MathDB
Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2003 Kazakhstan National Olympiad
2003 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(7)
2
1
Hide problems
sum x ^ 3/ (x + y) >= (xy + yz + zx)/2
For positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
, prove the inequality: \displaylines {\frac {x ^ 3} {x + y} + \frac {y ^ 3} {y + z} + \frac {z ^ 3} {z + x} \geq \frac {xy + yz + zx} {2}.}
6
1
Hide problems
circumcenter of triangle BCD lies on the circumcircle of triangle ABC
Let the point
B
B
B
lie on the circle
S
1
S_1
S
1
and let the point
A
A
A
, other than the point
B
B
B
, lie on the tangent to the circle
S
1
S_1
S
1
passing through the point
B
B
B
. Let a point
C
C
C
be chosen outside the circle
S
1
S_1
S
1
, so that the segment
A
C
AC
A
C
intersects
S
1
S_1
S
1
at two different points. Let the circle
S
2
S_2
S
2
touch the line
A
C
AC
A
C
at the point
C
C
C
and the circle
S
1
S_1
S
1
at the point
D
D
D
, on the opposite side from the point
B
B
B
with respect to the line
A
C
AC
A
C
. Prove that the center of the circumcircle of triangle
B
C
D
BCD
BC
D
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
.
4
1
Hide problems
concurrent lines starting with touchpoint of incircle with triangle
Let the inscribed circle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
touch the side
B
C
BC
BC
at the point
A
′
A '
A
′
. Let
A
A
′
AA '
A
A
′
intersect
ω
\omega
ω
at
P
≠
A
P \neq A
P
=
A
. Let
C
P
CP
CP
and
B
P
BP
BP
intersect
ω
\omega
ω
, respectively, at points
N
N
N
and
M
M
M
other than
P
P
P
. Prove that
A
A
′
,
B
N
AA ', BN
A
A
′
,
BN
and
C
M
CM
CM
intersect at one point.
5
1
Hide problems
national kazakhstan 2003
Prove that for all primes
p
>
3
p>3
p
>
3
,
(
2
p
p
)
−
2
\binom{2p}{p}-2
(
p
2
p
)
−
2
is divisible by
p
3
p^3
p
3
8
1
Hide problems
Function Equation German TST 2003 - f(f(x)+y)=2x+f(f(y)-x)
Determine all functions
f
:
R
→
R
f: \mathbb R \to \mathbb R
f
:
R
→
R
with the property f(f(x)+y)=2x+f(f(y)-x), \forall x,y \in \mathbb R.
3
1
Hide problems
Classical: Overlay two sheets and puncture them 2003 times
Two square sheets have areas equal to
2003
2003
2003
. Each of the sheets is arbitrarily divided into
2003
2003
2003
nonoverlapping polygons, besides, each of the polygons has an unitary area. Afterward, one overlays two sheets, and it is asked to prove that the obtained double layer can be punctured
2003
2003
2003
times, so that each of the
4006
4006
4006
polygons gets punctured precisely once.
1
1
Hide problems
Find all n
Find all natural numbers
n
n
n
,such that there exist x_1,x_2,\dots,x_{n\plus{}1}\in\mathbb{N},such that \frac{1}{x_1^2}\plus{}\frac{1}{x_2^2}\plus{}\dots\plus{}\frac{1}{x_n^2}\equal{}\frac{n\plus{}1}{x_{n\plus{}1}^2}.