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Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
2006 All-Russian Olympiad
2006 All-Russian Olympiad
Part of
All-Russian Olympiad
Subcontests
(8)
8
2
Hide problems
Four zeros of iterated quadratic trinomial
Given a quadratic trinomial
f
(
x
)
=
x
2
+
a
x
+
b
f\left(x\right)=x^2+ax+b
f
(
x
)
=
x
2
+
a
x
+
b
. Assume that the equation
f
(
f
(
x
)
)
=
0
f\left(f\left(x\right)\right)=0
f
(
f
(
x
)
)
=
0
has four different real solutions, and that the sum of two of these solutions is
−
1
-1
−
1
. Prove that
b
≤
−
1
4
b\leq -\frac14
b
≤
−
4
1
.
Tourists and t-shirts
At a tourist camp, each person has at least
50
50
50
and at most
100
100
100
friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most)
1331
1331
1331
different colors, and any person has
20
20
20
friends whose t-shirts all have pairwisely different colors.
7
2
Hide problems
Gluing a 100*100 chessboard - who wins?
A
100
×
100
100\times 100
100
×
100
chessboard is cut into dominoes (
1
×
2
1\times 2
1
×
2
rectangles). Two persons play the following game: At each turn, a player glues together two adjacent cells (which were formerly separated by a cut-edge). A player loses if, after his turn, the
100
×
100
100\times 100
100
×
100
chessboard becomes connected, i. e. between any two cells there exists a way which doesn't intersect any cut-edge. Which player has a winning strategy - the starting player or his opponent?
Polynomial (x+1)^n - 1 divisible by...
Assume that the polynomial
(
x
+
1
)
n
−
1
\left(x+1\right)^n-1
(
x
+
1
)
n
−
1
is divisible by some polynomial
P
(
x
)
=
x
k
+
c
k
−
1
x
k
−
1
+
c
k
−
2
x
k
−
2
+
.
.
.
+
c
1
x
+
c
0
P\left(x\right)=x^k+c_{k-1}x^{k-1}+c_{k-2}x^{k-2}+...+c_1x+c_0
P
(
x
)
=
x
k
+
c
k
−
1
x
k
−
1
+
c
k
−
2
x
k
−
2
+
...
+
c
1
x
+
c
0
, whose degree
k
k
k
is even and whose coefficients
c
k
−
1
c_{k-1}
c
k
−
1
,
c
k
−
2
c_{k-2}
c
k
−
2
, ...,
c
1
c_1
c
1
,
c
0
c_0
c
0
all are odd integers. Show that
k
+
1
∣
n
k+1\mid n
k
+
1
∣
n
.
6
3
Hide problems
AP = CQ and RPBQ is cyclic; prove that RX = RY
Let
P
P
P
,
Q
Q
Q
,
R
R
R
be points on the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
A
CA
C
A
of a triangle
A
B
C
ABC
A
BC
such that
A
P
=
C
Q
AP=CQ
A
P
=
CQ
and the quadrilateral
R
P
B
Q
RPBQ
RPBQ
is cyclic. The tangents to the circumcircle of triangle
A
B
C
ABC
A
BC
at the points
C
C
C
and
A
A
A
intersect the lines
R
Q
RQ
RQ
and
R
P
RP
RP
at the points
X
X
X
and
Y
Y
Y
, respectively. Prove that
R
X
=
R
Y
RX=RY
RX
=
R
Y
.
Incenters equidistant from midpoint of arc ABC
Let
K
K
K
and
L
L
L
be two points on the arcs
A
B
AB
A
B
and
B
C
BC
BC
of the circumcircle of a triangle
A
B
C
ABC
A
BC
, respectively, such that
K
L
∥
A
C
KL\parallel AC
K
L
∥
A
C
. Show that the incenters of triangles
A
B
K
ABK
A
B
K
and
C
B
L
CBL
CB
L
are equidistant from the midpoint of the arc
A
B
C
ABC
A
BC
of the circumcircle of triangle
A
B
C
ABC
A
BC
.
special tetrahedron; prove that S'A' = S'B' = S'C'
Consider a tetrahedron
S
A
B
C
SABC
S
A
BC
. The incircle of the triangle
A
B
C
ABC
A
BC
has the center
I
I
I
and touches its sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at the points
E
E
E
,
F
F
F
,
D
D
D
, respectively. Let
A
′
A^{\prime}
A
′
,
B
′
B^{\prime}
B
′
,
C
′
C^{\prime}
C
′
be the points on the segments
S
A
SA
S
A
,
S
B
SB
SB
,
S
C
SC
SC
such that
A
A
′
=
A
D
AA^{\prime}=AD
A
A
′
=
A
D
,
B
B
′
=
B
E
BB^{\prime}=BE
B
B
′
=
BE
,
C
C
′
=
C
F
CC^{\prime}=CF
C
C
′
=
CF
, and let
S
′
S^{\prime}
S
′
be the point diametrically opposite to the point
S
S
S
on the circumsphere of the tetrahedron
S
A
B
C
SABC
S
A
BC
. Assume that the line
S
I
SI
S
I
is an altitude of the tetrahedron
S
A
B
C
SABC
S
A
BC
. Show that
S
′
A
′
=
S
′
B
′
=
S
′
C
′
S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}
S
′
A
′
=
S
′
B
′
=
S
′
C
′
.
5
2
Hide problems
Numbers increasing, greatest divisors decreasing
Let
a
1
a_1
a
1
,
a
2
a_2
a
2
, ...,
a
10
a_{10}
a
10
be positive integers such that
a
1
<
a
2
<
.
.
.
<
a
10
a_1<a_2<...<a_{10}
a
1
<
a
2
<
...
<
a
10
. For every
k
k
k
, denote by
b
k
b_k
b
k
the greatest divisor of
a
k
a_k
a
k
such that
b
k
<
a
k
b_k<a_k
b
k
<
a
k
. Assume that
b
1
>
b
2
>
.
.
.
>
b
10
b_1>b_2>...>b_{10}
b
1
>
b
2
>
...
>
b
10
. Show that
a
10
>
500
a_{10}>500
a
10
>
500
.
X_k will win [Two sequences of positives: some x_k is > y_k]
Two sequences of positive reals,
(
x
n
)
\left(x_n\right)
(
x
n
)
and
(
y
n
)
\left(y_n\right)
(
y
n
)
, satisfy the relations x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2 and y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1} for all natural numbers
n
n
n
. Prove that, if the numbers
x
1
x_1
x
1
,
x
2
x_2
x
2
,
y
1
y_1
y
1
,
y
2
y_2
y
2
are all greater than
1
1
1
, then there exists a natural number
k
k
k
such that
x
k
>
y
k
x_k > y_k
x
k
>
y
k
.
4
3
Hide problems
BT = tangent from B to omega
Given a triangle
A
B
C
ABC
A
BC
. Let a circle
ω
\omega
ω
touch the circumcircle of triangle
A
B
C
ABC
A
BC
at the point
A
A
A
, intersect the side
A
B
AB
A
B
at a point
K
K
K
, and intersect the side
B
C
BC
BC
. Let
C
L
CL
C
L
be a tangent to the circle
ω
\omega
ω
, where the point
L
L
L
lies on
ω
\omega
ω
and the segment
K
L
KL
K
L
intersects the side
B
C
BC
BC
at a point
T
T
T
. Show that the segment
B
T
BT
BT
has the same length as the tangent from the point
B
B
B
to the circle
ω
\omega
ω
.
Isosceles triangle and tangent circles
Consider an isosceles triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB=AC
A
B
=
A
C
, and a circle
ω
\omega
ω
which is tangent to the sides
A
B
AB
A
B
and
A
C
AC
A
C
of this triangle and intersects the side
B
C
BC
BC
at the points
K
K
K
and
L
L
L
. The segment
A
K
AK
A
K
intersects the circle
ω
\omega
ω
at a point
M
M
M
(apart from
K
K
K
). Let
P
P
P
and
Q
Q
Q
be the reflections of the point
K
K
K
in the points
B
B
B
and
C
C
C
, respectively. Show that the circumcircle of triangle
P
M
Q
PMQ
PMQ
is tangent to the circle
ω
\omega
ω
.
A well-known circle hides in the dark
Given a triangle
A
B
C
ABC
A
BC
. The angle bisectors of the angles
A
B
C
ABC
A
BC
and
B
C
A
BCA
BC
A
intersect the sides
C
A
CA
C
A
and
A
B
AB
A
B
at the points
B
1
B_1
B
1
and
C
1
C_1
C
1
, and intersect each other at the point
I
I
I
. The line
B
1
C
1
B_1C_1
B
1
C
1
intersects the circumcircle of triangle
A
B
C
ABC
A
BC
at the points
M
M
M
and
N
N
N
. Prove that the circumradius of triangle
M
I
N
MIN
M
I
N
is twice as long as the circumradius of triangle
A
B
C
ABC
A
BC
.
3
2
Hide problems
A game with 2006 points and chords on a circle
Given a circle and
2006
2006
2006
points lying on this circle. Albatross colors these
2006
2006
2006
points in
17
17
17
colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common points (not even a common endpoint). Hereby, Frankinfueter intends to draw as many chords as possible, while Albatross is trying to hinder him as much as he can. What is the maximal number of chords Frankinfueter will always be able to draw?
Directing segments: who wins?
On a
49
×
69
49\times 69
49
×
69
rectangle formed by a grid of lattice squares, all
50
⋅
70
50\cdot 70
50
⋅
70
lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their interior.) Segments are drawn this way until all formerly blue points are colored red. At this moment, the first player directs all segments drawn - i. e., he takes every segment AB, and replaces it either by the vector
A
B
→
\overrightarrow{AB}
A
B
, or by the vector
B
A
→
\overrightarrow{BA}
B
A
. If the first player succeeds to direct all the segments drawn in such a way that the sum of the resulting vectors is
0
→
\overrightarrow{0}
0
, then he wins; else, the second player wins. Which player has a winning strategy?
2
3
Hide problems
Large a, b, c, d satisfying 1/a + 1/b + 1/c + 1/d = 1/(abcd)
Show that there exist four integers
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
whose absolute values are all
>
1000000
>1000000
>
1000000
and which satisfy
1
a
+
1
b
+
1
c
+
1
d
=
1
a
b
c
d
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{abcd}
a
1
+
b
1
+
c
1
+
d
1
=
ab
c
d
1
.
If (a-1)^3 + a^3 + (a+1)^3 is a cube, then 4 | a
If an integer
a
>
1
a > 1
a
>
1
is given such that
(
a
−
1
)
3
+
a
3
+
(
a
+
1
)
3
\left(a-1\right)^3+a^3+\left(a+1\right)^3
(
a
−
1
)
3
+
a
3
+
(
a
+
1
)
3
is the cube of an integer, then show that
4
∣
a
4\mid a
4
∣
a
.
Sum and product of purely periodic decimal fractions
The sum and the product of two purely periodic decimal fractions
a
a
a
and
b
b
b
are purely periodic decimal fractions of period length
T
T
T
. Show that the lengths of the periods of the fractions
a
a
a
and
b
b
b
are not greater than
T
T
T
. Note. A purely periodic decimal fraction is a periodic decimal fraction without a non-periodic starting part.
1
2
Hide problems
Symmetric non-self-intersecting path in 15*15 chessboard
Given a
15
×
15
15\times 15
15
×
15
chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is
≤
200
\leq 200
≤
200
.
Sin sqrt(x) < sqrt(sin x) for 0 < x < pi/2
Prove that
sin
x
<
sin
x
\sin\sqrt{x}<\sqrt{\sin x}
sin
x
<
sin
x
for every real
x
x
x
such that
0
<
x
<
π
2
0<x<\frac{\pi}{2}
0
<
x
<
2
π
.