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Contests
National and Regional Contests
Romania Contests
IMAR Test
2012 IMAR Test
2012 IMAR Test
Part of
IMAR Test
Subcontests
(4)
1
1
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vectors OX\pm OY,. OX\pm OZ, OY \pm OZ, in convex planar set S
Let
K
K
K
be a convex planar set, symmetric about a point
O
O
O
, and let
X
,
Y
,
Z
X, Y , Z
X
,
Y
,
Z
be three points in
K
K
K
. Show that
K
K
K
contains the head of one of the vectors
O
X
→
±
O
Y
→
,
O
X
→
±
O
Z
→
,
O
Y
→
±
O
Z
→
\overrightarrow{OX} \pm \overrightarrow{OY} , \overrightarrow{OX} \pm \overrightarrow{OZ}, \overrightarrow{OY} \pm \overrightarrow{OZ}
OX
±
O
Y
,
OX
±
OZ
,
O
Y
±
OZ
.
4
1
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2-colouring of S yields (at least) three monochromatic collinear points
Design a planar finite non-empty set
S
S
S
satisfying the following two conditions: (a) every line meets
S
S
S
in at most four points; and (b) every
2
2
2
-colouring of
S
S
S
- that is, each point of
S
S
S
is coloured one of two colours - yields (at least) three monochromatic collinear points.
2
1
Hide problems
\Sigma 1/pq where summation is over all coprime integers p,q under conditions
Given an integer
n
≥
2
n \ge 2
n
≥
2
, evaluate
Σ
1
p
q
\Sigma \frac{1}{pq}
Σ
pq
1
,where the summation is over all coprime integers
p
p
p
and
q
q
q
such that
1
≤
p
<
q
≤
n
1 \le p < q \le n
1
≤
p
<
q
≤
n
and
p
+
q
>
n
p + q > n
p
+
q
>
n
.
3
1
Hide problems
equal angles after reflecting an external bisector on 2 internal bisectors
Given a triangle
A
B
C
ABC
A
BC
, let
D
D
D
be a point different from
A
A
A
on the external bisectrix
ℓ
\ell
ℓ
of the angle
B
A
C
BAC
B
A
C
, and let
E
E
E
be an interior point of the segment
A
D
AD
A
D
. Reflect
ℓ
\ell
ℓ
in the internal bisectrices of the angles
B
D
C
BDC
B
D
C
and
B
E
C
BEC
BEC
to obtain two lines that meet at some point
F
F
F
. Show that the angles
A
B
D
ABD
A
B
D
and
E
B
F
EBF
EBF
are congruent.