✔ 最佳答案
3) 留意:
40^2 + 81 = 41^2, 因為 81 為第 41 個正單數.
所以答案為 1631
2010-04-30 23:24:05 補充：
1) 以正弦定律計:
a/sin A = b/sin B = c/sin C = k, 其中 k 為某常數.
所以:
a/(b + c) = k sin A/(k sin B + k sin C)
= sin A/(sin B + sin C)
= 2 sin (A/2) cos (A/2)/{2 sin [(B + C)/2] cos [(B - C)/2]}
= sin (A/2) cos (A/2)/{sin (90 - A/2) cos [(B - C)/2]}
= sin (A/2) cos (A/2)/{cos (A/2) cos [(B - C)/2]}
= sin (A/2)/cos [(B - C)/2]
由於 cos [(B - C)/2] <= 1, 1/cos [(B - C)/2] >= 1
所以:
sin (A/2)/cos [(B - C)/2] >= sin (A/2)
2) 留意 509 為質數, 所以 a 或 2a + b 其中一個必為 509.
若 a = 509, 則:
(b + 1018)2 = 511b + 2036
b2 + 2036b + 10182 = 511b + 2036
b2 + 1525b + (10182 - 2036) = 0
此方程式的判別式為負數, 所以 b 不是實數. 即 a = 509 的可能排除.
若 2a + b = 509, 則:
a (2a + b) = 4a + 511b
509a = 2(2a + b) + 509b
509a = 1018 + 509b
a - b = 2
4) Suppose that n6 + 3n5 - 9n4 - 11n3 + 8n2 + 12n + 3 is a perfect square, then it must be in the form:
n6 + 3n5 - 9n4 - 11n3 + 8n2 + 12n + 3 = (n3 + an2 + bn + c)2
where a, b and c are some constants.
Now since n3 + an2 + bn + c is an integer, all a, b and c must be rational. Hence when expanding the equality for:
n6 + 3n5 - 9n4 - 11n3 + 8n2 + 12n + 3 = (n3 + an2 + bn + c)2
we get c2 = 3, implying that c is irrational.
Hence the assumption leads to a contradiction.
5) Pls. allow some time for me to think further.
