vmt-award-2024_-46-2-_yellow

HEXA Challenge

Problems created by: Eric Choate

Challenge 1: The sequence n 1 , n 2 , n 3 , … is defined as the positive integers such that n k !(n k +1)! / 3 is a perfect square. Find the first five terms in this sequence.

Challenge 2: Depending on the value of a , the parabola y = x 2 + a can intersect the circle x 2 + y 2 = 4 at zero, one, two, three, or four points. Determine the number of points of intersection as a function of a .

Challenge 3: Can a number x be added to the set {9, 11, 14, 20, 31, 40} such that both the mean and the median of the new set are equal to x ? If so, find the value. If not, why not?

Challenge 4: The dimensions of a rectangle are A and B . Find the dimensions of a rectangle that has exactly half the area and exactly half the perimeter of the first rectangle.

Challenge 5: For two distinct positive real numbers a and b , denote the two solutions of the equation ( ln a x )( ln b x ) = 1 as x 1 and x 2 . What condition must a and b satisfy in order for | ln x 1 | = | ln x 2 |?

Virginia Mathematics Teacher vol. 46, no. 2

32