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Contests
National and Regional Contests
Argentina Contests
Argentina Team Selection Test
2005 Argentina Team Selection Test
2005 Argentina Team Selection Test
Part of
Argentina Team Selection Test
Subcontests
(3)
1
2
Hide problems
Is it possible to reach value of the expression?
We have
150
150
150
numbers
x
1
,
x
2
,
⋯
,
x
150
x_1,x_2, \cdots , x_{150}
x
1
,
x
2
,
⋯
,
x
150
each of which is either \sqrt 2 \plus{}1 or \sqrt 2 \minus{}1 We calculate the following sum: S\equal{}x_1x_2 \plus{}x_3x_4\plus{} x_5x_6\plus{} \cdots \plus{} x_{149}x_{150} Can we choose the
150
150
150
numbers such that S\equal{}121? And what about S\equal{}111?
covering a board...
Find all pairs of integers
(
m
,
n
)
(m,n)
(
m
,
n
)
such that an
m
×
n
m\times n
m
×
n
board can be totally covered with
1
×
3
1\times 3
1
×
3
and
2
×
5
2 \times 5
2
×
5
pieces.
2
2
Hide problems
Well known functional equation
Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that
∀
x
,
y
∈
R
\forall x,y \in \mathbb{R}
∀
x
,
y
∈
R
we have
f
(
x
f
(
x
)
+
f
(
y
)
)
=
f
(
x
)
2
+
y
f(xf(x)+f(y)) = f(x)^2 + y
f
(
x
f
(
x
)
+
f
(
y
))
=
f
(
x
)
2
+
y
4p-3 is a perfect square
Let
n
,
p
n,p
n
,
p
be integers such that
n
>
1
n>1
n
>
1
and
p
p
p
is a prime. If
n
∣
p
−
1
n\mid p-1
n
∣
p
−
1
and
p
∣
n
3
−
1
p\mid n^3-1
p
∣
n
3
−
1
, show that
4
p
−
3
4p-3
4
p
−
3
is a perfect square.
3
2
Hide problems
Equilaterals outside ABC
Given the triangle
A
B
C
ABC
A
BC
we consider the points
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
such that the triangles
A
B
Z
,
B
C
X
,
C
A
Z
ABZ,BCX,CAZ
A
BZ
,
BCX
,
C
A
Z
are equilateral, and they don't have intersection with
A
B
C
ABC
A
BC
. Let
B
′
B'
B
′
be the midpoint of
B
C
BC
BC
,
N
′
N'
N
′
the midpoint of
C
Y
CY
C
Y
, and
M
,
N
M,N
M
,
N
the midpoints of
A
Z
,
C
X
AZ,CX
A
Z
,
CX
, respectively. Prove that
B
′
N
′
⊥
M
N
B'N' \bot MN
B
′
N
′
⊥
MN
.
Graph problem, not so easy
We say that a group of
k
k
k
boys is
n
−
a
c
c
e
p
t
a
b
l
e
n-acceptable
n
−
a
cce
pt
ab
l
e
if removing any boy from the group one can always find, in the other
k
−
1
k-1
k
−
1
group, a group of
n
n
n
boys such that everyone knows each other. For each
n
n
n
, find the biggest
k
k
k
such that in any group of
k
k
k
boys that is
n
−
a
c
c
e
p
t
a
b
l
e
n-acceptable
n
−
a
cce
pt
ab
l
e
we must always have a group of
n
+
1
n+1
n
+
1
boys such that everyone knows each other.