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All-Russian Olympiad
1977 All Soviet Union Mathematical Olympiad
248
248
Part of
1977 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 248 All Soviet Union MO 1977 (x_1+...+x_n)=(y_1+...+y_m)<mn
Source:
7/6/2019
Given natural numbers
x
1
,
x
2
,
.
.
.
,
x
n
,
y
1
,
y
2
,
.
.
.
,
y
m
x_1,x_2,...,x_n,y_1,y_2,...,y_m
x
1
,
x
2
,
...
,
x
n
,
y
1
,
y
2
,
...
,
y
m
. The following condition is valid:
(
x
1
+
x
2
+
.
.
.
+
x
n
)
=
(
y
1
+
y
2
+
.
.
.
+
y
m
)
<
m
n
(
∗
)
(x_1+x_2+...+x_n)=(y_1+y_2+...+y_m)<mn \,\,\,\, (*)
(
x
1
+
x
2
+
...
+
x
n
)
=
(
y
1
+
y
2
+
...
+
y
m
)
<
mn
(
∗
)
Prove that it is possible to delete some terms from (*) (not all and at least one) and to obtain another valid condition.
combinatorics
algebra
Sum