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Problems
Contests
International Contests
Tuymaada Olympiad
2020 Tuymaada Olympiad
2020 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(8)
3
1
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Hamilton path weights zero
Each edge of a complete graph with
101
101
101
vertices is marked with
1
1
1
or
−
1
-1
−
1
. It is known that absolute value of the sum of numbers on all the edges is less than
150
150
150
. Prove that the graph contains a path visiting each vertex exactly once such that the sum of numbers on all edges of this path is zero.(Y. Caro, A. Hansberg, J. Lauri, C. Zarb)
8
2
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Partitioning strips with segments
In a horizontal strip
1
×
n
1 \times n
1
×
n
made of
n
n
n
unit squares the vertices of all squares are marked. The strip is partitioned into parts by segments connecting marked points and not lying on the sides of the strip. The segments can not have common inner points; the upper end of each segment must be either above the lower end or further to the right. Prove that the number of all partitions is divisible by
2
n
2^n
2
n
. (The partition where no segments are drawn, is counted too.)(E. Robeva, M. Sun)
Polynomial equation involving x^{n + 1} and (x+1)^{n + 1}
The degrees of polynomials
P
P
P
and
Q
Q
Q
with real coefficients do not exceed
n
n
n
. These polynomials satisfy the identity
P
(
x
)
x
n
+
1
+
Q
(
x
)
(
x
+
1
)
n
+
1
=
1.
P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1.
P
(
x
)
x
n
+
1
+
Q
(
x
)
(
x
+
1
)
n
+
1
=
1.
Determine all possible values of
Q
(
−
1
2
)
Q \left( - \frac{1}{2} \right)
Q
(
−
2
1
)
.
7
2
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Products of digits + 1 is N+1
How many positive integers
N
N
N
in the segment
[
10
,
1
0
20
]
\left[10, 10^{20} \right]
[
10
,
1
0
20
]
are such that if all their digits are increased by
1
1
1
and then multiplied, the result is
N
+
1
N+1
N
+
1
?(F. Bakharev)
Police and thieves under surveillance
Several policemen try to catch a thief who has
2
m
2m
2
m
accomplices. To that end they place the accomplices under surveillance. In the beginning, the policemen shadow nobody. Every morning each policeman places under his surveillance one of the accomplices. Every evening the thief stops trusting one of his accomplices The thief is caught if by the
m
m
m
-th evening some policeman shadows exactly those
m
m
m
accomplices who are still trusted by the thief. Prove that to guarantee the capture of the thief at least
2
m
2^m
2
m
policemen are needed.
6
2
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Angle chasings
A
K
AK
A
K
and
B
L
BL
B
L
are altitudes of an acute triangle
A
B
C
ABC
A
BC
. Point
P
P
P
is chosen on the segment
A
K
AK
A
K
so that
L
K
=
L
P
LK=LP
L
K
=
L
P
. The parallel to
B
C
BC
BC
through
P
P
P
meets the parallel to
P
L
PL
P
L
through
B
B
B
at point
Q
Q
Q
. Prove that
∠
A
Q
B
=
∠
A
C
B
\angle AQB = \angle ACB
∠
A
QB
=
∠
A
CB
.(S. Berlov)
Circumcenter on an isosceles triangle.
An isosceles triangle
A
B
C
ABC
A
BC
(
A
B
=
B
C
AB = BC
A
B
=
BC
) is given. Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
with centres
O
1
O_1
O
1
and
O
2
O_2
O
2
lie in the angle
A
B
C
ABC
A
BC
and touch the sides
A
B
AB
A
B
and
C
B
CB
CB
at
A
A
A
and
C
C
C
respectively, and touch each other externally at point
X
X
X
. The side
A
C
AC
A
C
meets the circles again at points
Y
Y
Y
and
Z
Z
Z
.
O
O
O
is the circumcenter of the triangle
X
Y
Z
XYZ
X
Y
Z
. Lines
O
2
O
O_2 O
O
2
O
and
O
1
O
O_1 O
O
1
O
intersect lines
A
B
AB
A
B
and
B
C
BC
BC
at points
C
1
C_1
C
1
and
A
1
A_1
A
1
respectively. Prove that
B
B
B
is the circumcentre of the triangle
A
1
O
C
1
A_1 OC_1
A
1
O
C
1
.
5
1
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Ruler and compass on parabola and axis
Coordinate axes (without any marks, with the same scale) and the graph of a quadratic trinomial
y
=
x
2
+
a
x
+
b
y = x^2 + ax + b
y
=
x
2
+
a
x
+
b
are drawn in the plane. The numbers
a
a
a
and
b
b
b
are not known. How to draw a unit segment using only ruler and compass?
4
2
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Ratio of perimeter and area with weird angle condition
Points
D
D
D
and
E
E
E
lie on the lines
B
C
BC
BC
and
A
C
AC
A
C
respectively so that
B
B
B
is between
C
C
C
and
D
D
D
,
C
C
C
is between
A
A
A
and
E
E
E
,
B
C
=
B
D
BC = BD
BC
=
B
D
and
∠
B
A
D
=
∠
C
D
E
\angle BAD = \angle CDE
∠
B
A
D
=
∠
C
D
E
. It is known that the ratio of the perimeters of the triangles
A
B
C
ABC
A
BC
and
A
D
E
ADE
A
D
E
is
2
2
2
. Find the ratio of the areas of these triangles.
Give me a bound of $g(k)$
For each positive integer
k
k
k
, let
g
(
k
)
g(k)
g
(
k
)
be the maximum possible number of points in the plane such that pairwise distances between these points have only
k
k
k
different values. Prove that there exists
k
k
k
such that
g
(
k
)
>
2
k
+
2020
g(k) > 2k + 2020
g
(
k
)
>
2
k
+
2020
.
2
2
Hide problems
Coefficients counting
All non-zero coefficients of the polynomial
f
(
x
)
f(x)
f
(
x
)
equal
1
1
1
, while the sum of the coefficients is
20
20
20
. Is it possible that thirteen coefficients of
f
2
(
x
)
f^2(x)
f
2
(
x
)
equal
9
9
9
?(S. Ivanov, K. Kokhas)
Inequality strikes again.
Given positive real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots, a_n
a
1
,
a
2
,
…
,
a
n
. Let
m
=
min
(
a
1
+
1
a
2
,
a
2
+
1
a
3
,
…
,
a
n
−
1
+
1
a
n
,
a
n
+
1
a
1
)
.
m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right).
m
=
min
(
a
1
+
a
2
1
,
a
2
+
a
3
1
,
…
,
a
n
−
1
+
a
n
1
,
a
n
+
a
1
1
)
.
Prove the inequality
a
1
a
2
…
a
n
n
+
1
a
1
a
2
…
a
n
n
≥
m
.
\sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m.
n
a
1
a
2
…
a
n
+
n
a
1
a
2
…
a
n
1
≥
m
.
1
2
Hide problems
Represent as m+t_m
For each positive integer
m
m
m
let
t
m
t_m
t
m
be the smallest positive integer not dividing
m
m
m
. Prove that there are infinitely many positive integers which can not be represented in the form
m
+
t
m
m + t_m
m
+
t
m
.(A. Golovanov)
absolute greater than 2020
Does the system of equation \begin{align*} \begin{cases} x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\ x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\ x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3 \end{cases} \end{align*} admit a solution in integers such that the absolute value of each of these integers is greater than
2020
2020
2020
?