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National and Regional Contests
Romania Contests
IMAR Test
2013 IMAR Test
2013 IMAR Test
Part of
IMAR Test
Subcontests
(4)
4
1
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Hard geometry with Miquel points!
Given a triangle
A
B
C
ABC
A
BC
, a circle centered at some point
O
O
O
meets the segments
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
in the pairs of points
X
X
X
and
X
′
X^{'}
X
′
,
Y
Y
Y
and
Y
′
Y^{'}
Y
′
,
Z
Z
Z
and
Z
′
Z^{'}
Z
′
, respectively ,labelled in circular order :
X
,
X
′
,
Y
,
Y
′
,
Z
,
Z
′
X,X^{'},Y,Y^{'},Z,Z^{'}
X
,
X
′
,
Y
,
Y
′
,
Z
,
Z
′
. Let
M
M
M
be the Miquel point of the triangle
X
Y
Z
XYZ
X
Y
Z
and let
M
′
M^{'}
M
′
be that of the triangle
X
′
Y
′
Z
′
X^{'}Y^{'}Z^{'}
X
′
Y
′
Z
′
. Prove that the segments
O
M
OM
OM
and
O
M
′
OM^{'}
O
M
′
have equal lehgths.
3
1
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Circles covering a quadrangle!
The closure (interior and boundary) of a convex quadrangle is covered by four closed discs centered at each vertex of the quadrangle each. Show that three of these discs cover the closure of the triangle determined by their centers.
2
1
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Sum of digits of the powers of 2 !
For every non-negative integer
n
n
n
, let
s
n
s_n
s
n
be the sum of digits in the decimal expansion of
2
n
2^n
2
n
. Is the sequence
(
s
n
)
n
∈
N
(s_n)_{n \in \mathbb{N}}
(
s
n
)
n
∈
N
eventually increasing ?
1
1
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Primes with modulo property!
Given a prime
p
≥
5
p \geq 5
p
≥
5
, show that there exist at least two distinct primes
q
q
q
and
r
r
r
in the range
2
,
3
,
…
p
−
2
2, 3, \ldots p-2
2
,
3
,
…
p
−
2
such that
q
p
−
1
≢
1
(
m
o
d
p
2
)
q^{p-1} \not\equiv 1 \pmod{p^2}
q
p
−
1
≡
1
(
mod
p
2
)
and
r
p
−
1
≢
1
(
m
o
d
p
2
)
r^{p-1} \not\equiv 1 \pmod{p^2}
r
p
−
1
≡
1
(
mod
p
2
)
.