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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
1997 Ukraine National Mathematical Olympiad
1997 Ukraine National Mathematical Olympiad
Part of
Ukraine National Mathematical Olympiad
Subcontests
(8)
8
3
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easy inequality
For every natural number
n
n
n
prove the inequality: \sqrt{2\plus{}\sqrt{2\plus{}\sqrt{2\plus{}...\plus{}\sqrt{2}}}}\plus{}\sqrt{6\plus{}\sqrt{6\plus{}\sqrt{6\plus{}...\plus{}\sqrt{6}}}}<5 where there are
n
n
n
twos and
n
n
n
sixs on the left hand side.
find k
Let
d
(
n
)
d(n)
d
(
n
)
denote the largest odd divisor of a positive integer
n
n
n
. The function
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
is defined by f(2n\minus{}1)\equal{}2^n and f(2n)\equal{}n\plus{}\frac{2n}{d(n)} for all
n
∈
N
n \in \mathbb{N}
n
∈
N
. Find all natural numbers
k
k
k
such that: f(f(...f(1)...))\equal{}1997. (where the paranthesis appear
k
k
k
times)
solid geometry
On the edges
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
of a parallelepiped
A
B
C
D
A
1
B
1
C
1
D
1
ABCDA_1 B_1 C_1 D_1
A
BC
D
A
1
B
1
C
1
D
1
points
K
,
L
,
M
,
N
K,L,M,N
K
,
L
,
M
,
N
are selected, respectively. Prove that the circumcenters of the tetrahedra
A
1
A
K
N
,
B
1
B
K
L
,
C
1
C
L
M
,
D
1
D
M
N
A_1 AKN, B_1 BKL, C_1 CLM, D_1 DMN
A
1
A
K
N
,
B
1
B
K
L
,
C
1
C
L
M
,
D
1
D
MN
are vertices of a parallelogram.
7
3
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inequality
Triangles
A
B
C
ABC
A
BC
and
A
1
B
1
C
1
A_1 B_1 C_1
A
1
B
1
C
1
are non-congruent, but AC\equal{}A_1 C_1\equal{}b, BC\equal{}B_1 C_1\equal{}a, and BH\equal{}B_1 H_1, where
B
H
BH
B
H
and
B
1
H
1
B_1 H_1
B
1
H
1
are the altitudes. Prove the inequality: a \cdot AB\plus{}b \cdot A_1 B_1 \le \sqrt{2}(a^2\plus{}b^2).
equal areas
In a parallelogram
A
B
C
D
ABCD
A
BC
D
,
M
M
M
is the midpoint of
B
C
BC
BC
and
N
N
N
an arbitrary point on the side
A
D
AD
A
D
. Let
P
P
P
be the intersection of
M
N
MN
MN
and
A
C
AC
A
C
, and
Q
Q
Q
the intersection of
A
M
AM
A
M
and
B
N
BN
BN
. Prove that the triangles
B
D
Q
BDQ
B
D
Q
and
D
M
P
DMP
D
MP
have equal areas.
find the smallest natural number
Determine the smallest natural number
n
n
n
such that among any
n
n
n
integers one can choose
18
18
18
integers whose sum is divisible by
18
18
18
.
6
4
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5
3
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construct the bisector
Construct the bisector of a given angle using a ruler and a compass, but without marking any auxiliary points inside the angle.
find the smallest number
Find the smallest natural number which can be represented in the form 19m\plus{}97n, where
m
,
n
m,n
m
,
n
are positive integers, in at least two different ways.
find all solutions
Find all real solutions to the equation: (2\plus{}\sqrt{3})^x\plus{}1\equal{}\left( 2\sqrt{2\plus{}\sqrt{3}} \right)^x.
4
4
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3
4
Show problems
2
4
Show problems
1
4
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