MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1993 Turkey Team Selection Test
1993 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(6)
6
1
Hide problems
Functional equation from Q^+ to Q^+
Determine all functions
f
:
Q
+
→
Q
+
f: \mathbb{Q^+} \rightarrow \mathbb{Q^+}
f
:
Q
+
→
Q
+
that satisfy:
f
(
x
+
y
x
)
=
f
(
x
)
+
f
(
y
x
)
+
2
y
for all
x
,
y
∈
Q
+
f\left(x+\frac{y}{x}\right) = f(x)+f\left(\frac{y}{x}\right)+2y \:\text{for all}\: x, y \in \mathbb{Q^+}
f
(
x
+
x
y
)
=
f
(
x
)
+
f
(
x
y
)
+
2
y
for all
x
,
y
∈
Q
+
5
1
Hide problems
Finding angle values such that quadrilateral is tangent
Points
E
E
E
and
C
C
C
are chosen on a semicircle with diameter
A
B
AB
A
B
and center
O
O
O
such that
O
E
⊥
A
B
OE \perp AB
OE
⊥
A
B
and the intersection point
D
D
D
of
A
C
AC
A
C
and
O
E
OE
OE
is inside the semicircle. Find all values of
∠
C
A
B
\angle{CAB}
∠
C
A
B
for which the quadrilateral
O
B
C
D
OBCD
OBC
D
is tangent.
4
1
Hide problems
Traveling between towns
Some towns are connected by roads, with at most one road between any two towns. Let
v
v
v
be the number of towns and
e
e
e
be the number of roads. Prove that
(
a
)
(a)
(
a
)
if
e
<
v
−
1
e<v-1
e
<
v
−
1
, then there are two towns such that one cannot travel between them;
(
b
)
(b)
(
b
)
if
2
e
>
(
v
−
1
)
(
v
−
2
)
2e>(v-1)(v-2)
2
e
>
(
v
−
1
)
(
v
−
2
)
, then one can travel between any two towns.
3
1
Hide problems
There exists natural K such that sum > 1993/1000
Let (
b
n
b_n
b
n
) be a sequence such that
b
n
≥
0
b_n \geq 0
b
n
≥
0
and
b
n
+
1
2
≥
b
1
2
1
3
+
⋯
+
b
n
2
n
3
b_{n+1}^2 \geq \frac{b_1^2}{1^3}+\cdots+\frac{b_n^2}{n^3}
b
n
+
1
2
≥
1
3
b
1
2
+
⋯
+
n
3
b
n
2
for all
n
≥
1
n \geq 1
n
≥
1
. Prove that there exists a natural number
K
K
K
such that
∑
n
=
1
K
b
n
+
1
b
1
+
b
2
+
⋯
+
b
n
≥
1993
1000
\sum_{n=1}^{K} \frac{b_{n+1}}{b_1+b_2+ \cdots + b_n} \geq \frac{1993}{1000}
n
=
1
∑
K
b
1
+
b
2
+
⋯
+
b
n
b
n
+
1
≥
1000
1993
2
1
Hide problems
Show that CM is perpendicular to PQ with circumcenter M
Let
M
M
M
be the circumcenter of an acute-angled triangle
A
B
C
ABC
A
BC
. The circumcircle of triangle
B
M
A
BMA
BM
A
intersects
B
C
BC
BC
at
P
P
P
and
A
C
AC
A
C
at
Q
Q
Q
. Show that
C
M
⊥
P
Q
CM \perp PQ
CM
⊥
PQ
.
1
1
Hide problems
Smallest common difference of arithmetic progressions
Show that there exists an infinite arithmetic progression of natural numbers such that the first term is
16
16
16
and the number of positive divisors of each term is divisible by
5
5
5
. Of all such sequences, find the one with the smallest possible common difference.