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National and Regional Contests
Russia Contests
All-Russian Olympiad
1962 All Russian Mathematical Olympiad
026
026
Part of
1962 All Russian Mathematical Olympiad
Problems
(1)
ASU 026 All Russian MO 1962 10.5 table
Source:
6/17/2019
Given positive numbers
a
1
,
a
2
,
.
.
.
,
a
m
,
b
1
,
b
2
,
.
.
.
,
b
n
a_1, a_2, ..., a_m, b_1, b_2, ..., b_n
a
1
,
a
2
,
...
,
a
m
,
b
1
,
b
2
,
...
,
b
n
. Is known that
a
1
+
a
2
+
.
.
.
+
a
m
=
b
1
+
b
2
+
.
.
.
+
b
n
.
a_1+a_2+...+a_m=b_1+b_2+...+b_n.
a
1
+
a
2
+
...
+
a
m
=
b
1
+
b
2
+
...
+
b
n
.
Prove that you can fill an empty table with
m
m
m
rows and
n
n
n
columns with no more than
(
m
+
n
−
1
)
(m+n-1)
(
m
+
n
−
1
)
positive number in such a way, that for all
i
,
j
i,j
i
,
j
the sum of the numbers in the
i
i
i
-th row will equal to
a
i
a_i
a
i
, and the sum of the numbers in the
j
j
j
-th column -- to
b
j
b_j
b
j
.
combinatorics
table