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International Contests
JBMO ShortLists
2013 JBMO Shortlist
2013 JBMO Shortlist
Part of
JBMO ShortLists
Subcontests
(6)
4
1
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latticed rectangles with given area with or without // sides to the axes
A rectangle in xy Cartesian System is called latticed if all it's vertices have integer coordinates. a) Find a latticed rectangle of area
2013
2013
2013
, whose sides are not parallel to the axes. b) Show that if a latticed rectangle has area
2011
2011
2011
, then their sides are parallel to the axes.
6
2
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Concurrent perpendiculars in a rectangle
Let
P
P
P
and
Q
Q
Q
be the midpoints of the sides
B
C
BC
BC
and
C
D
CD
C
D
, respectively in a rectangle
A
B
C
D
ABCD
A
BC
D
. Let
K
K
K
and
M
M
M
be the intersections of the line
P
D
PD
P
D
with the lines
Q
B
QB
QB
and
Q
A
QA
Q
A
, respectively, and let
N
N
N
be the intersection of the lines
P
A
PA
P
A
and
Q
B
QB
QB
. Let
X
X
X
,
Y
Y
Y
and
Z
Z
Z
be the midpoints of the segments
A
N
AN
A
N
,
K
N
KN
K
N
and
A
M
AM
A
M
, respectively. Let
ℓ
1
\ell_1
ℓ
1
be the line passing through
X
X
X
and perpendicular to
M
K
MK
M
K
,
ℓ
2
\ell_2
ℓ
2
be the line passing through
Y
Y
Y
and perpendicular to
A
M
AM
A
M
and
ℓ
3
\ell_3
ℓ
3
the line passing through
Z
Z
Z
and perpendicular to
K
N
KN
K
N
. Prove that the lines
ℓ
1
\ell_1
ℓ
1
,
ℓ
2
\ell_2
ℓ
2
and
ℓ
3
\ell_3
ℓ
3
are concurrent.
solve in integers: x^2-y^2=z and 3xy+(x-y)z=z^2
Solve in integers the system of equations:
x
2
−
y
2
=
z
x^2-y^2=z
x
2
−
y
2
=
z
3
x
y
+
(
x
−
y
)
z
=
z
2
3xy+(x-y)z=z^2
3
x
y
+
(
x
−
y
)
z
=
z
2
5
2
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Junior Geometric Inequality with <BAC=60
A circle passing through the midpoint
M
M
M
of the side
B
C
BC
BC
and the vertex
A
A
A
of the triangle
A
B
C
ABC
A
BC
intersects the segments
A
B
AB
A
B
and
A
C
AC
A
C
for the second time in the points
P
P
P
and
Q
Q
Q
, respectively. Prove that if
∠
B
A
C
=
6
0
∘
\angle BAC=60^{\circ}
∠
B
A
C
=
6
0
∘
, then
A
P
+
A
Q
+
P
Q
<
A
B
+
A
C
+
1
2
B
C
AP+AQ+PQ<AB+AC+\frac{1}{2} BC
A
P
+
A
Q
+
PQ
<
A
B
+
A
C
+
2
1
BC
.
\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}
Solve in positive integers:
1
x
2
+
y
x
z
+
1
z
2
=
1
2013
\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}
x
2
1
+
x
z
y
+
z
2
1
=
2013
1
.
4
1
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JBMO Shortlist 2013, G4
Let
I
I
I
be the incenter and
A
B
AB
A
B
the shortest side of the triangle
A
B
C
ABC
A
BC
. The circle centered at
I
I
I
passing through
C
C
C
intersects the ray
A
B
AB
A
B
in
P
P
P
and the ray
B
A
BA
B
A
in
Q
Q
Q
. Let
D
D
D
be the point of tangency of the
A
A
A
-excircle of the triangle
A
B
C
ABC
A
BC
with the side
B
C
BC
BC
. Let
E
E
E
be the reflection of
C
C
C
with respect to the point
D
D
D
. Prove that
P
E
⊥
C
Q
PE\perp CQ
PE
⊥
CQ
.
2
4
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1
4
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