MathDB
Problems
Contests
International Contests
Rioplatense Mathematical Olympiad, Level 3
2010 Rioplatense Mathematical Olympiad, Level 3
2010 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(3)
1
2
Hide problems
Min/max{a,b,c,d} if a^b divides b^c divides c^d divides d^a
Suppose
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
are distinct positive integers such that
a
b
a^b
a
b
divides
b
c
b^c
b
c
,
b
c
b^c
b
c
divides
c
d
c^d
c
d
, and
c
d
c^d
c
d
divides
d
a
d^a
d
a
. (a) Is it possible to determine which of the numbers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
is the smallest? (b) Is it possible to determine which of the numbers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
is the largest?
999 distinct remainders with one of them zero
Let
r
2
,
r
3
,
…
,
r
1000
r_2, r_3,\ldots, r_{1000}
r
2
,
r
3
,
…
,
r
1000
denote the remainders when a positive odd integer is divided by
2
,
3
,
…
,
1000
2,3,\ldots,1000
2
,
3
,
…
,
1000
, respectively. It is known that the remainders are pairwise distinct and one of them is
0
0
0
. Find all values of
k
k
k
for which it is possible that
r
k
=
0
r_k = 0
r
k
=
0
.
2
2
Hide problems
Minimum and maximum value of \frac{a}{b}+\frac{c}{d}
Find the minimum and maximum values of
S
=
a
b
+
c
d
S=\frac{a}{b}+\frac{c}{d}
S
=
b
a
+
d
c
where
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
are positive integers satisfying
a
+
c
=
20202
a + c = 20202
a
+
c
=
20202
and
b
+
d
=
20200
b + d = 20200
b
+
d
=
20200
.
Ratio in acute triangle with lines through feet of altitudes
Acute triangle
A
B
P
ABP
A
BP
, where
A
B
>
B
P
AB > BP
A
B
>
BP
, has altitudes
B
H
BH
B
H
,
P
Q
PQ
PQ
, and
A
S
AS
A
S
. Let
C
C
C
denote the intersection of lines
Q
S
QS
QS
and
A
P
AP
A
P
, and let
L
L
L
denote the intersection of lines
H
S
HS
H
S
and
B
C
BC
BC
. If
H
S
=
S
L
HS = SL
H
S
=
S
L
and
H
L
HL
H
L
is perpendicular to
B
C
BC
BC
, find the value of
S
L
S
C
\frac{SL}{SC}
SC
S
L
.
3
2
Hide problems
Functional equation f(x+y)=f(x)+f(y) on a restricted domain
Find all the functions
f
:
N
→
R
f:\mathbb{N}\to\mathbb{R}
f
:
N
→
R
that satisfy
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f(x+y)=f(x)+f(y)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
N
x,y\in\mathbb{N}
x
,
y
∈
N
satisfying
1
0
6
−
1
1
0
6
<
x
y
<
1
0
6
+
1
1
0
6
10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}
1
0
6
−
1
0
6
1
<
y
x
<
1
0
6
+
1
0
6
1
.Note:
N
\mathbb{N}
N
denotes the set of positive integers and
R
\mathbb{R}
R
denotes the set of real numbers.
Moving a pebble among 1, 2,..., n game
Alice and Bob play the following game. To start, Alice arranges the numbers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's turn consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number
k
k
k
at most
k
k
k
times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer
n
n
n
, determine who has a winning strategy.