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2017 Harvard-MIT Mathematics Tournament
33
33
Part of
2017 Harvard-MIT Mathematics Tournament
Problems
(1)
2017 Guts #33: USAYNO Algebra
Source:
2/21/2017
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
n
n
problems and get them all correct, you will receive
max
(
0
,
(
n
−
1
)
(
n
−
2
)
)
\max(0, (n-1)(n-2))
max
(
0
,
(
n
−
1
)
(
n
−
2
))
points. If any of them are wrong (or you leave them all blank), you will receive
0
0
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
12
12
points if all five answers are correct, 0 points if any are wrong).(a)
a
,
b
,
c
,
d
,
A
,
B
,
C
,
a,b,c,d,A,B,C,
a
,
b
,
c
,
d
,
A
,
B
,
C
,
and
D
D
D
are positive real numbers such that
a
b
>
A
B
\frac{a}{b} > \frac{A}{B}
b
a
>
B
A
and
c
d
>
C
D
\frac{c}{d} > \frac{C}{D}
d
c
>
D
C
. Is it necessarily true that
a
+
c
b
+
d
>
A
+
C
B
+
D
\frac{a+c}{b+d} > \frac{A+C}{B+D}
b
+
d
a
+
c
>
B
+
D
A
+
C
?(b) Do there exist irrational numbers
α
\alpha
α
and
β
\beta
β
such that the sequence
⌊
α
⌋
+
⌊
β
⌋
,
⌊
2
α
⌋
+
⌊
2
β
⌋
,
⌊
3
α
⌋
+
⌊
3
β
⌋
,
…
\lfloor\alpha\rfloor+\lfloor\beta\rfloor, \lfloor2\alpha\rfloor+\lfloor2\beta\rfloor, \lfloor3\alpha\rfloor+\lfloor3\beta\rfloor, \dots
⌊
α
⌋
+
⌊
β
⌋
,
⌊
2
α
⌋
+
⌊
2
β
⌋
,
⌊
3
α
⌋
+
⌊
3
β
⌋
,
…
is arithmetic?(c) For any set of primes
P
\mathbb{P}
P
, let
S
P
S_\mathbb{P}
S
P
denote the set of integers whose prime divisors all lie in
P
\mathbb{P}
P
. For instance
S
{
2
,
3
}
=
{
2
a
3
b
∣
a
,
b
≥
0
}
=
{
1
,
2
,
3
,
4
,
6
,
8
,
9
,
12
,
…
}
S_{\{2,3\}}=\{2^a3^b \; | \; a,b\ge 0\}=\{1,2,3,4,6,8,9,12,\dots\}
S
{
2
,
3
}
=
{
2
a
3
b
∣
a
,
b
≥
0
}
=
{
1
,
2
,
3
,
4
,
6
,
8
,
9
,
12
,
…
}
. Does there exist a finite set of primes
P
\mathbb{P}
P
and integer polynomials
P
P
P
and
Q
Q
Q
such that
gcd
(
P
(
x
)
,
Q
(
y
)
)
∈
S
P
\gcd(P(x), Q(y))\in S_\mathbb{P}
g
cd
(
P
(
x
)
,
Q
(
y
))
∈
S
P
for all
x
,
y
x,y
x
,
y
?(d) A function
f
f
f
is called P-recursive if there exists a positive integer
m
m
m
and real polynomials
p
0
(
n
)
,
p
1
(
n
)
,
…
,
p
m
(
n
)
p_0(n), p_1(n), \dots, p_m(n)
p
0
(
n
)
,
p
1
(
n
)
,
…
,
p
m
(
n
)
[color = red], not all zero, satisfying
p
m
(
n
)
f
(
n
+
m
)
=
p
m
−
1
(
n
)
f
(
n
+
m
−
1
)
+
⋯
+
p
0
(
n
)
f
(
n
)
p_m(n)f(n+m)=p_{m-1}(n)f(n+m-1)+\dots+p_0(n)f(n)
p
m
(
n
)
f
(
n
+
m
)
=
p
m
−
1
(
n
)
f
(
n
+
m
−
1
)
+
⋯
+
p
0
(
n
)
f
(
n
)
for all
n
n
n
. Does there exist a P-recursive function
f
f
f
satisfying
lim
n
→
∞
f
(
n
)
n
2
=
1
\lim_{n\to\infty} \frac{f(n)}{n^{\sqrt{2}}}=1
lim
n
→
∞
n
2
f
(
n
)
=
1
?(e) Does there exist a nonpolynomial function
f
:
Z
→
Z
f: \mathbb{Z}\to\mathbb{Z}
f
:
Z
→
Z
such that
a
−
b
a-b
a
−
b
divides
f
(
a
)
−
f
(
b
)
f(a)-f(b)
f
(
a
)
−
f
(
b
)
for all integers
a
≠
b
a\neq b
a
=
b
?(f) Do there exist periodic functions
f
,
g
:
R
→
R
f, g:\mathbb{R}\to\mathbb{R}
f
,
g
:
R
→
R
such that
f
(
x
)
+
g
(
x
)
=
x
f(x)+g(x)=x
f
(
x
)
+
g
(
x
)
=
x
for all
x
x
x
?[color = red]A clarification was issued for problem 33(d) during the test. I have included it above.
USAYNO
algebra