HMMT 二月 2017 · 冲刺赛 · 第 33 题
HMMT February 2017 — Guts Round — Problem 33
题目详情
- [ 20 ] Welcome to the USAYNO , where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer n problems and get them all correct, you will receive max(0 , ( n − 1)( n − 2)) points. If any of them are wrong, you will receive 0 points. Your answer should be a six-character string containing ’Y’ (for yes), ’N’ (for no), or ’B’ (for blank). For instance if you think 1, 2, and 6 are ’yes’ and 3 and 4 are ’no’, you would answer YYNNBY (and receive 12 points if all five answers are correct, 0 points if any are wrong). a A c C (a) a, b, c, d, A, B, C, and D are positive real numbers such that > and > . Is it necessarily b B d D a + c A + C true that > ? b + d B + D (b) Do there exist irrational numbers α and β such that the sequence b α c + b β c , b 2 α c + b 2 β c , b 3 α c + b 3 β c , . . . is arithmetic? (c) For any set of primes P , let S denote the set of integers whose prime divisors all lie in P . For P a b instance S = { 2 3 | a, b ≥ 0 } = { 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , . . . } . Does there exist a finite set of { 2 , 3 } primes P and integer polynomials P and Q such that gcd( P ( x ) , Q ( y )) ∈ S for all x, y ? P (d) A function f is called P-recursive if there exists a positive integer m and real polynomials p ( n ) , p ( n ) , . . . , p ( n ) satisfying 0 1 m p ( n ) f ( n + m ) = p ( n ) f ( n + m − 1) + . . . + p ( n ) f ( n ) m m − 1 0 f ( n ) √ for all n . Does there exist a P-recursive function f satisfying lim = 1? n →∞ 2 n (e) Does there exist a nonpolynomial function f : Z → Z such that a − b divides f ( a ) − f ( b ) for all integers a 6 = b ? (f) Do there exist periodic functions f, g : R → R such that f ( x ) + g ( x ) = x for all x ?
解析
- [ 20 ] Welcome to the USAYNO , where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer n problems and get them all correct, you will receive max(0 , ( n − 1)( n − 2)) points. If any of them are wrong, you will receive 0 points. Your answer should be a six-character string containing ’Y’ (for yes), ’N’ (for no), or ’B’ (for blank). For instance if you think 1, 2, and 6 are ’yes’ and 3 and 4 are ’no’, you would answer YYNNBY (and receive 12 points if all five answers are correct, 0 points if any are wrong). a A c C (a) a, b, c, d, A, B, C, and D are positive real numbers such that > and > . Is it necessarily b B d D a + c A + C true that > ? b + d B + D (b) Do there exist irrational numbers α and β such that the sequence b α c + b β c , b 2 α c + b 2 β c , b 3 α c + b 3 β c , . . . is arithmetic? (c) For any set of primes P , let S denote the set of integers whose prime divisors all lie in P . For P a b instance S = { 2 3 | a, b ≥ 0 } = { 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , . . . } . Does there exist a finite set of { 2 , 3 } primes P and integer polynomials P and Q such that gcd( P ( x ) , Q ( y )) ∈ S for all x, y ? P (d) A function f is called P-recursive if there exists a positive integer m and real polynomials p ( n ) , p ( n ) , . . . , p ( n ) satisfying 0 1 m p ( n ) f ( n + m ) = p ( n ) f ( n + m − 1) + . . . + p ( n ) f ( n ) m m − 1 0 f ( n ) √ for all n . Does there exist a P-recursive function f satisfying lim = 1? n →∞ 2 n (e) Does there exist a nonpolynomial function f : Z → Z such that a − b divides f ( a ) − f ( b ) for all integers a 6 = b ? (f) Do there exist periodic functions f, g : R → R such that f ( x ) + g ( x ) = x for all x ? Proposed by: Alexander Katz Answer: NNNYYY