兩條IMO Question

(1) If the base = 12cm, the two other sides can approach infinity => approaching infinite area.
If the two equal sides are 12cm, the angle between the sides ranges from 0 to 180 degrees
The area = (1/2)(12^2)sinθ = 72 sinθ
The maximum area = 72 cm^2
(2) 1 + n^2 + n^4
= n^4 + 2n^2 + 1 - n^2
= (n^2 + 1)^2 - n^2
= (n^2 - n + 1)(n^2 + n + 1)
n / (1 + n^2 + n^4)
= (An + B) / (n^2 + n + 1) + (Cn + D) / (n^2 - n + 1)
= [(A + C)n^3 + (A + B - C + D)n^2 + (A + B + C - D)n + (B + D)] / [(n^2 - n + 1)(n^2 + n + 1)]
Hence by comparing coefficients,
A + C = 0 ... (1)
A + B - C + D = 0 ... (2)
A + B + C - D = 1 ... (3)
B + D = ... (4)
These give A = C = 0; B = -1/2 and D = 1/2
Therefore n / (1 + n^2 + n^4) = (1/2)[1 / (n^2 - n + 1) - 1 / (n^2 + n + 1)]
And the required sum = (1/2)[(1 - 1/3) + (1/3 - 1/7) + (1/7 - 1/13) + )(1/13 - 1/21) + ... + (1/9901 - 1/10101)]
= (1/2)(1 - 1/10101)
= 5050 / 10101

第二條IMO有答案啦，提供埋步驟添。
第一條唔難，當佢係等腰直角三角形個陣面積最大，其值為 72 cm^2。