MathDB
Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2005 Kazakhstan National Olympiad
2005 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(4)
3
2
Hide problems
Good sets of points on the plane
Call a set of points in the plane
g
o
o
d
good
g
oo
d
if any three points of the set are the vertices of an isosceles triangle or if they are collinear. Determine all
a
)
a)
a
)
6-element
g
o
o
d
good
g
oo
d
sets
b
)
b)
b
)
7-element
g
o
o
d
good
g
oo
d
sets.
combinatorial number theory ?
Exactly one number from the set
{
−
1
,
0
,
1
}
\{ -1,0,1 \}
{
−
1
,
0
,
1
}
is written in each unit cell of a
2005
×
2005
2005 \times 2005
2005
×
2005
table, so that the sum of all the entries is
0
0
0
. Prove that there exist two rows and two columns of the table, such that the sum of the four numbers written at the intersections of these rows and columns is equal to
0
0
0
.
1
2
Hide problems
Solve the half-trigonometric equation
Solve equation
2
1
2
−
2
∣
x
∣
=
∣
tan
x
+
1
2
∣
+
∣
tan
x
−
1
2
∣
2^{\tfrac{1}{2}-2|x|} = \left| {\tan x + \frac{1}{2}} \right| + \left| {\tan x - \frac{1}{2}} \right|
2
2
1
−
2∣
x
∣
=
tan
x
+
2
1
+
tan
x
−
2
1
system of equations
Does there exist a solution in real numbers of the system of equations
{
(
x
−
y
)
(
z
−
t
)
(
z
−
x
)
(
z
−
t
)
2
=
A
,
(
y
−
z
)
(
t
−
x
)
(
t
−
y
)
(
x
−
z
)
2
=
B
,
(
x
−
z
)
(
y
−
t
)
(
z
−
t
)
(
y
−
z
)
2
=
C
,
\left\{ \begin{array}{rcl} (x - y)(z - t)(z - x)(z - t)^2 = A, \\ (y - z)(t - x)(t - y)(x - z)^2 = B,\\ (x - z)(y - t)(z - t)(y - z)^2 = C,\\ \end{array} \right.
⎩
⎨
⎧
(
x
−
y
)
(
z
−
t
)
(
z
−
x
)
(
z
−
t
)
2
=
A
,
(
y
−
z
)
(
t
−
x
)
(
t
−
y
)
(
x
−
z
)
2
=
B
,
(
x
−
z
)
(
y
−
t
)
(
z
−
t
)
(
y
−
z
)
2
=
C
,
when a)
A
=
2
,
B
=
8
,
C
=
6
;
A=2, B=8, C=6;
A
=
2
,
B
=
8
,
C
=
6
;
b)
A
=
2
,
B
=
6
,
C
=
8.
A=2, B=6, C=8.
A
=
2
,
B
=
6
,
C
=
8.
?
2
2
Hide problems
a+b+c+2=abc
Prove that
a
b
+
b
c
+
c
a
≥
2
(
a
+
b
+
c
)
ab+bc+ca\ge 2(a+b+c)
ab
+
b
c
+
c
a
≥
2
(
a
+
b
+
c
)
where
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive reals such that
a
+
b
+
c
+
2
=
a
b
c
a+b+c+2=abc
a
+
b
+
c
+
2
=
ab
c
.
Find the angle ∠MTB
The line parallel to side
A
C
AC
A
C
of a right triangle
A
B
C
ABC
A
BC
(
∠
C
=
9
0
∘
)
(\angle C=90^\circ)
(
∠
C
=
9
0
∘
)
intersects sides
A
B
AB
A
B
and
B
C
BC
BC
at
M
M
M
and
N
N
N
, respectively, so that the
C
N
/
B
N
=
A
C
/
B
C
=
2
CN / BN = AC / BC = 2
CN
/
BN
=
A
C
/
BC
=
2
. Let
O
O
O
be the intersection point of the segments
A
N
AN
A
N
and
C
M
CM
CM
and
K
K
K
be a point on the segment
O
N
ON
ON
such that
M
O
+
O
K
=
K
N
MO + OK = KN
MO
+
O
K
=
K
N
. The perpendicular line to
A
N
AN
A
N
at point
K
K
K
and the bisector of triangle
A
B
C
ABC
A
BC
of
∠
B
\angle B
∠
B
meet at point
T
T
T
. Find the angle
∠
M
T
B
\angle MTB
∠
MTB
.
4
2
Hide problems
Functional equation
Find all functions
f
:
R
→
R
f :\mathbb{R}\to\mathbb{R}
f
:
R
→
R
, satisfying the condition
f
(
f
(
x
)
+
x
+
y
)
=
2
x
+
f
(
y
)
f(f(x)+x+y)=2x+f(y)
f
(
f
(
x
)
+
x
+
y
)
=
2
x
+
f
(
y
)
for any real
x
x
x
and
y
y
y
.
Find all polynomials
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that for every positive integer
n
n
n
there exists a rational
r
r
r
with
P
(
r
)
=
n
P(r)=n
P
(
r
)
=
n
.