MathDB
Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2002 Kazakhstan National Olympiad
2002 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(6)
8
1
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N grasshoppers are lined up in a row, jumping over exactly 2
N
N
N
grasshoppers are lined up in a row. At any time, one grasshopper is allowed to jump over exactly two grasshoppers standing to the right or left of him. At what
n
n
n
can grasshoppers rearrange themselves in reverse order?
7
1
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for any integers n>m>0, number 2^n-1 has a prime divisor not dividing 2^m-1
Prove that for any integers
n
>
m
>
0
n> m> 0
n
>
m
>
0
the number
2
n
−
1
2 ^n-1
2
n
−
1
has a prime divisor not dividing
2
m
−
1
2 ^m-1
2
m
−
1
.
6
1
Hide problems
P (x^2) = P (x) P (x + 1)
Find all polynomials
P
(
x
)
P (x)
P
(
x
)
with real coefficients that satisfy the identity
P
(
x
2
)
=
P
(
x
)
P
(
x
+
1
)
P (x ^ 2) = P (x) P (x + 1)
P
(
x
2
)
=
P
(
x
)
P
(
x
+
1
)
.
4
1
Hide problems
product of all numbers from A-{a} , when divided by a remainder `
Prove that there is a set
A
A
A
consisting of
2002
2002
2002
different natural numbers satisfying the condition: for each
a
∈
A
a \in A
a
∈
A
, the product of all numbers from
A
A
A
, except
a
a
a
, when divided by
a
a
a
gives the remainder
1
1
1
.
1
1
Hide problems
equal segments if an angle is double of another angle, incenter related
Let
O
O
O
be the center of the inscribed circle of the triangle
A
B
C
ABC
A
BC
, tangent to the side of
B
C
BC
BC
. Let
M
M
M
be the midpoint of
A
C
AC
A
C
, and
P
P
P
be the intersection point of
M
O
MO
MO
and
B
C
BC
BC
. Prove that
A
B
=
B
P
AB = BP
A
B
=
BP
if
∠
B
A
C
=
2
∠
A
C
B
\angle BAC = 2 \angle ACB
∠
B
A
C
=
2∠
A
CB
.
5
1
Hide problems
concurrent circumcircles, tangents and bases of altitudes related
On the plane is given the acute triangle
A
B
C
ABC
A
BC
. Let
A
1
A_1
A
1
and
B
1
B_1
B
1
be the feet of the altitudes of
A
A
A
and
B
B
B
drawn from those vertices, respectively. Tangents at points
A
1
A_1
A
1
and
B
1
B_1
B
1
drawn to the circumscribed circle of the triangle
C
A
1
B
1
CA_1B_1
C
A
1
B
1
intersect at
M
M
M
. Prove that the circles circumscribed around the triangles
A
M
B
1
AMB_1
A
M
B
1
,
B
M
A
1
BMA_1
BM
A
1
and
C
A
1
B
1
CA_1B_1
C
A
1
B
1
have a common point.