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Contests
National and Regional Contests
Lithuania Contests
Lithuania Team Selection Test
2006 Lithuania Team Selection Test
2006 Lithuania Team Selection Test
Part of
Lithuania Team Selection Test
Subcontests
(5)
5
1
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5x5
Does the bellow depicted figure fit into a square
5
×
5
5\times5
5
×
5
.
4
1
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Trivial combinatorial geometry
Prove that in every polygon there is a diagonal that cuts off a triangle and lies within the polygon.
3
1
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Four triangles of equal area
Inside a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
there is a point
P
P
P
such that the triangles
P
A
B
,
P
B
C
,
P
C
D
,
P
D
A
PAB, PBC, PCD, PDA
P
A
B
,
PBC
,
PC
D
,
P
D
A
have equal areas. Prove that the area of
A
B
C
D
ABCD
A
BC
D
is bisected by one of the diagonals.
2
1
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Equation in integers
Solve in integers
x
x
x
and
y
y
y
the equation
x
3
−
y
3
=
2
x
y
+
8
x^3-y^3=2xy+8
x
3
−
y
3
=
2
x
y
+
8
.
1
1
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Quite well known and easy
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers, whose sum is
1
1
1
. Prove that
a
1
2
a
1
+
a
2
+
a
2
2
a
2
+
a
3
+
⋯
+
a
n
−
1
2
a
n
−
1
+
a
n
+
a
n
2
a
n
+
a
1
≥
1
2
\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+\dots+\frac{a_{n-1}^2}{a_{n-1}+a_n}+\frac{a_n^2}{a_n+a_1}\ge \frac{1}{2}
a
1
+
a
2
a
1
2
+
a
2
+
a
3
a
2
2
+
⋯
+
a
n
−
1
+
a
n
a
n
−
1
2
+
a
n
+
a
1
a
n
2
≥
2
1