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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
1997 Turkey MO (2nd round)
1997 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(3)
3
2
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Turkey NMO 1997 Problem 3
Let
n
n
n
and
k
k
k
be positive integers, where
n
>
1
n > 1
n
>
1
is odd. Suppose
n
n
n
voters are to elect one of the
k
k
k
cadidates from a set
A
A
A
according to the rule of "majoritarian compromise" described below. After each voter ranks the candidates in a column according to his/her preferences, these columns are concatenated to form a
k
k
k
x
n
n
n
voting matrix. We denote the number of ccurences of
a
∈
A
a \in A
a
∈
A
in the
i
i
i
-th row of the voting matrix by
a
i
a_{i}
a
i
. Let
l
a
l_{a}
l
a
stand for the minimum integer
l
l
l
for which
∑
i
=
1
l
a
i
>
n
2
\sum^{l}_{i=1}{a_{i}}> \frac{n}{2}
∑
i
=
1
l
a
i
>
2
n
. Setting
l
′
=
m
i
n
{
l
a
∣
a
∈
A
}
l'= min \{l_{a} | a \in A\}
l
′
=
min
{
l
a
∣
a
∈
A
}
, we will regard the voting matrices which make the set
{
a
∈
A
∣
l
a
=
l
′
}
\{a \in A | l_{a} = l' \}
{
a
∈
A
∣
l
a
=
l
′
}
as admissible. For each such matrix, the single candidate in this set will get elected according to majoritarian compromise. Moreover, if w_{1} \geq w_{2} \geq ... \geq w_{k} \geq 0 are given, for each admissible voting matrix,
∑
i
=
1
k
w
i
a
i
\sum^{k}_{i=1}{w_{i}a_{i}}
∑
i
=
1
k
w
i
a
i
is called the total weighted score of
a
∈
A
a \in A
a
∈
A
. We will say that the system
(
w
1
,
w
2
,
.
.
.
,
w
k
)
(w_{1},w_{2}, . . . , w_{k})
(
w
1
,
w
2
,
...
,
w
k
)
of weights represents majoritarian compromise if the total score of the elected candidate is maximum among the scores of all candidates. (a) Determine whether there is a system of weights representing majoritarian compromise if
k
=
3
k = 3
k
=
3
. (b) Show that such a system of weights does not exist for
k
>
3
k > 3
k
>
3
.
Turkey NMO 1997 Problem 6, Rectangular Parallelepipeds
Let
D
1
,
D
2
,
.
.
.
,
D
n
D_{1}, D_{2}, . . . , D_{n}
D
1
,
D
2
,
...
,
D
n
be rectangular parallelepipeds in space, with edges parallel to the
x
,
y
,
z
x, y, z
x
,
y
,
z
axes. For each
D
i
D_{i}
D
i
, let
x
i
,
y
i
,
z
i
x_{i} , y_{i} , z_{i}
x
i
,
y
i
,
z
i
be the lengths of its projections onto the
x
,
y
,
z
x, y, z
x
,
y
,
z
axes, respectively. Suppose that for all pairs
D
i
D_{i}
D
i
,
D
j
D_{j}
D
j
, if at least one of
x
i
<
x
j
x_{i} < x_{j}
x
i
<
x
j
,
y
i
<
y
j
y_{i} < y_{j}
y
i
<
y
j
,
z
i
<
z
j
z_{i} < z_{j}
z
i
<
z
j
holds, then
x
i
≤
x
j
x_{i} \leq x_{j}
x
i
≤
x
j
,
y
i
≤
y
j
y_{i} \leq y_{j}
y
i
≤
y
j
, and
z
i
<
z
j
z_{i} < z_{j}
z
i
<
z
j
. If the volume of the region
⋃
i
=
1
n
D
i
\bigcup^{n}_{i=1}{D_{i}}
⋃
i
=
1
n
D
i
equals 1997, prove that there is a subset
{
D
i
1
,
D
i
2
,
.
.
.
,
D
i
m
}
\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}
{
D
i
1
,
D
i
2
,
...
,
D
i
m
}
of the set
{
D
1
,
.
.
.
,
D
n
}
\{D_{1}, . . . , D_{n}\}
{
D
1
,
...
,
D
n
}
such that
(
i
)
(i)
(
i
)
if
k
≠
l
k \not= l
k
=
l
then
D
i
k
∩
D
i
l
=
∅
D_{i_{k}} \cap D_{i_{l}} = \emptyset
D
i
k
∩
D
i
l
=
∅
, and
(
i
i
)
(ii)
(
ii
)
the volume of
⋃
k
=
1
m
D
i
k
\bigcup^{m}_{k=1}{D_{i_{k}}}
⋃
k
=
1
m
D
i
k
is at least 73.
2
2
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Turkey NMO 1997 Problem 2, Convex Pentagon
Let
F
F
F
be a point inside a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
, and let
a
1
a_{1}
a
1
,
a
2
a_{2}
a
2
,
a
3
a_{3}
a
3
,
a
4
a_{4}
a
4
,
a
5
a_{5}
a
5
denote the distances from
F
F
F
to the lines
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
E
DE
D
E
,
E
A
EA
E
A
, respectively. The points
F
1
F_{1}
F
1
,
F
2
F_{2}
F
2
,
F
3
F_{3}
F
3
,
F
4
F_{4}
F
4
,
F
5
F_{5}
F
5
are chosen on the inner bisectors of the angles
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
of the pentagon respectively, so that
A
F
1
=
A
F
AF_{1} = AF
A
F
1
=
A
F
,
B
F
2
=
B
F
BF_{2} = BF
B
F
2
=
BF
,
C
F
3
=
C
F
CF_{3} = CF
C
F
3
=
CF
,
D
F
4
=
D
F
DF_{4} = DF
D
F
4
=
D
F
and
E
F
5
=
E
F
EF_{5} = EF
E
F
5
=
EF
. If the distances from
F
1
F_{1}
F
1
,
F
2
F_{2}
F
2
,
F
3
F_{3}
F
3
,
F
4
F_{4}
F
4
,
F
5
F_{5}
F
5
to the lines
E
A
EA
E
A
,
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
E
DE
D
E
are
b
1
b_{1}
b
1
,
b
2
b_{2}
b
2
,
b
3
b_{3}
b
3
,
b
4
b_{4}
b
4
,
b
5
b_{5}
b
5
, respectively. Prove that
a
1
+
a
2
+
a
3
+
a
4
+
a
5
≤
b
1
+
b
2
+
b
3
+
b
4
+
b
5
a_{1} + a_{2} + a_{3} + a_{4} + a_{5} \leq b_{1} + b_{2} + b_{3} + b_{4} + b_{5}
a
1
+
a
2
+
a
3
+
a
4
+
a
5
≤
b
1
+
b
2
+
b
3
+
b
4
+
b
5
Turkey NMO 1997 Problem 5, PQ=m
In a triangle
A
B
C
ABC
A
BC
, the inner and outer bisectors of the
∠
A
\angle A
∠
A
meet the line
B
C
BC
BC
at
D
D
D
and
E
E
E
, respectively. Let
d
d
d
be a common tangent of the circumcircle
(
O
)
(O)
(
O
)
of
△
A
B
C
\triangle ABC
△
A
BC
and the circle with diameter
D
E
DE
D
E
and center
F
F
F
. The projections of the tangency points onto
F
O
FO
FO
are denoted by
P
P
P
and
Q
Q
Q
, and the length of their common chord is denoted by
m
m
m
. Prove that
P
Q
=
m
PQ = m
PQ
=
m
1
2
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Turkey NMO 1997 Problem 1, Diophant Equation
Find all pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
5
x
2
−
6
x
y
+
7
y
2
=
383
5x^{2}-6xy+7y^{2}=383
5
x
2
−
6
x
y
+
7
y
2
=
383
.
Turkey NMO 1997 Problem 4, least value of f(e)
Let
e
>
0
e > 0
e
>
0
be a given real number. Find the least value of
f
(
e
)
f(e)
f
(
e
)
(in terms of
e
e
e
only) such that the inequality
a
3
+
b
3
+
c
3
+
d
3
≤
e
2
(
a
2
+
b
2
+
c
2
+
d
2
)
+
f
(
e
)
(
a
4
+
b
4
+
c
4
+
d
4
)
a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})
a
3
+
b
3
+
c
3
+
d
3
≤
e
2
(
a
2
+
b
2
+
c
2
+
d
2
)
+
f
(
e
)
(
a
4
+
b
4
+
c
4
+
d
4
)
holds for all real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
.