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Problems
Contests
National and Regional Contests
Estonia Contests
Estonia Team Selection Test
2001 Estonia Team Selection Test
2001 Estonia Team Selection Test
Part of
Estonia Team Selection Test
Subcontests
(6)
6
1
Hide problems
C_1,C_2 are incircle,circumcircle of triangle A'B'C' besides triangle ABC
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be the incircle and the circumcircle of the triangle
A
B
C
ABC
A
BC
, respectively. Prove that, for any point
A
′
A'
A
′
on
C
2
C_2
C
2
, there exist points
B
′
B'
B
′
and
C
′
C'
C
′
such that
C
1
C_1
C
1
and
C
2
C_2
C
2
are the incircle and the circumcircle of triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
, respectively.
5
1
Hide problems
exponent of 37 in representation of 111...... 11
Find the exponent of
37
37
37
in the representation of the number
111......11
111...... 11
111......11
with
3
⋅
3
7
2000
3\cdot 37^{2000}
3
⋅
3
7
2000
digits equals to
1
1
1
, as product of prime powers
4
1
Hide problems
sum of products of 2, 4, 6, ..., 2000 of elements of {1/2, ..., 1/2001}
Consider all products by
2
,
4
,
6
,
.
.
.
,
2000
2, 4, 6, ..., 2000
2
,
4
,
6
,
...
,
2000
of the elements of the set
A
=
{
1
2
,
1
3
,
1
4
,
.
.
.
,
1
2000
,
1
2001
}
A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}
A
=
{
2
1
,
3
1
,
4
1
,
...
,
2000
1
,
2001
1
}
. Find the sum of all these products.
3
1
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f(x)+ (f(y))^2 = kf(x + y^2) where k is a fixed real number
Let
k
k
k
be a fixed real number. Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
such that
f
(
x
)
+
(
f
(
y
)
)
2
=
k
f
(
x
+
y
2
)
f(x)+ (f(y))^2 = kf(x + y^2)
f
(
x
)
+
(
f
(
y
)
)
2
=
k
f
(
x
+
y
2
)
for all real numbers
x
x
x
and
y
y
y
.
2
1
Hide problems
1/h_1+1h_2+...+1/h_n> 2\pi /a inside a regular n-gon
Point
X
X
X
is taken inside a regular
n
n
n
-gon of side length
a
a
a
. Let
h
1
,
h
2
,
.
.
.
,
h
n
h_1,h_2,...,h_n
h
1
,
h
2
,
...
,
h
n
be the distances from
X
X
X
to the lines defined by the sides of the
n
n
n
-gon. Prove that
1
h
1
+
1
h
2
+
.
.
.
+
1
h
n
>
2
π
a
\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}
h
1
1
+
h
2
1
+
...
+
h
n
1
>
a
2
π
1
1
Hide problems
coloring rectangles on the coordinate plane
Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is
2
k
2^k
2
k
, where
k
=
0
,
1
,
2....
k = 0,1,2....
k
=
0
,
1
,
2....
Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?