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Problems
Contests
National and Regional Contests
Estonia Contests
Estonia Team Selection Test
2007 Estonia Team Selection Test
2007 Estonia Team Selection Test
Part of
Estonia Team Selection Test
Subcontests
(6)
6
1
Hide problems
on every move, we colour 4 unit squares in a 10x10 grid
Consider a
10
×
10
10 \times 10
10
×
10
grid. On every move, we colour
4
4
4
unit squares that lie in the intersection of some two rows and two columns. A move is allowed if at least one of the
4
4
4
squares is previously uncoloured. What is the largest possible number of moves that can be taken to colour the whole grid?
5
1
Hide problems
f(x+f(y)) = y+f(x+1) continuous
Find all continuous functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
such that for all reals
x
x
x
and
y
y
y
,
f
(
x
+
f
(
y
)
)
=
y
+
f
(
x
+
1
)
f(x+f(y)) = y+f(x+1)
f
(
x
+
f
(
y
))
=
y
+
f
(
x
+
1
)
.
4
1
Hide problems
anglechasing candidate, <EKF =?, when AK= EF, inside a square
In square
A
B
C
D
,
ABCD,
A
BC
D
,
points
E
E
E
and
F
F
F
are chosen in the interior of sides
B
C
BC
BC
and
C
D
CD
C
D
, respectively. The line drawn from
F
F
F
perpendicular to
A
E
AE
A
E
passes through the intersection point
G
G
G
of
A
E
AE
A
E
and diagonal
B
D
BD
B
D
. A point
K
K
K
is chosen on
F
G
FG
FG
such that
∣
A
K
∣
=
∣
E
F
∣
|AK|= |EF|
∣
A
K
∣
=
∣
EF
∣
. Find
∠
E
K
F
.
\angle EKF.
∠
E
K
F
.
3
1
Hide problems
(b^n-1)(b-1) is prime power then n is prime
Let
n
n
n
be a natural number,
n
>
2
n > 2
n
>
2
. Prove that if
b
n
−
1
b
−
1
\frac{b^n-1}{b-1}
b
−
1
b
n
−
1
is a prime power for some positive integer
b
b
b
then
n
n
n
is prime.
2
1
Hide problems
|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|, circumradii, inradii
Let
D
D
D
be the foot of the altitude of triangle
A
B
C
ABC
A
BC
drawn from vertex
A
A
A
. Let
E
E
E
and
F
F
F
be points symmetric to
D
D
D
w.r.t. lines
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Let
R
1
R_1
R
1
and
R
2
R_2
R
2
be the circumradii of triangles
B
D
E
BDE
B
D
E
and
C
D
F
CDF
C
D
F
, respectively, and let
r
1
r_1
r
1
and
r
2
r_2
r
2
be the inradii of the same triangles. Prove that
∣
S
A
B
D
−
S
A
C
D
∣
>
∣
R
1
r
1
−
R
2
r
2
∣
|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|
∣
S
A
B
D
−
S
A
C
D
∣
>
∣
R
1
r
1
−
R
2
r
2
∣
1
1
Hide problems
n electric switches on control board of a nuclear station
On the control board of a nuclear station, there are
n
n
n
electric switches (
n
>
0
n > 0
n
>
0
), all in one row. Each switch has two possible positions: up and down. The switches are connected to each other in such a way that, whenever a switch moves down from its upper position, its right neighbour (if it exists) automatically changes position. At the beginning, all switches are down. The operator of the board first changes the position of the leftmost switch once, then the position of the second leftmost switch twice etc., until eventually he changes the position of the rightmost switch n times. How many switches are up after all these operations?