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Problems
Contests
International Contests
Baltic Way
1990 Baltic Way
1990 Baltic Way
Part of
Baltic Way
Subcontests
(20)
20
1
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Baltic Way 1990 Q20 - A Creative Task!
A creative task: propose an original competition problem together with its solution.
19
1
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How many subsets can have this intersection property
What is the largest possible number of subsets of the set
{
1
,
2
,
…
,
2
n
+
1
}
\{1, 2, \dots , 2n+1\}
{
1
,
2
,
…
,
2
n
+
1
}
such that the intersection of any two subsets consists of one or several consecutive integers?
18
1
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Some row or column must contain 11 different numbers
Numbers
1
,
2
,
…
,
101
1, 2,\dots , 101
1
,
2
,
…
,
101
are written in the cells of a
101
×
101
101\times 101
101
×
101
square board so that each number is repeated
101
101
101
times. Prove that there exists either a column or a row containing at least
11
11
11
different numbers.
16
1
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Four divides the number of edges of closed curve
A closed polygonal line is drawn on a unit squared paper so that its vertices lie at lattice points and its sides have odd lengths. Prove that its number of sides is divisible by
4
4
4
.
17
1
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Game involving taking candy from piles
There are two piles with
72
72
72
and
30
30
30
candies. Two students alternate taking candies from one of the piles. Each time the number of candies taken from a pile must be a multiple of the number of candies in the other pile. Which student can always assure taking the last candy from one of the piles?
15
1
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None of these numbers are cubes
Prove that none of the numbers
2
2
n
+
1
2^{2^n}+ 1
2
2
n
+
1
,
n
=
0
,
1
,
2
,
…
n = 0, 1, 2, \dots
n
=
0
,
1
,
2
,
…
is a perfect cube.
14
1
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Coprime integers with composite sums
Do there exist
1990
1990
1990
pairwise coprime positive integers such that all sums of two or more of these numbers are composite numbers?
13
1
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Quadratic Diophantine Equation with infinitely many solution
Show that the equation
x
2
−
7
y
2
=
1
x^2-7y^2 = 1
x
2
−
7
y
2
=
1
has infinitely many solutions in natural numbers.
12
1
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Two linear forms are simultaneously divisible by 83
Let
m
m
m
and
n
n
n
be positive integers. Show that
25
m
+
3
n
25m+ 3n
25
m
+
3
n
is divisible by
83
83
83
if and only if so is
3
m
+
7
n
3m+ 7n
3
m
+
7
n
.
11
1
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An integer root of a polynomial cannot be too large
Prove that the modulus of an integer root of a polynomial with integer coefficients cannot exceed the maximum of the moduli of the coefficients.
10
1
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How far can you move this segment?
A segment
A
B
AB
A
B
is marked on a line
t
t
t
. The segment is moved on the plane so that it remains parallel to
t
t
t
and that the traces of points
A
A
A
and
B
B
B
do not intersect. The segment finally returns onto
t
t
t
. How far can point
A
A
A
now be from its initial position?
9
1
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Congruent triangles in an ellipse
Two congruent triangles are inscribed in an ellipse. Are they necessarily symmetric with respect to an axis or the center of the ellipse?
8
1
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Simson lines of opposite points are perpendicular
It is known that for any point
P
P
P
on the circumcircle of a triangle
A
B
C
ABC
A
BC
, the orthogonal projections of
P
P
P
onto
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
lie on a line, called a Simson line of
P
P
P
. Show that the Simson lines of two diametrically opposite points
P
1
P_1
P
1
and
P
2
P_2
P
2
are perpendicular.
7
1
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Five lines in a pentagon are concurrent
The midpoint of each side of a convex pentagon is connected by a segment with the centroid of the triangle formed by the remaining three vertices of the pentagon. Prove that these five segments have a common point.
6
1
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Equilateral triangle erected on a side defines another
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
A
D
=
B
C
AD = BC
A
D
=
BC
and
∠
D
A
B
+
∠
A
B
C
=
12
0
∘
\angle DAB + \angle ABC = 120^\circ
∠
D
A
B
+
∠
A
BC
=
12
0
∘
. An equilateral triangle
D
P
C
DPC
D
PC
is erected in the exterior of the quadrilateral. Prove that the triangle
A
P
B
APB
A
PB
is also equilateral.
5
1
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True expression for an associative or commutative operation
Let
∗
*
∗
be an operation, assigning a real number a * b to each pair of real numbers
(
a
,
b
)
(a, b)
(
a
,
b
)
. Find an equation which is true (for all possible values of variables) provided the operation
∗
*
∗
is commutative or associative and which can be false otherwise.
4
1
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This sum is always positive
Prove that, for any real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots , a_n
a
1
,
a
2
,
…
,
a
n
,
∑
i
,
j
=
1
n
a
i
a
j
i
+
j
−
1
≥
0.
\sum_{i,j=1}^n \frac{a_ia_j}{i+j-1}\ge 0.
i
,
j
=
1
∑
n
i
+
j
−
1
a
i
a
j
≥
0.
3
1
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Can the sequence become negative at a specified time?
Given
a
0
>
0
a_0 > 0
a
0
>
0
and
c
>
0
c > 0
c
>
0
, the sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by a_{n+1}=\frac{a_n+c}{1-ca_n} \text{for }n=1,2,\dots Is it possible that
a
0
,
a
1
,
…
,
a
1989
a_0, a_1, \dots , a_{1989}
a
0
,
a
1
,
…
,
a
1989
are all positive but
a
1990
a_{1990}
a
1990
is negative?
2
1
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Which polynomial represents the numbers in the array?
The squares of a squared paper are enumerated as shown on the picture.
⋱
10
⋱
6
9
⋱
3
5
8
12
⋱
1
2
4
7
11
⋱
\begin{array}{|c|c|c|c|c|c} \ddots &&&&&\\ \hline 10&\ddots&&&&\\ \hline 6&9&\ddots&&&\\ \hline 3&5&8&12&\ddots&\\ \hline 1&2&4&7&11&\ddots\\ \hline \end{array}
⋱
10
6
3
1
⋱
9
5
2
⋱
8
4
12
7
⋱
11
⋱
Devise a polynomial
p
(
m
,
n
)
p(m, n)
p
(
m
,
n
)
in two variables such that for any
m
,
n
∈
N
m, n \in \mathbb{N}
m
,
n
∈
N
the number written in the square with coordinates
(
m
,
n
)
(m, n)
(
m
,
n
)
is equal to
p
(
m
,
n
)
p(m, n)
p
(
m
,
n
)
.
1
1
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Smallest sum of differences around a circle
Numbers
1
,
2
,
…
,
n
1, 2, \dots , n
1
,
2
,
…
,
n
are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?