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1987 All Soviet Union Mathematical Olympiad
441
441
Part of
1987 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 441 All Soviet Union MO 1987 tennis x_1^2+...+x_{10}^2=y_1^2+...+y_{10}^2
Source:
8/7/2019
Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let
x
i
xi
x
i
be the number of
i
i
i
-th player victories,
y
i
yi
y
i
-- losses. Prove that
x
1
2
+
.
.
.
+
x
10
2
=
y
1
2
+
.
.
.
+
y
10
2
x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2
x
1
2
+
...
+
x
10
2
=
y
1
2
+
...
+
y
10
2
Sum of Squares
algebra
combinatorics
Tournament