MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2002 Turkey Team Selection Test
2002 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
3
2
Hide problems
Cutting a seq. into two s.t. |Sum_1 - Sum_2| <= |some term|
A positive integer
n
n
n
and real numbers
a
1
,
…
,
a
n
a_1,\dots, a_n
a
1
,
…
,
a
n
are given. Show that there exists integers
m
m
m
and
k
k
k
such that
∣
∑
i
=
1
m
a
i
−
∑
i
=
m
+
1
n
a
i
∣
≤
∣
a
k
∣
.
|\sum\limits_{i=1}^m a_i -\sum\limits_{i=m+1}^n a_i | \leq |a_k|.
∣
i
=
1
∑
m
a
i
−
i
=
m
+
1
∑
n
a
i
∣
≤
∣
a
k
∣.
Connected Monochromatic k-subset
Consider
2
n
+
1
2n+1
2
n
+
1
points in space, no four of which are coplanar where
n
>
1
n>1
n
>
1
. Each line segment connecting any two of these points is either colored red, white or blue. A subset
M
M
M
of these points is called a connected monochromatic subset, if for each
a
,
b
∈
M
a,b \in M
a
,
b
∈
M
, there are points
a
=
x
0
,
x
1
,
…
,
x
l
=
b
a=x_0,x_1, \dots, x_l = b
a
=
x
0
,
x
1
,
…
,
x
l
=
b
that belong to
M
M
M
such that the line segments
x
0
x
1
,
x
1
x
2
,
…
,
x
l
−
1
x
1
x_0x_1, x_1x_2, \dots, x_{l-1}x_1
x
0
x
1
,
x
1
x
2
,
…
,
x
l
−
1
x
1
are all have the same color. No matter how the points are colored, if there always exists a connected monochromatic
k
−
k-
k
−
subset, find the largest value of
k
k
k
. (
l
>
1
l > 1
l
>
1
)
2
2
Hide problems
Given cut ratio of two angle bisectors
In a triangle
A
B
C
ABC
A
BC
, the angle bisector of
A
B
C
^
\widehat{ABC}
A
BC
meets
[
A
C
]
[AC]
[
A
C
]
at
D
D
D
, and the angle bisector of
B
C
A
^
\widehat{BCA}
BC
A
meets
[
A
B
]
[AB]
[
A
B
]
at
E
E
E
. Let
X
X
X
be the intersection of the lines
B
D
BD
B
D
and
C
E
CE
CE
where
∣
B
X
∣
=
3
∣
X
D
∣
|BX|=\sqrt 3|XD|
∣
BX
∣
=
3
∣
X
D
∣
ve
∣
X
E
∣
=
(
3
−
1
)
∣
X
C
∣
|XE|=(\sqrt 3 - 1)|XC|
∣
XE
∣
=
(
3
−
1
)
∣
XC
∣
. Find the angles of triangle
A
B
C
ABC
A
BC
.
Internally tangent circles
Two circles are internally tangent at a point
A
A
A
. Let
C
C
C
be a point on the smaller circle other than
A
A
A
. The tangent line to the smaller circle at
C
C
C
meets the bigger circle at
D
D
D
and
E
E
E
; and the line
A
C
AC
A
C
meets the bigger circle at
A
A
A
and
P
P
P
. Show that the line
P
E
PE
PE
is tangent to the circle through
A
A
A
,
C
C
C
, and
E
E
E
.
1
2
Hide problems
ab(a+b) | a^2+ab+b^2
If
a
b
(
a
+
b
)
ab(a+b)
ab
(
a
+
b
)
divides
a
2
+
a
b
+
b
2
a^2 + ab+ b^2
a
2
+
ab
+
b
2
for different integers
a
a
a
and
b
b
b
, prove that
∣
a
−
b
∣
>
a
b
3
.
|a-b|>\sqrt[3]{ab}.
∣
a
−
b
∣
>
3
ab
.
function as sum of a linear function and a periodic function
If a function
f
f
f
defined on all real numbers has at least two centers of symmetry, show that this function can be written as sum of a linear function and a periodic function.[For every real number
x
x
x
, if there is a real number
a
a
a
such that
f
(
a
−
x
)
+
f
(
a
+
x
)
=
2
f
(
a
)
f(a-x) + f(a+x) =2f(a)
f
(
a
−
x
)
+
f
(
a
+
x
)
=
2
f
(
a
)
, the point
(
a
,
f
(
a
)
)
(a,f(a))
(
a
,
f
(
a
))
is called a center of symmetry of the function
f
f
f
.]