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Problems
Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2008 239 Open Mathematical Olympiad
2008 239 Open Mathematical Olympiad
Part of
239 Open Math Olympiad
Subcontests
(8)
8
1
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n numbers with different partial sums
The natural numbers
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots , x_n
x
1
,
x
2
,
…
,
x
n
are such that all their
2
n
2^n
2
n
partial sums are distinct. Prove that:
x
1
2
+
x
2
2
+
…
+
x
n
2
≥
4
n
–
1
3
.
{x_1}^2 + {x_2}^2 + \ldots + {x_n}^2 \geq \frac{4^n – 1}{3}.
x
1
2
+
x
2
2
+
…
+
x
n
2
≥
3
4
n
–1
.
7
1
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Number theory equation
Find all natural numbers
n
,
k
n, k
n
,
k
such that
2
n
–
5
k
=
7.
2^n – 5^k = 7.
2
n
–
5
k
=
7.
6
2
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Two variable polynomial and function
Given a polynomial
P
(
x
,
y
)
P(x,y)
P
(
x
,
y
)
with real coefficients, suppose that some real function
f
:
R
→
R
f:\mathbb R \to \mathbb R
f
:
R
→
R
satisfies
P
(
x
,
y
)
=
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
P(x,y) = f(x+y)-f(x)-f(y)
P
(
x
,
y
)
=
f
(
x
+
y
)
−
f
(
x
)
−
f
(
y
)
for all
x
,
y
∈
R
x,y\in\mathbb R
x
,
y
∈
R
. Show that some polynomial
q
q
q
satisfies
P
(
x
,
y
)
=
q
(
x
+
y
)
−
q
(
x
)
−
q
(
y
)
P(x,y) = q(x+y)-q(x)-q(y)
P
(
x
,
y
)
=
q
(
x
+
y
)
−
q
(
x
)
−
q
(
y
)
Three circles and a chord
A
B
AB
A
B
is the chord of the circle
S
S
S
. Circles
S
1
S_1
S
1
and
S
2
S_2
S
2
touch the circle
S
S
S
at points
P
P
P
and
Q
Q
Q
, respectively, and the segment
A
B
AB
A
B
at point
K
K
K
. It turned out that
∠
P
B
A
=
∠
Q
B
A
\angle{PBA}=\angle{QBA}
∠
PB
A
=
∠
QB
A
. Prove that
A
B
AB
A
B
is the diameter of the circle
S
S
S
.
5
2
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Geometric Inequality
In the triangle
A
B
C
ABC
A
BC
,
∠
B
=
12
0
∘
\angle{B} = 120^{\circ}
∠
B
=
12
0
∘
, point
M
M
M
is the midpoint of side
A
C
AC
A
C
. On the sides
A
B
AB
A
B
and
B
C
BC
BC
, the points
K
K
K
and
L
L
L
are chosen such that
K
L
∥
A
C
KL \parallel AC
K
L
∥
A
C
. Prove that
M
K
+
M
L
≥
M
A
MK + ML \geq MA
M
K
+
M
L
≥
M
A
.
Checkered square
You are given a checkered square, the side of which is
n
–
1
n – 1
n
–1
long and contains
n
≥
10
n \geq 10
n
≥
10
nodes. A non-return path is a path along edges, the intersection of which with any horizontal or vertical line is a segment, point or empty set, and which does not pass along any edge more than once. What is the smallest number of non-return paths that can cover all the edges? (An edge is a unit segment between adjacent nodes.)
4
2
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n numbers around the circle
For what natural number
n
>
100
n> 100
n
>
100
can
n
n
n
pairwise distinct numbers be arranged on a circle such that each number is either greater than
100
100
100
numbers following it clockwise or less than all of them? and would any property be violated when deleting any of those numbers?
Inscribed hexagon
Point
P
P
P
is located inside an acute-angled triangle
A
B
C
ABC
A
BC
.
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
are points symmetric to
P
P
P
with respect to the sides of triangle
A
B
C
ABC
A
BC
. It turned out that the hexagon
A
B
1
C
A
1
B
C
1
AB_1CA_1BC_1
A
B
1
C
A
1
B
C
1
is inscribed. Prove that
P
P
P
is the Torricelli point of triangle
A
B
C
ABC
A
BC
.
3
2
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Directed graph
Prove that you can arrange arrows on the edges of a convex polyhedron such that each vertex contains at most three arrows.
Connected graph w/ p=100
A connected graph has
100
100
100
vertices, the degrees of all the vertices do not exceed
4
4
4
and no two vertices of degree
4
4
4
are adjacent. Prove that it is possible to remove several edges that have no common vertices from this graph such that there would be no triangles in the resulting graph.
2
2
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Equal radii lengths
A circumscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
is given.
E
E
E
and
F
F
F
are the intersection points of opposite sides of the
A
B
C
D
ABCD
A
BC
D
. It turned out that the radii of the inscribed circles of the triangles
A
E
F
AEF
A
EF
and
C
E
F
CEF
CEF
are equal. Prove that
A
C
⊥
B
D
AC \bot BD
A
C
⊥
B
D
.
1/a + 1/b + 1/c = 1 Inequality
For all positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfying
1
a
+
1
b
+
1
c
=
1
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1
a
1
+
b
1
+
c
1
=
1
, prove that:
a
a
+
b
c
+
b
b
+
c
a
+
c
c
+
a
b
≥
3
4
.
\frac{a}{a+bc} + \frac{b}{b+ca} + \frac{c}{c+ab} \geq \frac{3}{4} .
a
+
b
c
a
+
b
+
c
a
b
+
c
+
ab
c
≥
4
3
.
1
2
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Ascending proper divisors
Composite numbers
a
a
a
and
b
b
b
have equal number of divisors. All proper divisors of
a
a
a
were written in ascending order and all proper divisors of
b
b
b
were written under them in ascending order, then the numbers that are below each other were added together. It turned out that the resulting numbers formed a set of all proper divisors of a certain number. What are the smallest values that
a
a
a
and
b
b
b
take?
Proper divisors set
An odd natural number
k
k
k
is given. Consider a composite number
n
n
n
. We define
d
(
n
)
d(n)
d
(
n
)
the set of proper divisors of number
n
n
n
. If for some number
m
m
m
,
d
(
m
)
d(m)
d
(
m
)
is equal to
d
(
n
)
∪
{
k
}
d(n) \cup \{ k \}
d
(
n
)
∪
{
k
}
, we call
n
n
n
a good number. prove that there exist only finitely many good numbers. (A proper divisor of a number is any divisor other than one and the number itself.)