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(5)
5
1
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Labelling points
There are
n
n
n
dots on the plane such that no three dots are collinear. Each dot is assigned a
0
0
0
or a
1
1
1
. Each pair of dots is connected by a line segment. If the endpoints of a line segment are two dots with the same number, then the segment is assigned a
0
0
0
. Otherwise, the segment is assigned a
1
1
1
. Find all
n
n
n
such that it is possible to assign
0
0
0
's and
1
1
1
's to the
n
n
n
dots in a way that the corresponding line segments are assigned equally many
0
0
0
's as
1
1
1
's.
4
1
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Divisor multiplicity
For each positive integer
n
n
n
, let
τ
(
n
)
\tau\left(n\right)
τ
(
n
)
be the number of positive divisors of
n
n
n
. It is well-known that if
a
a
a
and
b
b
b
are relatively prime positive integers then
τ
(
a
b
)
=
τ
(
a
)
τ
(
b
)
\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)
τ
(
ab
)
=
τ
(
a
)
τ
(
b
)
. Does the converse hold? That is, if
a
a
a
and
b
b
b
are positive integers such that
τ
(
a
b
)
=
τ
(
a
)
τ
(
b
)
\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)
τ
(
ab
)
=
τ
(
a
)
τ
(
b
)
, then is it necessarily true that
a
a
a
and
b
b
b
are relatively prime? Either give a proof, or find a counter-example.
3
1
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Diophantine with cubes
Find all integers
x
x
x
and
y
y
y
such that
x
3
−
117
y
3
=
5.
x^3-117y^3=5.
x
3
−
117
y
3
=
5.
2
1
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French accents
There are
5
5
5
accents in French, each applicable to only specific letters as follows: [*] The cédille: ç [*] The accent aigu: é [*] The accent circonflexe: â, ê, î, ô, û [*] The accent grave: à, è, ù [*] The accent tréma: ë, ö, ü Cédric needs to write down a phrase in French. He knows that there are
3
3
3
words in the phrase and that the letters appear in the order:
c
e
s
o
n
t
o
i
s
e
a
u
x
.
cesontoiseaux.
ceso
n
t
o
i
se
a
ux
.
He does not remember what the words are and which letters have what accents in the phrase. If
n
n
n
is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of
n
n
n
.
1
1
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Circles within circles
As shown in the diagram, three circles of radius
1
1
1
are all externally tangent to each other, and there are two circles that are tangent to all three of these circles. Find the area of the shaded region.[asy] size(5cm); real r=2/sqrt(3)-1, R=2/sqrt(3)+1; pair O=(0,0), C_1=O+(0,2/sqrt(3)), C_2=O+(-1,-1/sqrt(3)), C_3=O+(1,-1/sqrt(3)); fill(circle(O,R),rgb(0.5,0.5,0.5)); draw(circle(O,R)); fill(circle(C_1,1),rgb(1,1,1)); draw(circle(C_1,1)); fill(circle(C_2,1),rgb(1,1,1)); draw(circle(C_2,1)); fill(circle(C_3,1),rgb(1,1,1)); draw(circle(C_3,1)); fill(circle(O,r),rgb(1,1,1)); draw(circle(O,r)); [/asy]