Two athletes, A and B, each shot five arrows. 1764 has six prime numbers multiplied, so at least one ring is the product of two.

Moreover, each ring cannot be greater than 10 ring, so if we only shoot 1 ring, the product of the number of 5 arrow rings is 1764.

1 6 6 7 7 and 27.

1 4 9 7 7 and 28.

2, 3, 6, 7, 7 and 25.

2, 2, 9, 7, 7 and 27.

Three, three, four, seven, seven and 24.

A has four more rings than B, so

A is a 28-ring, 1 4 9 7 7.

B is 24 rings, 3 3 4 7 7.