# Key Problems

For any two positive real numbers a and b, find the limit of the sequence {x_n/n} as n approaches infinity. The sequence {x_n} consists of positive integer multiples of a and b, arranged in non-decreasing order with repetitions.

Let A={13k-5: k=1,2,3,...,2023} and B={17k-10: k=1,2,3,...,2023} How many elements are in AUB?

3928

3926

3905

None of these

What is the area of the image of the unit disk under the map T:R^2-->R^2 given by T(x,y)=(x+2y,3x+7y+1)?

pi

pi/2

2pi

3

How many fixed points a bijection f:[0,1]-->[0,1] with f(0)=1 can have?

Either 0 or exactly 1

At least 1

As many as one wants

None of these

If f(x) is a continuous function on the closed interval [a, b], then which of the following statements is true?

f(x) is differentiable on (a, b).

f(x) is uniformly continuous on (a, b).

f(x) is necessarily monotonic on [a, b].

f(x) must have a local minimum on (a, b).