MathDB
Problems
Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2013 239 Open Mathematical Olympiad
2013 239 Open Mathematical Olympiad
Part of
239 Open Math Olympiad
Subcontests
(8)
8
2
Hide problems
Chords of a circle with 10^100 points
Prove that if you choose
1
0
100
10^{100}
1
0
100
points on a circle and arrange numbers from
1
1
1
to
1
0
100
10^{100}
1
0
100
on them in some order, then you can choose
100
100
100
pairwise disjoint chords with ends at the selected points such that the sums of the numbers at the ends of all of them are equal to each other.
5 variable inequality with product of 1
The product of the positive numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
and
e
e
e
is equal to
1
1
1
. Prove that
a
2
b
2
+
b
2
c
2
+
c
2
d
2
+
d
2
e
2
+
e
2
a
2
≥
a
+
b
+
c
+
d
+
e
.
\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{e^2}+\frac{e^2}{a^2} \geq a+b+c+d+e .
b
2
a
2
+
c
2
b
2
+
d
2
c
2
+
e
2
d
2
+
a
2
e
2
≥
a
+
b
+
c
+
d
+
e
.
7
2
Hide problems
Geometric inequality in convex quadrilateral
Point
M
M
M
is the midpoint of side
B
C
BC
BC
of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. If
∠
A
M
D
<
12
0
∘
\angle{AMD} < 120^{\circ}
∠
A
M
D
<
12
0
∘
. Prove that
(
A
B
+
A
M
)
2
+
(
C
D
+
D
M
)
2
>
A
D
⋅
B
C
+
2
A
B
⋅
C
D
.
(AB+AM)^2 + (CD+DM)^2 > AD \cdot BC + 2AB \cdot CD.
(
A
B
+
A
M
)
2
+
(
C
D
+
D
M
)
2
>
A
D
⋅
BC
+
2
A
B
⋅
C
D
.
Removing numbers on the board
Dima wrote several natural numbers on the blackboard and underlined some of them. Misha wants to erase several numbers (but not all) such that a multiple of three underlined numbers remain and the total amount of the remaining numbers would be divisible by
2013
2013
2013
; but after trying for a while he realizes that it's impossible to do this. What is the largest number of the numbers on the board?
6
2
Hide problems
Cutting a convex polyhedron into pieces
Convex polyhedron
M
M
M
with triangular faces is cut into tetrahedrons; all the vertices of the tetrahedrons are the vertices of the polyhedron, and any two tetrahedrons either do not intersect, or they intersect along a common vertex, common edge, or common face. Prove that it it's not possible that each tetrahedron has exactly one face on the surface of
M
M
M
.
Infinite checkered plane
A quarter of an checkered plane is given, infinite to the right and up. All its rows and columns are numbered starting from
0
0
0
. All cells with coordinates
(
2
n
,
n
)
(2n, n)
(
2
n
,
n
)
, were cut out from this figure, starting from
n
=
1
n = 1
n
=
1
. In each of the remaining cells they wrote a number, the number of paths from the corner cell to this one (you can only walk up and to the right and you cannot pass through the removed cells). Prove that for each removed cell the numbers to the left and below it differ by exactly
2
2
2
.
5
1
Hide problems
Infinite nuts in some bags
A squirrel has infinitely many nuts; one nut of each of the masses
1
g
,
2
g
,
3
g
,
…
1g, 2g, 3g, \ldots
1
g
,
2
g
,
3
g
,
…
. The squirrel took
100
100
100
bags, in each put a finite number of nuts, after which wrote on each bag the total mass of the nuts inside it. Prove that it is possible to create bags of the same mass using no more than
500
500
500
nuts.
4
2
Hide problems
Sqrt inequality with a+b+c<2
For positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying condition
a
+
b
+
c
<
2
a+b+c<2
a
+
b
+
c
<
2
, Prove that
a
2
+
b
c
+
b
2
+
c
a
+
c
2
+
a
b
<
3.
\sqrt{a^2 +bc}+\sqrt{b^2 +ca}+\sqrt{c^2 + ab}<3.
a
2
+
b
c
+
b
2
+
c
a
+
c
2
+
ab
<
3.
Lesser degree on the edges
We are given a graph
G
G
G
with
n
n
n
edges. For each edge, we write down the lesser degree of two vertices at the end of that edge. Prove that the sum of the resulting
n
n
n
numbers is at most
100
n
n
100n\sqrt{n}
100
n
n
.
3
2
Hide problems
A point on the incircle
Inside a regular triangle
A
B
C
ABC
A
BC
, points
X
X
X
and
Y
Y
Y
are chosen such that
∠
A
X
C
=
12
0
∘
\angle{AXC} = 120^{\circ}
∠
A
XC
=
12
0
∘
,
2
∠
X
A
C
+
∠
Y
B
C
=
9
0
∘
2\angle{XAC} + \angle{YBC} = 90^{\circ}
2∠
X
A
C
+
∠
Y
BC
=
9
0
∘
and
X
Y
=
Y
B
=
A
C
3
XY = YB = \frac{AC}{\sqrt{3}}
X
Y
=
Y
B
=
3
A
C
. Prove that point
Y
Y
Y
lies on the incircle of triangle
A
B
C
ABC
A
BC
.
Intersection points of two circles
The altitudes
A
A
1
AA_1
A
A
1
and
C
C
1
CC_1
C
C
1
of an acute-angled triangle
A
B
C
ABC
A
BC
intersect at point
H
H
H
. A straight line passing through
H
H
H
parallel to line
A
1
C
1
A_1C_1
A
1
C
1
intersects the circumscribed circles of triangles
A
H
C
1
AHC_1
A
H
C
1
and
C
H
A
1
CHA_1
C
H
A
1
at points
X
X
X
and
Y
Y
Y
, respectively. Prove that points
X
X
X
and
Y
Y
Y
are equidistant from the midpoint of segment
B
H
BH
B
H
.
2
2
Hide problems
kn = 1+2+...+n (mod 10^100)
For some
99
99
99
-digit number
k
k
k
, there exist two different
100
100
100
-digit numbers
n
n
n
such that the sum of all natural numbers from
1
1
1
to
n
n
n
ends in the same
100
100
100
digits as the number
k
n
kn
kn
, but is not equal to it. Prove that
k
−
3
k-3
k
−
3
is divisible by
5
5
5
.
Union of subsets
In the set
A
A
A
with
n
n
n
elements,
[
2
n
]
+
2
[\sqrt{2n}]+2
[
2
n
]
+
2
subsets are chosen such that the union of any three of them is equal to
A
A
A
. Prove that the union of any two of them is equal to
A
A
A
as well.
1
2
Hide problems
Double permutations
Consider all permutations of natural numbers from
1
1
1
to
100
100
100
. A permutation is called \emph{double} when it has the following property: If you write this permutation twice in a row, then delete
100
100
100
numbers from them you get the remaining numbers
1
,
2
,
3
,
…
,
100
1, 2, 3, \ldots , 100
1
,
2
,
3
,
…
,
100
in order. How many \emph{double} permutations are there?
Divisors mod 2013
Among the divisors of a natural number
n
n
n
, we have numbers such that when they are devided by
2013
2013
2013
, give us remainders
1001
,
1002
,
…
,
2012
1001, 1002, \ldots, 2012
1001
,
1002
,
…
,
2012
. Prove that among the divisors of the number
n
2
n^2
n
2
, there exist numbers such that when they are divided by
2013
2013
2013
, give us reminders
1
,
2
,
3
,
…
,
2012
1, 2, 3, \ldots, 2012
1
,
2
,
3
,
…
,
2012
.