hkmo question(urgent)

1)

For the roots of a quadratic equation are integers , △ is a complete square.

So we let △ = m² where m is an integer.

△ = 4(2P-3)² - 4(4P² - 14P + 8) = m²

(2P-3)² - (4P² - 14P + 8) = m²/4 = (m/2)²

2)

0 ≤ x - 1/2 < 1
==>
2*0 ≤ 2(x - 1/2) < 2*1
==>
0 ≤ 2x - 1 < 2
==>
0+(2.5) ≤ 2x - 1 + 2.5 < 2 + 2.5
==>
2.5 ≤ 2x + 3/2 < 4.5

3)

Let a^(1/3) = A , b^(1/3) = B ,
then
a^(2/3) = A² , b^(2/3) = B²
and
a = A³ , b = B³
So

P = (A³ + 3A²B + 3AB² + B³)^(2/3) + (A³ - 3A²B + 3AB² - B³)^(2/3)

P = (A + B)³ ^(2/3) + (A - B)³ ^(2/3)

P = (A + B)^ 3(2/3) + (A - B)^ 3(2/3)

P = (a^(1/3) + b^(1/3)) ^ 3(2/3) + (a^(1/3) - b^(1/3)) ^ 3(2/3)

4)

Since 2 points can form 1 line ,
Q sides can form QC2 lines.

But QC2 lines including Q sides which are not diagonals ,

So the number of diagonals = QC2 - Q











2011-05-25 20:23:58 補充：
for q1, why 2P + 1 = 5^2 ?

It is because only when P = 12 (5 < P < 20),
2P + 1 = 25 is a perfect square.