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National and Regional Contests
Estonia Contests
Estonia Math Open Senior Contests
2007 Estonia Math Open Senior Contests
2007 Estonia Math Open Senior Contests
Part of
Estonia Math Open Senior Contests
Subcontests
(10)
6
1
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bluetooth network consisting of 1 master and many connected slaves.
A Bluetooth device can connect to any other Bluetooth device that is not more than
10
10
10
meters from him. A piconet is called a bluetooth network consisting of one master and a plurality of connected slaves. What is the greatest number of slaves, what can be on the pickup provided that all devices are on the same level and all slaves are out of range of each other?
5
1
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a labyrinth on a nxn square
Let
n
n
n
be a fixed natural number. The maze is a grid of dimensions
n
×
n
n \times n
n
×
n
, with a gate to the sky on one of the squares and some adjacent squares with partitions separated from each other so that it is still possible to move from one square to another. The program is in the UP, DOWN, RIGHT, LEFT final sequence, With each command, the Creature moves from its current square to the corresponding neighboring square, unless the partition or the outer boundary of the labyrinth prevents execution of the command (otherwise it does nothing), upon entering the gate, the Creature moves on to heaven. God creates a program, then Satan creates a labyrinth and places it on a square. Prove that God can make such a program that, independently of Satan's labyrinth and selected from the source square, the Creature always reaches heaven by following this program.
10
1
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Estonian Math Competitions 2006/2007
Consider triangles whose each side length squared is a rational number. Is it true that (a) the square of the circumradius of every such triangle is rational; (b) the square of the inradius of every such triangle is rational?
9
1
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Estonian Math Competitions 2006/2007
Find all positive integers n such that one can write an integer 1 to
n
2
n^2
n
2
into each unit square of a
n
2
×
n
2
n^2 \times n^2
n
2
×
n
2
table in such a way that, in each row, each column and each
n
×
n
n \times n
n
×
n
block of unit squares, each number 1 to
n
2
n^2
n
2
occurs exactly once.
8
1
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Estonian Math Competitions 2006/2007
Tangents
l
1
l_1
l
1
and
l
2
l_2
l
2
common to circles
c
1
c_1
c
1
and
c
2
c_2
c
2
intersect at point
P
P
P
, whereby tangent points remain to different sides from
P
P
P
on both tangent lines. Through some point
T
T
T
, tangents
p
1
p_1
p
1
and
p
2
p_2
p
2
to circle
c
1
c_1
c
1
and tangents
p
3
p_3
p
3
and
p
4
p_4
p
4
to circle
c
2
c_2
c
2
are drawn. The intersection points of
l
1
l_1
l
1
with lines
p
1
,
p
2
,
p
3
,
p
4
p_1, p_2, p_3, p_4
p
1
,
p
2
,
p
3
,
p
4
are
A
1
,
B
1
,
C
1
,
D
1
A_1, B_1, C_1, D_1
A
1
,
B
1
,
C
1
,
D
1
, respectively, whereby the order of points on
l
1
l_1
l
1
is:
A
1
,
B
1
,
P
,
C
1
,
D
1
A_1, B_1, P, C_1, D_1
A
1
,
B
1
,
P
,
C
1
,
D
1
. Analogously, the intersection points of
l
2
l_2
l
2
with lines
p
1
,
p
2
,
p
3
,
p
4
p_1, p_2, p_3, p_4
p
1
,
p
2
,
p
3
,
p
4
are
A
2
,
B
2
,
C
2
,
D
2
A_2, B_2, C_2, D_2
A
2
,
B
2
,
C
2
,
D
2
, respectively. Prove that if both quadrangles
A
1
A
2
D
1
D
2
A_1A_2D_1D_2
A
1
A
2
D
1
D
2
and
B
1
B
2
C
1
C
2
B_1B_2C_1C_2
B
1
B
2
C
1
C
2
are cyclic then radii of
c
1
c_1
c
1
and
c
2
c_2
c
2
are equal.
7
1
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Estonian Math Competitions 2006/2007
Does there exist a natural number
n
n
n
such that
n
>
2
n>2
n
>
2
and the sum of squares of some
n
n
n
consecutive integers is a perfect square?
4
1
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Estonian Math Competitions 2006/2007
The Fibonacci sequence is determined by conditions F_0 \equal{} 0, F1 \equal{} 1, and F_k\equal{}F_{k\minus{}1}\plus{}F_{k\minus{}2} for all
k
≥
2
k \ge 2
k
≥
2
. Let
n
n
n
be a positive integer and let P(x) \equal{} a_mx^m \plus{}. . .\plus{} a_1x\plus{} a_0 be a polynomial that satisfies the following two conditions: (1) P(F_n) \equal{} F_{n}^{2} ; (2) P(F_k) \equal{} P(F_{k\minus{}1}) \plus{} P(F_{k\minus{}2} for all
k
≥
2
k \ge 2
k
≥
2
. Find the sum of the coefficients of P.
3
1
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Estonian Math Competitions 2006/2007
Let
b
b
b
be an even positive integer for which there exists a natural number n such that
n
>
1
n>1
n
>
1
and \frac{b^n\minus{}1}{b\minus{}1} is a perfect square. Prove that
b
b
b
is divisible by 8.
2
1
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Estonian Math Competitions 2006/2007
Three circles with centres A, B, C touch each other pairwise externally, and touch circle c from inside. Prove that if the centre of c coincideswith the orthocentre of triangle ABC, then ABC is equilateral.
1
1
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Estonian Math Competitions 2006/2007
Let a_n \equal{} 1 \plus{} 2 \plus{} ... \plus{} n for every
n
≥
1
n \ge 1
n
≥
1
; the numbers
a
n
a_n
a
n
are called triangular. Prove that if 2a_m \equal{} a_n then a_{2m \minus{} n} is a perfect square.