MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2000 Turkey Team Selection Test
2000 Turkey Team Selection Test
Part of
Turkey Team Selection Test
Subcontests
(3)
2
1
Hide problems
ABCD is a rhombus, M,N,K,L∈AB,BC,CD,DA and MN||LK
Points
M
,
N
,
K
,
L
M,\ N,\ K,\ L
M
,
N
,
K
,
L
are taken on the sides
A
B
,
B
C
,
C
D
,
D
A
AB,\ BC,\ CD,\ DA
A
B
,
BC
,
C
D
,
D
A
of a rhombus
A
B
C
D
,
ABCD,
A
BC
D
,
respectively, in such a way that
M
N
∥
L
K
MN\parallel LK
MN
∥
L
K
and the distance between
M
N
MN
MN
and
K
L
KL
K
L
is equal to the height of
A
B
C
D
.
ABCD.
A
BC
D
.
Show that the circumcircles of the triangles
A
L
M
ALM
A
L
M
and
N
C
K
NCK
NC
K
intersect each other, while those of
L
D
K
LDK
L
DK
and
M
B
N
MBN
MBN
do not.
3
2
Hide problems
Sequences (x_k, y_k) and expressions P(x)=x+1,Q(x)=x^2+1
Let
P
(
x
)
=
x
+
1
P(x)=x+1
P
(
x
)
=
x
+
1
and
Q
(
x
)
=
x
2
+
1.
Q(x)=x^2+1.
Q
(
x
)
=
x
2
+
1.
Consider all sequences
⟨
(
x
k
,
y
k
)
⟩
k
∈
N
\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}
⟨(
x
k
,
y
k
)
⟩
k
∈
N
such that
(
x
1
,
y
1
)
=
(
1
,
3
)
(x_1,y_1)=(1,3)
(
x
1
,
y
1
)
=
(
1
,
3
)
and
(
x
k
+
1
,
y
k
+
1
)
(x_{k+1},y_{k+1})
(
x
k
+
1
,
y
k
+
1
)
is either
(
P
(
x
k
)
,
Q
(
y
k
)
)
(P(x_k), Q(y_k))
(
P
(
x
k
)
,
Q
(
y
k
))
or
(
Q
(
x
k
)
,
P
(
y
k
)
)
(Q(x_k),P(y_k))
(
Q
(
x
k
)
,
P
(
y
k
))
for each
k
.
k.
k
.
We say that a positive integer
n
n
n
is nice if
x
n
=
y
n
x_n=y_n
x
n
=
y
n
holds in at least one of these sequences. Find all nice numbers.
f:R→R; f(x+y)=f(x)+f(y)+1, Show that there exists g:R→R
Suppose
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
is a function such that
∣
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
∣
≤
1
for all
x
,
y
∈
R
.
|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R.
∣
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
∣
≤
1
for all
x
,
y
∈
R
.
Prove that there is a function
g
:
R
→
R
g:\mathbb{R}\to\mathbb{R}
g
:
R
→
R
such that
∣
f
(
x
)
−
g
(
x
)
∣
≤
1
|f(x)-g(x)|\le 1
∣
f
(
x
)
−
g
(
x
)
∣
≤
1
and
g
(
x
+
y
)
=
g
(
x
)
+
g
(
y
)
g(x+y)=g(x)+g(y)
g
(
x
+
y
)
=
g
(
x
)
+
g
(
y
)
for all
x
,
y
∈
R
.
x,y \in\mathbb R.
x
,
y
∈
R
.
1
2
Hide problems
Prove that number of ordered pairs (x,y) is divisible by 3
(
a
)
(a)
(
a
)
Prove that for every positive integer
n
n
n
, the number of ordered pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of integers satisfying
x
2
−
x
y
+
y
2
=
n
x^2-xy+y^2 = n
x
2
−
x
y
+
y
2
=
n
is divisible by
3.
3.
3.
(
b
)
(b)
(
b
)
Find all ordered pairs of integers satisfying
x
2
−
x
y
+
y
2
=
727.
x^2-xy+y^2=727.
x
2
−
x
y
+
y
2
=
727.
Triangular prism of infinite length cut by a plane
Show that any triangular prism of infinite length can be cut by a plane such that the resulting intersection is an equilateral triangle.