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International Contests
Tuymaada Olympiad
2011 Tuymaada Olympiad
2011 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(4)
4
4
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4
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Making an 11x11 hole in a 2011x2011 board tiled with dominos
How many ways are there to remove an
11
×
11
11\times11
11
×
11
square from a
2011
×
2011
2011\times2011
2011
×
2011
square so that the remaining part can be tiled with dominoes (
1
×
2
1\times 2
1
×
2
rectangles)?
Parallelism in a cyclic quad, circle & diagonals diagram
A circle passing through the vertices
A
A
A
and
B
B
B
of a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
intersects diagonals
A
C
AC
A
C
and
B
D
BD
B
D
at
E
E
E
and
F
F
F
, respectively. The lines
A
F
AF
A
F
and
B
C
BC
BC
meet at a point
P
P
P
, and the lines
B
E
BE
BE
and
A
D
AD
A
D
meet at a point
Q
Q
Q
. Prove that
P
Q
PQ
PQ
is parallel to
C
D
CD
C
D
.
Lines thru the midpoint of the common chord of two circles
Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
intersect at points
A
A
A
and
B
B
B
, and
M
M
M
is the midpoint of
A
B
AB
A
B
. Points
S
1
S_1
S
1
and
S
2
S_2
S
2
lie on the line
A
B
AB
A
B
(but not between
A
A
A
and
B
B
B
). The tangents drawn from
S
1
S_1
S
1
to
ω
1
\omega_1
ω
1
touch it at
X
1
X_1
X
1
and
Y
1
Y_1
Y
1
, and the tangents drawn from
S
2
S_2
S
2
to
ω
2
\omega_2
ω
2
touch it at
X
2
X_2
X
2
and
Y
2
Y_2
Y
2
. Prove that if the line
X
1
X
2
X_1X_2
X
1
X
2
passes through
M
M
M
, then line
Y
1
Y
2
Y_1Y_2
Y
1
Y
2
also passes through
M
M
M
.
1
3
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Red, blue, and green children raising their hands
Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands,
20
20
20
children raised their hands. When she asked the blue children that have a green neighbor to raise their hands,
25
25
25
children raised their hands. Prove that some child that raised her hand had two green neighbors.
Reals a,b such that a+b and ab are different colours
Each real number greater than
1
1
1
is coloured red or blue with both colours being used. Prove that there exist real numbers
a
a
a
and
b
b
b
such that the numbers
a
+
b
a+b
a
+
b
and
a
b
ab
ab
are of different colours.
Two-coloring (1,infty) implies a+1/b & b+1/a are different
Each real number greater than 1 is colored red or blue with both colors being used. Prove that there exist real numbers
a
a
a
and
b
b
b
such that the numbers
a
+
1
b
a+\frac1b
a
+
b
1
and
b
+
1
a
b+\frac1a
b
+
a
1
are different colors.