MaxWong

Solution removed. This person is cheating in a competition.

Let t hours be the time it took for Hany to drive from home to work.

Let x km be the length of his route to work.

We have the following relations according to the information given in the question:

\(\dfrac x{70} = t + \dfrac1{60}\\ \dfrac{x}{75} = t - \dfrac1{60}\)

Subtracting, we have \(\dfrac{x}{1050} = \dfrac1{30}\) so \(x = 35\).

His route to work is 35 km long.

As the 'A's cannot be at the same position it was before, the resulting arrangement must be AXAXAX, where X is one of the remaining letters.

We can only change the arrangement by considering permutations of "BNN", as the A's are already fixed.

As same letters are indistinguishable, the order does not matter. That means the only possible ways to rearrange the letters are "ABANAN", "ANABAN", and "ANANAB". There are only 3 ways in total.

Notice that \(a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - (ab + bc +ca))\)

The only missing info we need is the value of \(a^2 + b^2 + c^2\).

Now, because \((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\), plugging in values gives \(a^2 + b^2 + c^2 = 11^2 - 2(25) = 71\)

You can plug in the values on your own.

Product of digits is 0 \(\Longleftrightarrow\) One or more of the digits are 0.

By principles of counting, there are 272 integers satisfying the property.

\(\quad (2x(y + 3))(y(2x - 1))\\ = 2xy(2x - 1)(y + 3)\\ =2xy(2xy - y + 6x - 3)\\ =2(2x^2 y^2 - xy^2 + 6x^2 y - 3xy)\)

The answer is D.

a)
Center = (2, -3).

Using the point-line distance formula: 

\(\quad \text{Distance between }(x_0, y_0)\text{ and the line }ax+by+c=0\\ = \dfrac{\left|ax_0 + by_0 + c\right|}{\sqrt{a^2 + b^2}}\)

Plugging in \((x_0, y_0, a, b, c) = (2, -3, -4, 5, 20)\) gives the answer.

b)

In this case, you plot a straight line perpendicular to 5y - 4x + 20 = 0, which passes through the center of the circle. This straight line intersects the circle twice. For each intersection, calculate the point-line distance between the intersection and the straight line 5y - 4x + 20 = 0. The smaller one is the minimum distance, and the larger one is the required answer.

Use a calculator. And I am surprised because none of the options is the correct answer.

The answer is 5.

The graph touches the x-axis at only 1 point. This means the equation of the graph must be a product of a perfect square and a real constant, in this case:

\(y = k(x + 1)^2\)

When x = 0, y = 2.

\(2=k\cdot 1^2\\ k = 2\)

Therefore, the equation of the parabola is \(y = 2(x + 1)^2\).

If it is a square with one of the vertices being the origin, then two of the vertices lies on |y| = |x|, in this case, two of the vertices lies on y = -x.

We solve the system of simultaneous equations

\(\begin{cases}y = 2(x + 1)^2\\y = -x\end{cases}\)

I will leave the rest for you.