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National and Regional Contests
Estonia Contests
Estonia Team Selection Test
2014 Estonia Team Selection Test
2014 Estonia Team Selection Test
Part of
Estonia Team Selection Test
Subcontests
(6)
6
1
Hide problems
x^2 + y + z^2 = nxyz in positive integers
Find all natural numbers
n
n
n
such that the equation
x
2
+
y
2
+
z
2
=
n
x
y
z
x^2 + y^2 + z^2 = nxyz
x
2
+
y
2
+
z
2
=
n
x
yz
has solutions in positive integers
5
1
Hide problems
in Wonderland there are at least 5 towns, gold coins related
In Wonderland there are at least
5
5
5
towns. Some towns are connected directly by roads or railways. Every town is connected to at least one other town and for any four towns there exists some direct connection between at least three pairs of towns among those four. When entering the public transportation network of this land, the traveller must insert one gold coin into a machine, which lets him use a direct connection to go to the next town. But if the traveller continues travelling from some town with the same method of transportation that took him there, and he has paid a gold coin to get to this town, then going to the next town does not cost anything, but instead the traveller gains the coin he last used back. In other cases he must pay just like when starting travelling. Prove that it is possible to get from any town to any other town by using at most
2
2
2
gold coins.
4
1
Hide problems
MP = MQ, 2 circumcircles related
In an acute triangle the feet of altitudes drawn from vertices
A
A
A
and
B
B
B
are
D
D
D
and
E
E
E
, respectively. Let
M
M
M
be the midpoint of side
A
B
AB
A
B
. Line
C
M
CM
CM
intersects the circumcircle of
C
D
E
CDE
C
D
E
again in point
P
P
P
and the circumcircle of
C
A
B
CAB
C
A
B
again in point
Q
Q
Q
. Prove that
∣
M
P
∣
=
∣
M
Q
∣
|MP| = |MQ|
∣
MP
∣
=
∣
MQ
∣
.
3
1
Hide problems
maximum area of the convex hull of a figure by 3 line segments
Three line segments, all of length
1
1
1
, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.
2
1
Hide problems
sum a^2/(b^3 + c^4 + 1) > 1/5
Let
a
,
b
a, b
a
,
b
and
c
c
c
be positive real numbers for which
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
. Prove that
a
2
b
3
+
c
4
+
1
+
b
2
c
3
+
a
4
+
1
+
c
2
a
3
+
b
4
+
1
>
1
5
\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}
b
3
+
c
4
+
1
a
2
+
c
3
+
a
4
+
1
b
2
+
a
3
+
b
4
+
1
c
2
>
5
1
1
1
Hide problems
government of each country consists of exactly a men and b women
In Wonderland, the government of each country consists of exactly
a
a
a
men and
b
b
b
women, where
a
a
a
and
b
b
b
are fixed natural numbers and
b
>
1
b > 1
b
>
1
. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least
n
n
n
among whom are women, are formed (where
n
n
n
is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.