MathDB
Problems
Contests
National and Regional Contests
Estonia Contests
Estonia Team Selection Test
2010 Estonia Team Selection Test
2010 Estonia Team Selection Test
Part of
Estonia Team Selection Test
Subcontests
(6)
6
1
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red or blue unit squares in a nxn board
Every unit square of a
n
×
n
n \times n
n
×
n
board is colored either red or blue so that among all 2
×
2
\times 2
×
2
squares on this board all possible colorings of
2
×
2
2 \times 2
2
×
2
squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different). a) Find the least possible value of
n
n
n
. b) For the least possible value of
n
n
n
find the least possible number of red unit squares
5
1
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P(x, y) = (x^2 + y^2)^k when P(sin t, cos t) = 1
Let
P
(
x
,
y
)
P(x, y)
P
(
x
,
y
)
be a non-constant homogeneous polynomial with real coefficients such that
P
(
sin
t
,
cos
t
)
=
1
P(\sin t, \cos t) = 1
P
(
sin
t
,
cos
t
)
=
1
for every real number
t
t
t
. Prove that there exists a positive integer
k
k
k
such that
P
(
x
,
y
)
=
(
x
2
+
y
2
)
k
P(x, y) = (x^2 + y^2)^k
P
(
x
,
y
)
=
(
x
2
+
y
2
)
k
.
4
1
Hide problems
CE bisects the <BCD, circumcircles related
In an acute triangle
A
B
C
ABC
A
BC
the angle
C
C
C
is greater than the angle
A
A
A
. Let
A
E
AE
A
E
be a diameter of the circumcircle of the triangle. Let the intersection point of the ray
A
C
AC
A
C
and the tangent of the circumcircle through the vertex
B
B
B
be
K
K
K
. The perpendicular to
A
E
AE
A
E
through
K
K
K
intersects the circumcircle of the triangle
B
C
K
BCK
BC
K
for the second time at point
D
D
D
. Prove that
C
E
CE
CE
bisects the angle
B
C
D
BCD
BC
D
.
3
1
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cot^2 a +cot^2 b + cot^2 c >= 3 ( 9R^2/p^2 -1)
Let the angles of a triangle be
α
,
β
\alpha, \beta
α
,
β
, and
γ
\gamma
γ
, the perimeter
2
p
2p
2
p
and the radius of the circumcircle
R
R
R
. Prove the inequality
cot
2
α
+
cot
2
β
+
cot
2
γ
≥
3
(
9
R
2
p
2
−
1
)
\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)
cot
2
α
+
cot
2
β
+
cot
2
γ
≥
3
(
p
2
9
R
2
−
1
)
. When is the equality achieved?
2
1
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determine mass of all bodies with masses of 1, 2, ..., N using balance scale
Let
n
n
n
be a positive integer. Find the largest integer
N
N
N
for which there exists a set of
n
n
n
weights such that it is possible to determine the mass of all bodies with masses of
1
,
2
,
.
.
.
,
N
1, 2, ..., N
1
,
2
,
...
,
N
using a balance scale . (i.e. to determine whether a body with unknown mass has a mass
1
,
2
,
.
.
.
,
N
1, 2, ..., N
1
,
2
,
...
,
N
, and which namely).
1
1
Hide problems
a @ b =(a-b)/ gcd(a,b)
For arbitrary positive integers
a
,
b
a, b
a
,
b
, denote
a
@
b
=
a
−
b
g
c
d
(
a
,
b
)
a @ b =\frac{a-b}{gcd(a,b)}
a
@
b
=
g
c
d
(
a
,
b
)
a
−
b
Let
n
n
n
be a positive integer. Prove that the following conditions are equivalent: (i)
g
c
d
(
n
,
n
@
m
)
=
1
gcd(n, n @ m) = 1
g
c
d
(
n
,
n
@
m
)
=
1
for every positive integer
m
<
n
m < n
m
<
n
, (ii)
n
=
p
k
n = p^k
n
=
p
k
where
p
p
p
is a prime number and
k
k
k
is a non-negative integer.