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Brazil Contests
Brazil Team Selection Test
2011 Brazil Team Selection Test
2011 Brazil Team Selection Test
Part of
Brazil Team Selection Test
Subcontests
(2)
1
2
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trinomial P_1 + P_2 + P_3 has real roots if every 2 have common root
Let
P
1
P_1
P
1
,
P
2
P_2
P
2
and
P
3
P_3
P
3
be polynomials of degree two with positive coefficient leader and real roots . Prove that if each pair of polynomials has a common root , then the polynomial
P
1
+
P
2
+
P
3
P_1 + P_2 + P_3
P
1
+
P
2
+
P
3
has also real roots.
min n to paint 8x8 board such that each 3x1 tile has different colours
Find the smallest positive integer
n
n
n
such that it is possible to paint each of the
64
64
64
squares of an
8
×
8
8 \times 8
8
×
8
board of one of
n
n
n
colors so that any four squares that form an
L
L
L
as in the following figure (or congruent figures obtained through rotations and/or reflections) have different colors. https://cdn.artofproblemsolving.com/attachments/a/2/c8049b1be8f37657c058949e11faf041856da4.png
2
2
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n not prime if a^{n-1}=1 mod n, a^{{n-1}/q} not 1 mod n
Let
n
≥
3
n\ge 3
n
≥
3
be an integer such that for every prime factor
q
q
q
of
n
−
1
n-1
n
−
1
exists an integer
a
>
1
a > 1
a
>
1
such that
a
n
−
1
≡
1
(
m
o
d
n
)
a^{n-1} \equiv 1 \,(\mod n \, )
a
n
−
1
≡
1
(
mod
n
)
and
a
n
−
1
q
≢
1
(
m
o
d
n
)
a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, )
a
q
n
−
1
≡
1
(
mod
n
)
. Prove that
n
n
n
is not prime.
MS = MT iff X,Y,Z,W are concyclic, starting with 2 intersecting circles
Given two circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
, with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively intesrecting at two points
A
A
A
and
B
B
B
. Let
X
X
X
and
Y
Y
Y
be points on
ω
1
\omega_1
ω
1
. The lines
X
A
XA
X
A
and
Y
A
YA
Y
A
intersect
ω
2
\omega_2
ω
2
again in
Z
Z
Z
and
W
W
W
, respectively, such that
A
A
A
is between
X
,
Z
X,Z
X
,
Z
and
A
A
A
is between
Y
,
W
Y,W
Y
,
W
. Let
M
M
M
be the midpoint of
O
1
O
2
O_1O_2
O
1
O
2
, S be the midpoint of
X
A
XA
X
A
and
T
T
T
be the midpoint of
W
A
WA
W
A
. Prove that
M
S
=
M
T
MS = MT
MS
=
MT
if, and only if, the points
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
and
W
W
W
are concyclic.