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International Contests
Rioplatense Mathematical Olympiad, Level 3
2009 Rioplatense Mathematical Olympiad, Level 3
2009 Rioplatense Mathematical Olympiad, Level 3
Part of
Rioplatense Mathematical Olympiad, Level 3
Subcontests
(3)
2
2
Hide problems
Nine points, three pentagons, and one triangle in space
Let
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
,
F
F
F
,
G
G
G
,
H
H
H
,
I
I
I
be nine points in space such that
A
B
C
D
E
ABCDE
A
BC
D
E
,
A
B
F
G
H
ABFGH
A
BFG
H
, and
G
F
C
D
I
GFCDI
GFC
D
I
are each regular pentagons with side length
1
1
1
. Determine the lengths of the sides of triangle
E
H
I
EHI
E
H
I
.
Find all (a,b) with a divides b+1 and b divides a^3-1
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of integers with
a
>
1
a>1
a
>
1
and
b
>
1
b>1
b
>
1
such that
a
a
a
divides
b
+
1
b+1
b
+
1
and
b
b
b
divides
a
3
−
1
a^3-1
a
3
−
1
.
3
2
Hide problems
Number of decreasing subsequences of 1,2,...,n permutation
Call a permutation of the integers
(
1
,
2
,
…
,
n
)
(1,2,\ldots,n)
(
1
,
2
,
…
,
n
)
d
d
d
-ordered if it does not contains a decreasing subsequence of length
d
d
d
. Prove that for every
d
=
2
,
3
,
…
,
n
d=2,3,\ldots,n
d
=
2
,
3
,
…
,
n
, the number of
d
d
d
-ordered permutations of
(
1
,
2
,
…
,
n
)
(1,2,\ldots,n)
(
1
,
2
,
…
,
n
)
is at most
(
d
−
1
)
2
n
(d-1)^{2n}
(
d
−
1
)
2
n
.
Game: Combining rectangles to make bigger rectangles
Alice and Bob play the following game. It begins with a set of
1000
1000
1000
1
×
2
1\times 2
1
×
2
rectangles. A move consists of choosing two rectangles (a rectangle may consist of one or several
1
×
2
1\times 2
1
×
2
rectangles combined together) that share a common side length and combining those two rectangles into one rectangle along those sides sharing that common length. The first player who cannot make a move loses. Alice moves first. Describe a winning strategy for Bob.
1
2
Hide problems
Real roots of quadratics imply real roots of other quadratic
Find all pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of real numbers with the following property: Given any real numbers
c
c
c
and
d
d
d
, if both of the equations
x
2
+
a
x
+
1
=
c
x^2+ax+1=c
x
2
+
a
x
+
1
=
c
and
x
2
+
b
x
+
1
=
d
x^2+bx+1=d
x
2
+
b
x
+
1
=
d
have real roots, then the equation
x
2
+
(
a
+
b
)
x
+
1
=
c
d
x^2+(a+b)x+1=cd
x
2
+
(
a
+
b
)
x
+
1
=
c
d
has real roots.
Functional equation: f(xy)=max{f(x+y),f(x)f(y)}
Find all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
f
(
x
y
)
=
max
{
f
(
x
+
y
)
,
f
(
x
)
f
(
y
)
}
f(xy)=\max\{f(x+y),f(x) f(y)\}
f
(
x
y
)
=
max
{
f
(
x
+
y
)
,
f
(
x
)
f
(
y
)}
for all real numbers
x
x
x
and
y
y
y
.