MathDB

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 (or you leave them all blank), 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 should answer YYNNBY (and receive

12

points if all five answers are correct, 0 points if any are wrong).
(a)

a,b,c,d,A,B,C,

and

D

are positive real numbers such that

\frac{a}{b} > \frac{A}{B}

and

\frac{c}{d} > \frac{C}{D}

. Is it necessarily true that

\frac{a+c}{b+d} > \frac{A+C}{B+D}

?
(b) Do there exist irrational numbers

\alpha

and

\beta

such that the sequence

\lfloor\alpha\rfloor+\lfloor\beta\rfloor, \lfloor2\alpha\rfloor+\lfloor2\beta\rfloor, \lfloor3\alpha\rfloor+\lfloor3\beta\rfloor, \dots

is arithmetic?
(c) For any set of primes

\mathbb{P}

, let

S_\mathbb{P}

denote the set of integers whose prime divisors all lie in

\mathbb{P}

. For instance

S_{\{2,3\}}=\{2^a3^b \; | \; a,b\ge 0\}=\{1,2,3,4,6,8,9,12,\dots\}

. Does there exist a finite set of primes

\mathbb{P}

and integer polynomials

P

and

Q

such that

\gcd(P(x), Q(y))\in S_\mathbb{P}

for all

x,y

?
(d) A function

f

is called P-recursive if there exists a positive integer

m

and real polynomials

p_0(n), p_1(n), \dots, p_m(n)

[color = red], not all zero, satisfying

p_m(n)f(n+m)=p_{m-1}(n)f(n+m-1)+\dots+p_0(n)f(n)

for all

n

. Does there exist a P-recursive function

f

satisfying

\lim_{n\to\infty} \frac{f(n)}{n^{\sqrt{2}}}=1

?
(e) Does there exist a nonpolynomial function

f: \mathbb{Z}\to\mathbb{Z}

such that

a-b

divides

f(a)-f(b)

for all integers

a\neq b

?
(f) Do there exist periodic functions

f, g:\mathbb{R}\to\mathbb{R}

such that

f(x)+g(x)=x

for all

x

?
[color = red]A clarification was issued for problem 33(d) during the test. I have included it above.

Problem Statement