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Problems
Contests
International Contests
Baltic Way
1999 Baltic Way
1999 Baltic Way
Part of
Baltic Way
Subcontests
(20)
20
1
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Determine all values of a^2+b^2+c^2+d^2 for primes a,b,c,d
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
and
d
d
d
be prime numbers such that
a
>
3
b
>
6
c
>
12
d
a>3b>6c>12d
a
>
3
b
>
6
c
>
12
d
and
a
2
−
b
2
+
c
2
−
d
2
=
1749
a^2-b^2+c^2-d^2=1749
a
2
−
b
2
+
c
2
−
d
2
=
1749
. Determine all possible values of
a
2
+
b
2
+
c
2
+
d
2
a^2+b^2+c^2+d^2
a
2
+
b
2
+
c
2
+
d
2
.
18
1
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At most one factorization of m into the integers ab
Let
m
m
m
be a positive integer such that
m
=
2
(
m
o
d
4
)
m=2\pmod{4}
m
=
2
(
mod
4
)
. Show that there exists at most one factorization
m
=
a
b
m=ab
m
=
ab
where
a
a
a
and
b
b
b
are positive integers satisfying
0
<
a
−
b
<
5
+
4
4
m
+
1
0<a-b<\sqrt{5+4\sqrt{4m+1}}
0
<
a
−
b
<
5
+
4
4
m
+
1
17
1
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Adding a to each integer in the sequence gives a prime
Does there exist a finite sequence of integers
c
1
,
c
2
,
…
,
c
n
c_1,c_2,\ldots ,c_n
c
1
,
c
2
,
…
,
c
n
such that all the numbers
a
+
c
1
,
a
+
c
2
,
…
,
a
+
c
n
a+c_1,a+c_2,\ldots ,a+c_n
a
+
c
1
,
a
+
c
2
,
…
,
a
+
c
n
are primes for more than one but not infinitely many different integers
a
a
a
?
16
1
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Smallest k representable as 19^n-5^m
Find the smallest positive integer
k
k
k
which is representable in the form
k
=
1
9
n
−
5
m
k=19^n-5^m
k
=
1
9
n
−
5
m
for some positive integers
m
m
m
and
n
n
n
.
14
1
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Area ratio is equal to length ratio
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB=AC
A
B
=
A
C
. Points
D
D
D
and
E
E
E
lie on the sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. The line passing through
B
B
B
and parallel to
A
C
AC
A
C
meets the line
D
E
DE
D
E
at
F
F
F
. The line passing through
C
C
C
and parallel to
A
B
AB
A
B
meets the line
D
E
DE
D
E
at
G
G
G
. Prove that
[
D
B
C
G
]
[
F
B
C
E
]
=
A
D
D
E
\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE}
[
FBCE
]
[
D
BCG
]
=
D
E
A
D
13
1
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Determine the angle C if AE+BD=AB
The bisectors of the angles
A
A
A
and
B
B
B
of the triangle
A
B
C
ABC
A
BC
meet the sides
B
C
BC
BC
and
C
A
CA
C
A
at the points
D
D
D
and
E
E
E
, respectively. Assuming that
A
E
+
B
D
=
A
B
AE+BD=AB
A
E
+
B
D
=
A
B
, determine the angle
C
C
C
.
12
1
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Incentre, circumcentre and midpoints of AC,BC are concyclic
In a triangle
A
B
C
ABC
A
BC
it is given that
2
A
B
=
A
C
+
B
C
2AB=AC+BC
2
A
B
=
A
C
+
BC
. Prove that the incentre of
△
A
B
C
\triangle ABC
△
A
BC
, the circumcentre of
△
A
B
C
\triangle ABC
△
A
BC
, and the midpoints of
A
C
AC
A
C
and
B
C
BC
BC
are concyclic.
11
1
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Four points are either concyclic or circle contains one
Prove that for any four points in the plane, no three of which are collinear, there exists a circle such that three of the four points are on the circumference and the fourth point is either on the circumference or inside the circle.
10
1
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Partitioning circle into 3 subsets
May the points of a disc of radius
1
1
1
(including its circumference) be partitioned into three subsets in such a way that no subset contains two points separated by a distance
1
1
1
?
9
1
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Maximum number of odd sum rows in a cube
A cube with edge length
3
3
3
is divided into
27
27
27
unit cubes. The numbers
1
,
2
,
…
,
27
1, 2,\ldots ,27
1
,
2
,
…
,
27
are distributed arbitrarily over the unit cubes, with one number in each cube. We form the
27
27
27
possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). At most how many of the
27
27
27
row sums can be odd?
8
1
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1999 coins with distinct weights
We are given
1999
1999
1999
coins. No two coins have the same weight. A machine is provided which allows us with one operation to determine, for any three coins, which one has the middle weight. Prove that the coin that is the
1000
1000
1000
th by weight can be determined using no more than
1000000
1000000
1000000
operations and that this is the only coin whose position by weight can be determined using this machine.
7
1
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Tke king visiting each square exactly once on a chessboard
Two squares on an
8
×
8
8\times 8
8
×
8
chessboard are called adjacent if they have a common edge or common corner. Is it possible for a king to begin in some square and visit all squares exactly once in such a way that all moves except the first are made into squares adjacent to an even number of squares already visited?
6
1
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Knight moving from one corner to the opposite
What is the least number of moves it takes a knight to get from one corner of an
n
×
n
n\times n
n
×
n
chessboard, where
n
≥
4
n\ge 4
n
≥
4
, to the diagonally opposite corner?
5
1
Hide problems
A bit of algebra... Tangent to circle meets parabola
The point
(
a
,
b
)
(a,b)
(
a
,
b
)
lies on the circle
x
2
+
y
2
=
1
x^2+y^2=1
x
2
+
y
2
=
1
. The tangent to the circle at this point meets the parabola
y
=
x
2
+
1
y=x^2+1
y
=
x
2
+
1
at exactly one point. Find all such points
(
a
,
b
)
(a,b)
(
a
,
b
)
.
4
1
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There exists x_0 and y_0
For all positive real numbers
x
x
x
and
y
y
y
let
f
(
x
,
y
)
=
min
(
x
,
y
x
2
+
y
2
)
f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right)
f
(
x
,
y
)
=
min
(
x
,
x
2
+
y
2
y
)
Show that there exist
x
0
x_0
x
0
and
y
0
y_0
y
0
such that
f
(
x
,
y
)
≤
f
(
x
0
,
y
0
)
f(x, y)\le f(x_0, y_0)
f
(
x
,
y
)
≤
f
(
x
0
,
y
0
)
for all positive
x
x
x
and
y
y
y
, and find
f
(
x
0
,
y
0
)
f(x_0,y_0)
f
(
x
0
,
y
0
)
.
3
1
Hide problems
For which n does the inequality hold?
Determine all positive integers
n
≥
3
n\ge 3
n
≥
3
such that the inequality
a
1
a
2
+
a
2
a
3
+
…
a
n
−
1
a
n
≤
0
a_1a_2+a_2a_3+\ldots a_{n-1}a_n\le 0
a
1
a
2
+
a
2
a
3
+
…
a
n
−
1
a
n
≤
0
holds for all real numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots , a_n
a
1
,
a
2
,
…
,
a
n
which satisfy
a
1
+
a
2
+
…
+
a
n
=
0
a_1+a_2+\ldots +a_n=0
a
1
+
a
2
+
…
+
a
n
=
0
.
2
1
Hide problems
Cube root of n is its last three digits
Determine all positive integers
n
n
n
with the property that the third root of
n
n
n
is obtained by removing its last three decimal digits.
1
1
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Going through the sums of a,b,c,d
Determine all real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
that satisfy the following equations
{
a
b
c
+
a
b
+
b
c
+
c
a
+
a
+
b
+
c
=
1
b
c
d
+
b
c
+
c
d
+
d
b
+
b
+
c
+
d
=
9
c
d
a
+
c
d
+
d
a
+
a
c
+
c
+
d
+
a
=
9
d
a
b
+
d
a
+
a
b
+
b
d
+
d
+
a
+
b
=
9
\begin{cases} abc + ab + bc + ca + a + b + c = 1\\ bcd + bc + cd + db + b + c + d = 9\\ cda + cd + da + ac + c + d + a = 9\\ dab + da + ab + bd + d + a + b = 9\end{cases}
⎩
⎨
⎧
ab
c
+
ab
+
b
c
+
c
a
+
a
+
b
+
c
=
1
b
c
d
+
b
c
+
c
d
+
d
b
+
b
+
c
+
d
=
9
c
d
a
+
c
d
+
d
a
+
a
c
+
c
+
d
+
a
=
9
d
ab
+
d
a
+
ab
+
b
d
+
d
+
a
+
b
=
9
15
1
Hide problems
AB = DE ,Nice
Let
A
B
C
ABC
A
BC
be a triangle with
∠
C
=
6
0
∘
\angle C=60^\circ
∠
C
=
6
0
∘
and
A
C
<
B
C
AC<BC
A
C
<
BC
. The point
D
D
D
lies on the side
B
C
BC
BC
and satisfies
B
D
=
A
C
BD=AC
B
D
=
A
C
. The side
A
C
AC
A
C
is extended to the point
E
E
E
where
A
C
=
C
E
AC=CE
A
C
=
CE
. Prove that
A
B
=
D
E
AB=DE
A
B
=
D
E
.
19
1
Hide problems
baltic99
Prove that there exist infinitely many even positive integers
k
k
k
such that for every prime
p
p
p
the number
p
2
+
k
p^2+k
p
2
+
k
is composite.