≥Mn\ge M. Prove that the above equation is true for all positive integers nn. (Here Sq(x)S_q(x) is the sum of digits of xx taken in base qq).
number theory