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Problems
Contests
International Contests
Baltic Way
1991 Baltic Way
1991 Baltic Way
Part of
Baltic Way
Subcontests
(20)
20
1
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Geometry of certain points on a hyperbola
Consider two points
A
(
x
1
,
y
1
)
A(x_1, y_1)
A
(
x
1
,
y
1
)
and
B
(
x
2
,
y
2
)
B(x_2, y_2)
B
(
x
2
,
y
2
)
on the graph of the function
y
=
1
x
y = \frac{1}{x}
y
=
x
1
such that
0
<
x
1
<
x
2
0 < x_1 < x_2
0
<
x
1
<
x
2
and
A
B
=
2
⋅
O
A
AB = 2 \cdot OA
A
B
=
2
⋅
O
A
, where
O
=
(
0
,
0
)
O = (0, 0)
O
=
(
0
,
0
)
. Let
C
C
C
be the midpoint of the segment
A
B
AB
A
B
. Prove that the angle between the
x
x
x
-axis and the ray
O
A
OA
O
A
is equal to three times the angle between the
x
x
x
-axis and the ray
O
C
OC
OC
.
19
1
Hide problems
Product formula for three pairs of intersecting circles
Three circles in the plane, whose interiors have no common point, meet each other at three pairs of points:
A
1
A_1
A
1
and
A
2
A_2
A
2
,
B
1
B_1
B
1
and
B
2
B_2
B
2
, and
C
1
C_1
C
1
and
C
2
C_2
C
2
, where points
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
lie inside the triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
. Prove that
A
1
B
2
⋅
B
1
C
2
⋅
C
1
A
2
=
A
1
C
2
⋅
C
1
B
2
⋅
B
1
A
2
.
A_1B_2 \cdot B_1C_2 \cdot C_1A_2 = A_1C_2 \cdot C_1B_2 \cdot B_1A_2 .
A
1
B
2
⋅
B
1
C
2
⋅
C
1
A
2
=
A
1
C
2
⋅
C
1
B
2
⋅
B
1
A
2
.
18
1
Hide problems
Can you place two tetrahedra in a sphere?
Is it possible to place two non-intersecting tetrahedra of volume
1
2
\frac{1}{2}
2
1
into a sphere with radius
1
1
1
?
17
1
Hide problems
How does a ray reflect off coordinate axes?
Let the coordinate planes have the reflection property. A ray falls onto one of them. How does the final direction of the ray after reflecting from all three coordinate planes depend on its initial direction?
16
1
Hide problems
Equation between three tangent circles
Two circles
C
1
C_1
C
1
and
C
2
C_2
C
2
with radii
r
1
r_1
r
1
and
r
2
r_2
r
2
touch each other externally and both touch a line
l
l
l
. A circle
C
3
C_3
C
3
with radius
r
3
<
r
1
,
r
2
r_3 < r_1, r_2
r
3
<
r
1
,
r
2
is tangent to
l
l
l
and externally to
C
1
C_1
C
1
and
C
2
C_2
C
2
. Prove that
1
r
3
=
1
r
2
+
1
r
2
.
\frac{1}{\sqrt{r_3}}=\frac{1}{\sqrt{r_2}}+\frac{1}{\sqrt{r_2}}.
r
3
1
=
r
2
1
+
r
2
1
.
15
1
Hide problems
Can the king change all the numbers in the squares
In each of the squares of a chessboard an arbitrary integer is written. A king starts to move on the board. Whenever the king moves to some square, the number in that square is increased by
1
1
1
. Is it always possible to make the numbers on the chessboard: (a) all even; (b) all divisible by
3
3
3
; (c) all equal?
14
1
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Knight leaving a castle by visiting its halls
A castle has a number of halls and
n
n
n
doors. Every door leads into another hall or outside. Every hall has at least two doors. A knight enters the castle. In any hall, he can choose any door for exit except the one he just used to enter that hall. Find a strategy allowing the knight to get outside after visiting no more than
2
n
2n
2
n
halls (a hall is counted each time it is entered).
13
1
Hide problems
Labelling triangles
An equilateral triangle is divided into
25
25
25
equal equilateral triangles labelled by
1
1
1
through
25
25
25
. Prove that one can find two triangles having a common side whose labels differ by more than
3
3
3
.
12
1
Hide problems
Two diagonals have the same colouring before and after
The vertices of a convex
1991
1991
1991
-gon are enumerated with integers from
1
1
1
to
1991
1991
1991
. Each side and diagonal of the
1991
1991
1991
-gon is colored either red or blue. Prove that, for an arbitrary renumeration of vertices, one can find integers
k
k
k
and
l
l
l
such that the segment connecting the vertices numbered
k
k
k
and
l
l
l
before the renumeration has the same color as the segment connecting the vertices numbered
k
k
k
and
l
l
l
after the renumeration.
11
1
Hide problems
Are there more numbers whose digits sums are even or odd?
The integers from
1
1
1
to
1000000
1000000
1000000
are divided into two groups consisting of numbers with odd or even sums of digits respectively. Which group contains more numbers?
10
1
Hide problems
Compute sin 3 degrees
Express the value of
sin
3
∘
\sin 3^\circ
sin
3
∘
in radicals.
9
1
Hide problems
How many solutions does this exponential equation have?
Find the number of real solutions of the equation
a
e
x
=
x
3
a e^x = x^3
a
e
x
=
x
3
, where
a
a
a
is a real parameter.
8
1
Hide problems
A polynomial with four distinct roots
Let
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
be distinct real numbers. Prove that the equation
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
+
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
e
)
(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
+
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
e
)
+
(
x
−
a
)
(
x
−
b
)
(
x
−
d
)
(
x
−
e
)
+
(
x
−
a
)
(
x
−
c
)
(
x
−
d
)
(
x
−
e
)
+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)
+
(
x
−
a
)
(
x
−
b
)
(
x
−
d
)
(
x
−
e
)
+
(
x
−
a
)
(
x
−
c
)
(
x
−
d
)
(
x
−
e
)
+
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
(
x
−
e
)
=
0
+(x - b)(x - c)(x - d)(x - e) = 0
+
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
(
x
−
e
)
=
0
has four distinct real solutions.
7
1
Hide problems
Trigonometric inequality
If
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
are the angles of an acute-angled triangle, prove that \sin \alpha + \sin \beta > \cos \alpha + \cos\beta + \cos\gamma.
6
1
Hide problems
Equation with integer part and fractional part
Solve the equation
[
x
]
⋅
{
x
}
=
1991
x
[x] \cdot \{x\} = 1991x
[
x
]
⋅
{
x
}
=
1991
x
. (Here
[
x
]
[x]
[
x
]
denotes the greatest integer less than or equal to
x
x
x
, and
{
x
}
=
x
−
[
x
]
\{x\}=x-[x]
{
x
}
=
x
−
[
x
]
.)
5
1
Hide problems
Inequality involving harmonic sums
For any positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
prove the inequalities
1
a
+
1
b
+
1
c
≥
2
a
+
b
+
2
b
+
c
+
2
c
+
a
≥
9
a
+
b
+
c
.
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.
a
1
+
b
1
+
c
1
≥
a
+
b
2
+
b
+
c
2
+
c
+
a
2
≥
a
+
b
+
c
9
.
4
1
Hide problems
Inequality of a polynomial
A polynomial
p
p
p
with integer coefficients is such that
p
(
−
n
)
<
p
(
n
)
<
n
p(-n) < p(n) < n
p
(
−
n
)
<
p
(
n
)
<
n
for some integer
n
n
n
. Prove that
p
(
−
n
)
<
−
n
p(-n) < -n
p
(
−
n
)
<
−
n
.
3
1
Hide problems
Cats and sacks for sale
There are
20
20
20
cats priced from
$
12
\$12
$12
to
$
15
\$15
$15
and
20
20
20
sacks priced from
10
10
10
cents to
$
1
\$1
$1
for sale, all of different prices. Prove that John and Peter can each buy a cat in a sack paying the same amount of money.
2
1
Hide problems
A sum of powers is not a power
Prove that
10
2
1991
+
10
3
1991
102^{1991} + 103^{1991}
10
2
1991
+
10
3
1991
is not a proper power of an integer.
1
1
Hide problems
Product of differences divisible by 1991
Find the smallest positive integer
n
n
n
having the property that for any
n
n
n
distinct integers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots , a_n
a
1
,
a
2
,
…
,
a
n
the product of all differences
a
i
−
a
j
a_i-a_j
a
i
−
a
j
(
i
<
j
)
(i < j)
(
i
<
j
)
is divisible by
1991
1991
1991
.