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National and Regional Contests
Russia Contests
All-Russian Olympiad
1975 All Soviet Union Mathematical Olympiad
219
219
Part of
1975 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 219 All Soviet Union MO 1975 numbers in a table
Source:
7/5/2019
a) Given real numbers
a
1
,
a
2
,
b
1
,
b
2
a_1,a_2,b_1,b_2
a
1
,
a
2
,
b
1
,
b
2
and positive
p
1
,
p
2
,
q
1
,
q
2
p_1,p_2,q_1,q_2
p
1
,
p
2
,
q
1
,
q
2
. Prove that in the table
2
×
2
2\times 2
2
×
2
(
a
1
+
b
1
)
/
(
p
1
+
q
1
)
,
(
a
1
+
b
2
)
/
(
p
1
+
q
2
)
(a_1 + b_1)/(p_1 + q_1) , (a_1 + b_2)/(p_1 + q_2)
(
a
1
+
b
1
)
/
(
p
1
+
q
1
)
,
(
a
1
+
b
2
)
/
(
p
1
+
q
2
)
(
a
2
+
b
1
)
/
(
p
2
+
q
1
)
,
(
a
2
+
b
2
)
/
(
p
2
+
q
2
)
(a_2 + b_1)/(p_2 + q_1) , (a_2 + b_2)/(p_2 + q_2)
(
a
2
+
b
1
)
/
(
p
2
+
q
1
)
,
(
a
2
+
b
2
)
/
(
p
2
+
q
2
)
there is a number in the table, that is not less than another number in the same row and is not greater than another number in the same column (a saddle point). b) Given real numbers
a
1
,
a
2
,
.
.
.
,
a
n
,
b
1
,
b
2
,
.
.
.
,
b
n
a_1, a_2, ... , a_n, b_1, b_2, ... , b_n
a
1
,
a
2
,
...
,
a
n
,
b
1
,
b
2
,
...
,
b
n
and positive
p
1
,
p
2
,
.
.
.
,
p
n
,
q
1
,
q
2
,
.
.
.
,
q
n
p_1, p_2, ... , p_n, q_1, q_2, ... , q_n
p
1
,
p
2
,
...
,
p
n
,
q
1
,
q
2
,
...
,
q
n
. We construct the table
n
×
n
n\times n
n
×
n
, with the numbers (
0
<
i
,
j
≤
n
0 < i,j \le n
0
<
i
,
j
≤
n
)
(
a
i
+
b
j
)
/
(
p
i
+
q
j
)
(a_i + b_j)/(p_i + q_j)
(
a
i
+
b
j
)
/
(
p
i
+
q
j
)
in the intersection of the
i
i
i
-th row and
j
j
j
-th column. Prove that there is a number in the table, that is not less than arbitrary number in the same row and is not greater than arbitrary number in the same column (a saddle point).
table
combinatorics