If the line that passes through the points (2, 7) and (a, 3a) has a slope of 2, the value of a is

 The number of zeros in the integer equal to $\left(10^{100}\right)\times\left(100^{10}\right)$  is

When three positive integers are added in pairs, the resulting sums are 998, 1050, and 1234. What is the difference between the largest and smallest of the three original positive integers?

A total of n points are equally spaced around a circle and are labelled with the integers 1 to n, in order. Two points are called diametrically opposite if the line segment joining them is a diameter of the circle. If the points labelled 7 and 35 are diametrically opposite, then n equals

Alain and Louise are driving on a circular track with radius 25km. Alain leaves the starting line first, going clockwise around the track at a speed of 80km/h. Fifteen minutes after Alain starts, Louise leaves the same starting line, going counter-clockwise around the track at a speed of 100km/h. For how many hours will Louise have been driving when the two of them pass each other for the fourth time?

The value of  $3+3^2$  is

If n > 0 and  $\sqrt{n^2+n^2+n^2+n^2}=64$ then  $n$  equals

The number of integers x for which the value of  $\frac{-6}{x+1}$  is an integer is

If $2^x=15$ and  $15^{y\ }=32$ , the value of xy is

Suppose that a, b, c, and d are positive integers that satisfy the equations

At Matilda's birthday party, the ratio of people who ate ice cream to the people who ate cake was 3:2. People who are both ice cream and cake were included in both categories. If 120 people were at the party, what is the maximum number of people who could have eaten both ice cream and cake?

Suppose m and n are positive integers with m < n. The value of  $\frac{m+3}{n+3}$  will be