19^2 + 1^1 = 362
18^2 + 2^2 = 328
So 18 and 2 may be the numbers you require....
The two values are close to 1 and 19
Let the numbers be x and y
If their sum is 20, then;
x + y = 20 .................... 1
If the sum of their squares is less than 362, then;
x² + y² < 362 ...................2
From equation 1:
x = 20 - y ................... 3
Substitute equation 3 into 2
(20-y)² + y² < 362
400 - 40y +y²+y² < 362
2y²-40y < 362 - 400
2y²-40y < -38
2y²-40y + 38 <0
y²-20y + 19 < 0
y²-y -19y+ 19 < 0
y(y-1) - 19(y-1) < 0
y-1 < 0 and y - 19 < 0
y < 1 and y < 19
Recall that:
x + y = 20
x + 1 = 20
x = 19
Hence the two values are close to 1 and 19.
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