MathDB
Problems
Contests
International Contests
Romanian Masters of Mathematics Collection
2016 Romanian Master of Mathematics Shortlist
2016 Romanian Master of Mathematics Shortlist
Part of
Romanian Masters of Mathematics Collection
Subcontests
(9)
A2
1
Hide problems
|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil} and P(x, y) = P(y,- x -y)
Let
p
>
3
p > 3
p
>
3
be a prime number, and let
F
p
F_p
F
p
denote the (fi nite) set of residue classes modulo
p
p
p
. Let
S
d
S_d
S
d
denote the set of
2
2
2
-variable polynomials
P
(
x
,
y
)
P(x, y)
P
(
x
,
y
)
with coefficients in
F
p
F_p
F
p
, total degree
≤
d
\le d
≤
d
, and satisfying
P
(
x
,
y
)
=
P
(
y
,
−
x
−
y
)
P(x, y) = P(y,- x -y)
P
(
x
,
y
)
=
P
(
y
,
−
x
−
y
)
. Show that
∣
S
d
∣
=
p
⌈
(
d
+
1
)
(
d
+
2
)
/
6
⌉
|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}
∣
S
d
∣
=
p
⌈(
d
+
1
)
(
d
+
2
)
/6
⌉
. The total degree of a
2
2
2
-variable polynomial
P
(
x
,
y
)
P(x, y)
P
(
x
,
y
)
is the largest value of
i
+
j
i + j
i
+
j
among monomials
x
i
y
j
x^iy^j
x
i
y
j
appearing in
P
P
P
.
C4
1
Hide problems
u+v=x+y (mod 2016) in 46-element integer set with distinct {u, v}, {x,y}
Prove that a
46
46
46
-element set of integers contains two distinct doubletons
{
u
,
v
}
\{u, v\}
{
u
,
v
}
and
{
x
,
y
}
\{x,y\}
{
x
,
y
}
such that
u
+
v
≡
x
+
y
u + v \equiv x + y
u
+
v
≡
x
+
y
(mod
2016
2016
2016
).
C2
1
Hide problems
one frog at each vertex of an equilateral triangle, frogs jumping
A frog trainer places one frog at each vertex of an equilateral triangle
A
B
C
ABC
A
BC
of unit sidelength. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let
M
M
M
and
N
N
N
be two points on the rays
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, emanating from
A
A
A
, such that
A
M
=
A
N
=
ℓ
AM = AN = \ell
A
M
=
A
N
=
ℓ
, where
ℓ
\ell
ℓ
is a positive integer. After a fi nite number of jumps, the three frogs all lie in the triangle
A
M
N
AMN
A
MN
(inside or on the boundary), and no more jumps are performed. Determine the number of final positions the three frogs may reach in the triangle
A
M
N
AMN
A
MN
. (During the process, the frogs may leave the triangle
A
M
N
AMN
A
MN
, only their nal positions are to be in that triangle.)
C1
1
Hide problems
replacing n, (n + 1) by (n-2) and n, (n + 4) by (n - 1), min negative wanted
We start with any finite list of distinct positive integers. We may replace any pair
n
,
n
+
1
n, n + 1
n
,
n
+
1
(not necessarily adjacent in the list) by the single integer
n
−
2
n-2
n
−
2
, now allowing negatives and repeats in the list. We may also replace any pair
n
,
n
+
4
n, n + 4
n
,
n
+
4
by
n
−
1
n - 1
n
−
1
. We may repeat these operations as many times as we wish. Either determine the most negative integer which can appear in a list, or prove that there is no such minimum.
A1
1
Hide problems
f(a + b) = f(a) + f(b) + f(c) + f(d) in N-{O}, with 2ab = c^2 + d^2
Determine all functions
f
f
f
from the set of non-negative integers to itself such that
f
(
a
+
b
)
=
f
(
a
)
+
f
(
b
)
+
f
(
c
)
+
f
(
d
)
f(a + b) = f(a) + f(b) + f(c) + f(d)
f
(
a
+
b
)
=
f
(
a
)
+
f
(
b
)
+
f
(
c
)
+
f
(
d
)
, whenever
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
, are non-negative integers satisfying
2
a
b
=
c
2
+
d
2
2ab = c^2 + d^2
2
ab
=
c
2
+
d
2
.
N1
1
Hide problems
every quadratic non-residue modulo n is a primitive root modulo n.
Determine all integers
n
≥
3
n \ge 3
n
≥
3
whose decimal expansion has less than
20
20
20
digits, such that every quadratic non-residue modulo
n
n
n
is a primitive root modulo
n
n
n
.An integer
a
a
a
is a quadratic non-residue modulo
n
n
n
, if there is no integer
b
b
b
such that
a
−
b
2
a - b^2
a
−
b
2
is divisible by
n
n
n
. An integer
a
a
a
is a primitive root modulo
n
n
n
, if for every integer
b
b
b
relatively prime to n there is a positive integer
k
k
k
such that
a
k
−
b
a^k - b
a
k
−
b
is divisible by
n
n
n
.
G1
1
Hide problems
length of segment is equal to diameter , starting wiith intersecting circles
Two circles,
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
, centred at
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, meet at points
A
A
A
and
B
B
B
. A line through
B
B
B
meets
ω
1
\omega_1
ω
1
again at
C
C
C
, and
ω
2
\omega_2
ω
2
again at
D
D
D
. The tangents to
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
at
C
C
C
and
D
D
D
, respectively, meet at
E
E
E
, and the line
A
E
AE
A
E
meets the circle
ω
\omega
ω
through
A
,
O
1
,
O
2
A, O_1, O_2
A
,
O
1
,
O
2
again at
F
F
F
. Prove that the length of the segment
E
F
EF
EF
is equal to the diameter of
ω
\omega
ω
.
N2
1
Hide problems
Prove that sum is not divisible by q for finitely many q
Given a prime
p
p
p
, prove that the sum
∑
k
=
1
⌊
q
p
⌋
k
p
−
1
\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}
∑
k
=
1
⌊
p
q
⌋
k
p
−
1
is not divisible by
q
q
q
for all but finitely many primes
q
q
q
.
C3
1
Hide problems
Polygonal
A set
S
=
{
s
1
,
s
2
,
.
.
.
,
s
k
}
S=\{ s_1,s_2,...,s_k\}
S
=
{
s
1
,
s
2
,
...
,
s
k
}
of positive real numbers is "polygonal" if
k
≥
3
k\geq 3
k
≥
3
and there is a non-degenerate planar
k
−
k-
k
−
gon whose side lengths are exactly
s
1
,
s
2
,
.
.
.
,
s
k
s_1,s_2,...,s_k
s
1
,
s
2
,
...
,
s
k
; the set
S
S
S
is multipolygonal if in every partition of
S
S
S
into two subsets,each of which has at least three elements, exactly one of these two subsets in polygonal. Fix an integer
n
≥
7
n\geq 7
n
≥
7
. (a) Does there exist an
n
−
n-
n
−
element multipolygonal set, removal of whose maximal element leaves a multipolygonal set? (b) Is it possible that every
(
n
−
1
)
−
(n-1)-
(
n
−
1
)
−
element subset of an
n
−
n-
n
−
element set of positive real numbers be multipolygonal?