MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1995 Turkey Team Selection Test
1995 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
3
2
Hide problems
Find the locus of the intersection point
Let
D
D
D
be a point on the small arc
A
C
AC
A
C
of the circumcircle of an equilateral triangle
A
B
C
ABC
A
BC
, different from
A
A
A
and
C
C
C
. Let
E
E
E
and
F
F
F
be the projections of
D
D
D
onto
B
C
BC
BC
and
A
C
AC
A
C
respectively. Find the locus of the intersection point of
E
F
EF
EF
and
O
D
OD
O
D
, where
O
O
O
is the center of
A
B
C
ABC
A
BC
.
Show there exist a, b such that limit is 1
The sequence
{
x
n
}
\{x_n\}
{
x
n
}
of real numbers is defined by x_1=1 \text{and} x_{n+1}=x_n+\sqrt[3]{x_n} \text{for} n\geq 1.Show that there exist real numbers
a
,
b
a, b
a
,
b
such that
lim
n
→
∞
x
n
a
n
b
=
1
\lim_{n \rightarrow \infty}\frac{x_n}{an^b} = 1
lim
n
→
∞
a
n
b
x
n
=
1
.
2
2
Hide problems
Number of permutations such that \sigma(j) \geq j for 2 j
Let
n
n
n
be a positive integer. Find the number of permutations
σ
\sigma
σ
of the set
{
1
,
2
,
.
.
.
,
n
}
\{1, 2, ..., n\}
{
1
,
2
,
...
,
n
}
such that
σ
(
j
)
≥
j
\sigma(j) \geq j
σ
(
j
)
≥
j
holds for exactly two values of
j
j
j
.
Prove the two conditions are equivalent
Let
n
∈
N
n\in\mathbb{N}
n
∈
N
be given. Prove that the following two conditions are equivalent: (\text{i})\: n|a^n-a for any positive integer
a
a
a
; (\text{ii})\: For any prime divisor
p
p
p
of
n
n
n
,
p
2
∤
n
p^2 \nmid n
p
2
∤
n
and
p
−
1
∣
n
−
1
p-1|n-1
p
−
1∣
n
−
1
.
1
2
Hide problems
Find all solutions to the system
Given real numbers
b
≥
a
>
0
b \geq a>0
b
≥
a
>
0
, find all solutions of the system \begin{align*} &x_1^2+2ax_1+b^2=x_2,\\ &x_2^2+2ax_2+b^2=x_3,\\ &\qquad\cdots\cdots\cdots\\ &x_n^2+2ax_n+b^2=x_1. \end{align*}
Find angle measure in convex quadrilateral
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
it is given that
∠
C
A
B
=
4
0
∘
,
∠
C
A
D
=
3
0
∘
,
∠
D
B
A
=
7
5
∘
\angle{CAB} = 40^{\circ}, \angle{CAD} = 30^{\circ}, \angle{DBA} = 75^{\circ}
∠
C
A
B
=
4
0
∘
,
∠
C
A
D
=
3
0
∘
,
∠
D
B
A
=
7
5
∘
, and
∠
D
B
C
=
2
5
∘
\angle{DBC}=25^{\circ}
∠
D
BC
=
2
5
∘
. Find
∠
B
D
C
\angle{BDC}
∠
B
D
C
.