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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
1999 Ukraine National Mathematical Olympiad
1999 Ukraine National Mathematical Olympiad
Part of
Ukraine National Mathematical Olympiad
Subcontests
(8)
Problem 4
2
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compass+straightedge construction of triangle given circumcircle
The bisectors of angles
A
,
B
,
C
A,B,C
A
,
B
,
C
of a triangle
A
B
C
ABC
A
BC
intersect the circumcircle of the triangle at
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
, respectively. Let
P
P
P
be the intersection of the lines
B
1
C
1
B_1C_1
B
1
C
1
and
A
B
AB
A
B
, and
Q
Q
Q
be the intersection of the lines
B
1
A
1
B_1A_1
B
1
A
1
and
B
C
BC
BC
. Show how to construct the triangle
A
B
C
ABC
A
BC
by a ruler and a compass, given its circumcircle, points
P
P
P
and
Q
Q
Q
, and the halfplane determined by
P
Q
PQ
PQ
in which point
B
B
B
lies.
writing fibonacci-like sequence
Two players alternately write integers on a blackboard as follows: the first player writes
a
1
a_1
a
1
arbitrarily, then the second player writes
a
2
a_2
a
2
arbitrarily, and thereafter a player writes a number that is equal to the sum of the two preceding numbers. The player after whose move the obtained sequence contains terms such that
a
i
−
a
j
a_i-a_j
a
i
−
a
j
and
a
i
+
1
−
a
j
+
1
(
i
≠
j
)
a_{i+1}-a_{j+1}~(i\ne j)
a
i
+
1
−
a
j
+
1
(
i
=
j
)
are divisible by
1999
1999
1999
, wins the game. Which of the players has a winning strategy?
Problem 3
3
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2000-digit square with 1999 fives
Is there a
2000
2000
2000
-digit number which is a perfect square and
1999
1999
1999
of whose digits are fives?
9999999+1999000 is composite
Show that the number
9999999
+
1999000
9999999+1999000
9999999
+
1999000
is composite.
angle of intersection of planes
All faces of a parallelepiped
A
B
C
D
A
1
B
1
C
1
D
1
ABCDA_1B_1C_1D_1
A
BC
D
A
1
B
1
C
1
D
1
are rhombi, and their angles at
A
A
A
are all equal to
α
\alpha
α
. Points
M
,
N
,
P
,
Q
M,N,P,Q
M
,
N
,
P
,
Q
are selected on the edges
A
1
B
1
,
D
C
,
B
C
,
A
1
D
1
A_1B_1,DC,BC,A_1D_1
A
1
B
1
,
D
C
,
BC
,
A
1
D
1
, respectively, such that
A
1
M
=
B
P
A_1M=BP
A
1
M
=
BP
and
D
N
=
A
1
Q
DN=A_1Q
D
N
=
A
1
Q
. Find the angle between the intersection lines of the plane
A
1
B
D
A_1BD
A
1
B
D
with the planes
A
M
N
AMN
A
MN
and
A
P
Q
APQ
A
PQ
.
Problem 2
4
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Problem 1
4
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Problem 6
4
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Problem 7
3
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constant quantity of chords inside circle
Let
M
M
M
be a fixed point inside a given circle. Two perpendicular chords
A
C
AC
A
C
and
B
D
BD
B
D
are drawn through
M
M
M
, and
K
K
K
and
L
L
L
are the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
, respectively. Prove that the quantity
A
B
2
+
C
D
2
−
2
K
L
2
AB^2+CD^2-2KL^2
A
B
2
+
C
D
2
−
2
K
L
2
is independent of the chords
A
C
AC
A
C
and
B
D
BD
B
D
.
Ukraine 1999
If
x
1
x_1
x
1
,
x
2
x_2
x
2
, ...,
x
6
x_6
x
6
∈
\in
∈
[
0
,
1
]
[0,1]
[
0
,
1
]
, prove that the cyclic sum of
x
1
3
x
2
5
+
x
3
5
+
x
4
5
+
x
5
5
+
x
6
5
+
5
\frac{x_1^3}{x_2^5+x_3^5+x_4^5+x_5^5+x_6^5+5}
x
2
5
+
x
3
5
+
x
4
5
+
x
5
5
+
x
6
5
+
5
x
1
3
is less than
3
5
\frac{3}{5}
5
3
.
bounding period of trig function
Suppose that the function
f
(
x
)
=
tan
(
a
1
x
+
1
)
+
…
+
tan
(
a
10
x
+
1
)
f(x)=\tan(a_1x+1)+\ldots+\tan(a_{10}x+1)
f
(
x
)
=
tan
(
a
1
x
+
1
)
+
…
+
tan
(
a
10
x
+
1
)
has the period
T
>
0
T>0
T
>
0
, where
a
1
,
…
,
a
10
a_1,\ldots,a_{10}
a
1
,
…
,
a
10
are positive numbers. Prove that
T
≥
π
10
min
{
1
a
1
,
…
,
1
a
10
}
.
T\ge\frac\pi{10}\min\left\{\frac1{a_1},\ldots,\frac1{a_{10}}\right\}.
T
≥
10
π
min
{
a
1
1
,
…
,
a
10
1
}
.
Problem 8
2
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f(f(n))+f(n)=2n
find all function such that
f
:
N
−
>
N
f:N->N
f
:
N
−
>
N
for every
n
∈
N
n\in\mathbb{N}
n
∈
N
,
f
(
f
(
n
)
)
+
f
(
n
)
=
2
n
f(f(n))+f(n)=2n
f
(
f
(
n
))
+
f
(
n
)
=
2
n
PLZ->(solution not to use characterastic equation)<-
concurrency of lines formed by perpendiculars
Let
A
A
1
,
B
B
1
,
C
C
1
AA_1,BB_1,CC_1
A
A
1
,
B
B
1
,
C
C
1
be the altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
, and let
O
O
O
be an arbitrary interior point. Let
M
,
N
,
P
,
Q
,
R
,
S
M,N,P,Q,R,S
M
,
N
,
P
,
Q
,
R
,
S
be the feet of the perpendiculars from
O
O
O
to the lines
A
A
1
,
B
C
,
B
B
1
,
C
A
,
C
C
1
,
A
B
AA_1,BC,BB_1,CA,CC_1,AB
A
A
1
,
BC
,
B
B
1
,
C
A
,
C
C
1
,
A
B
, respectively. Prove that the lines
M
N
,
P
Q
,
R
S
MN,PQ,RS
MN
,
PQ
,
RS
are concurrent.
Problem 5
4
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