MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2014 Ukraine National Mathematical Olympiad
2014 Ukraine National Mathematical Olympiad
Part of
Ukraine National Mathematical Olympiad
Subcontests
(4)
3
2
Hide problems
Natural numbers satisfying an inequality
It is known that for natural numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
and
n
n
n
the following inequalities hold:
a
+
c
<
n
a+c<n
a
+
c
<
n
and
a
/
b
+
c
/
d
<
1
a/b+c/d<1
a
/
b
+
c
/
d
<
1
. Prove that
a
/
b
+
c
/
d
<
1
−
1
/
n
3
a/b+c/d<1-1/n^3
a
/
b
+
c
/
d
<
1
−
1/
n
3
.
Points and planes in space
Is it possible to choose
24
24
24
points in the space,such that no three of them lie on the same line and choose
2013
2013
2013
planes in such a way that each plane passes through at least
3
3
3
of the chosen points and each triple of points belongs to at least one of the chosen planes?
2
2
Hide problems
Parallel lines and midpoints
Let
M
M
M
be the midpoint of the side
B
C
BC
BC
of
△
A
B
C
\triangle ABC
△
A
BC
. On the side
A
B
AB
A
B
and
A
C
AC
A
C
the points
E
E
E
and
F
F
F
are chosen. Let
K
K
K
be the point of the intersection of
B
F
BF
BF
and
C
E
CE
CE
and
L
L
L
be chosen in a way that
C
L
∥
A
B
CL\parallel AB
C
L
∥
A
B
and
B
L
∥
C
E
BL\parallel CE
B
L
∥
CE
. Let
N
N
N
be the point of intersection of
A
M
AM
A
M
and
C
L
CL
C
L
. Show that
K
N
KN
K
N
is parallel to
F
L
FL
F
L
. Edit:Fixed typographical error.
Diophantine equation involving prime powers
Find all pairs of prime numbers
p
p
p
and
q
q
q
that satisfy the equation
3
p
q
−
2
q
p
−
1
=
19
3p^{q}-2q^{p-1}=19
3
p
q
−
2
q
p
−
1
=
19
.
4
2
Hide problems
Card distribution
There are
100
100
100
cards with numbers from
1
1
1
to
100
100
100
on the table.Andriy and Nick took the same number of cards in a way such that the following condition holds:if Andriy has a card with a number
n
n
n
then Nick has a card with a number
2
n
+
2
2n+2
2
n
+
2
.What is the maximal number of cards that could be taken by the two guys?
Midpoint of internal bisector
Let
M
M
M
be the midpoint of the internal bisector
A
D
AD
A
D
of
△
A
B
C
\triangle ABC
△
A
BC
.Circle
ω
1
\omega_1
ω
1
with diameter
A
C
AC
A
C
intersects
B
M
BM
BM
at
E
E
E
and circle
ω
2
\omega_2
ω
2
with diameter
A
B
AB
A
B
intersects
C
M
CM
CM
at
F
F
F
.Show that
B
,
E
,
F
,
C
B,E,F,C
B
,
E
,
F
,
C
are concyclic.
1
2
Hide problems
Equality of fractional parts
Suppose that for real
x
,
y
,
z
,
t
x,y,z,t
x
,
y
,
z
,
t
the following equalities hold:
{
x
+
y
+
z
}
=
{
y
+
z
+
t
}
=
{
z
+
t
+
x
}
=
{
t
+
x
+
y
}
=
1
/
4
\{x+y+z\}=\{y+z+t\}=\{z+t+x\}=\{t+x+y\}=1/4
{
x
+
y
+
z
}
=
{
y
+
z
+
t
}
=
{
z
+
t
+
x
}
=
{
t
+
x
+
y
}
=
1/4
. Find all possible values of
{
x
+
y
+
z
+
t
}
\{x+y+z+t\}
{
x
+
y
+
z
+
t
}
.(Here
{
x
}
=
x
−
[
x
]
\{x\}=x-[x]
{
x
}
=
x
−
[
x
]
)
Trigonometric inequality regarding minimums
Find the values of
x
x
x
such that the following inequality holds:
min
{
sin
x
,
cos
x
}
<
min
{
1
−
sin
x
,
1
−
cos
x
}
\min\{\sin x,\cos x\}<\min\{1-\sin x,1-\cos x\}
min
{
sin
x
,
cos
x
}
<
min
{
1
−
sin
x
,
1
−
cos
x
}