PART 1

QUESTION 01

The total number of positive divisors of 2019 is  \(\bbox[10px, border: 2px solid red]{(1)}\) and the whole sum of those divisors is \(\bbox[10px, border: 2px solid red]{(2)}\) .

QUESTION 02

For the three points O(0, 0), A(3, 0), and B(3, 4) on the xy-plane, the equation of the angle bisector of ∠AOB is \(y = \bbox[10px, border: 2px solid red]{(1)} \: x\).

QUESTION 03

For parabola \(y = x^2\) and two points (−1,1) and (3,9) on it, its tangent line parallel to the line through the two points is the line \(y = \bbox[10px, border: 2px solid red]{(1)} \: x + \bbox[10px, border: 2px solid red]{(2)}\), whose point of tangency is the point \(\left( \: \bbox[10px, border: 2px solid red]{(3)}, \bbox[10px, border: 2px solid red]{(4)} \: \right)\).

QUESTION 04

When the line \(y = m(x - 5) + 3\) intersects the circle \(x^2 + y^2 = r^2 \: (r > 0)\) if and only if \(0 \leq m \leq \bbox[10px, border: 2px solid red]{(1)}\), \(r = \bbox[10px, border: 2px solid red]{(2)}\) .

QUESTION 05

When \(| x | \leq \dfrac {\pi}{2}\) the maksimum of \(\sin x + \cos x\) is \(\bbox[10px, border: 2px solid red]{(1)}\) and the minimum of that is \(\bbox[10px, border: 2px solid red]{(2)}\) .

QUESTION 06

By \(\log_{10} 2 \approx 0.3010\) and \(\log_{10} 3 \approx 0.4771\), the number of digits of \(6^{100}\) is \(\bbox[10px, border: 2px solid red]{(1)}\), and its leading digit is \(\bbox[10px, border: 2px solid red]{(2)}\) .