MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2008 Turkey Team Selection Test
2008 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(6)
4
1
Hide problems
Turkey TST 2008 Q4
The sequence
(
x
n
)
(x_n)
(
x
n
)
is defined as; x_1\equal{}a, x_2\equal{}b and for all positive integer
n
n
n
, x_{n\plus{}2}\equal{}2008x_{n\plus{}1}\minus{}x_n. Prove that there are some positive integers
a
,
b
a,b
a
,
b
such that 1\plus{}2006x_{n\plus{}1}x_n is a perfect square for all positive integer
n
n
n
.
3
1
Hide problems
Turkey TST 2008 Q3
The equation x^3\minus{}ax^2\plus{}bx\minus{}c\equal{}0 has three (not necessarily different) positive real roots. Find the minimal possible value of \frac{1\plus{}a\plus{}b\plus{}c}{3\plus{}2a\plus{}b}\minus{}\frac{c}{b}.
5
1
Hide problems
Turkey TST 2008 Q5
D
D
D
is a point on the edge
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
such that AD\equal{}\frac{BD^2}{AB\plus{}AD}\equal{}\frac{CD^2}{AC\plus{}AD}.
E
E
E
is a point such that
D
D
D
is on
[
A
E
]
[AE]
[
A
E
]
and CD\equal{}\frac{DE^2}{CD\plus{}CE}. Prove that AE\equal{}AB\plus{}AC.
1
1
Hide problems
TurkeyTST 2008 Q1
In an
A
B
C
ABC
A
BC
triangle such that
m
(
∠
B
)
>
m
(
∠
C
)
m(\angle B)>m(\angle C)
m
(
∠
B
)
>
m
(
∠
C
)
, the internal and external bisectors of vertice
A
A
A
intersects
B
C
BC
BC
respectively at points
D
D
D
and
E
E
E
.
P
P
P
is a variable point on
E
A
EA
E
A
such that
A
A
A
is on
[
E
P
]
[EP]
[
EP
]
.
D
P
DP
D
P
intersects
A
C
AC
A
C
at
M
M
M
and
M
E
ME
ME
intersects
A
D
AD
A
D
at
Q
Q
Q
. Prove that all
P
Q
PQ
PQ
lines have a common point as
P
P
P
varies.
6
1
Hide problems
Turkey TST 2008 Q6
There are
n
n
n
voters and
m
m
m
candidates. Every voter makes a certain arrangement list of all candidates (there is one person in every place
1
,
2
,
.
.
.
m
1,2,...m
1
,
2
,
...
m
) and votes for the first
k
k
k
people in his/her list. The candidates with most votes are selected and say them winners. A poll profile is all of this
n
n
n
lists. If
a
a
a
is a candidate,
R
R
R
and
R
′
R'
R
′
are two poll profiles.
R
′
R'
R
′
is a\minus{}good for
R
R
R
if and only if for every voter; the people which in a worse position than
a
a
a
in
R
R
R
is also in a worse position than
a
a
a
in
R
′
R'
R
′
. We say positive integer
k
k
k
is monotone if and only if for every
R
R
R
poll profile and every winner
a
a
a
for
R
R
R
poll profile is also a winner for all a\minus{}good
R
′
R'
R
′
poll profiles. Prove that
k
k
k
is monotone if and only if k>\frac{m(n\minus{}1)}{n}.
2
1
Hide problems
Turkey TST 2008 Q2
A graph has
30
30
30
vertices,
105
105
105
edges and
4822
4822
4822
unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph.