11th Bay Area Mathematical Olympiad : BAMO-8 Exam
A square grid of 16 dots (see the figure) contains the corners of nine 1×1 squares, four 2×2 squares, and one 3×3 square, for a total of 14 squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the 14 squares is missing at least one corner?
Justify your answer by showing both that the number of dots you claim is sufficient and by explaining why no smaller number of dots will work.
The Fibonacci sequence is the list of numbers that begins 1, 2, 3, 5, 8, 13 and continues with each subsequent number being the sum of the previous two.
Prove that for every positive integer n, when the first n elements of the Fibonacci sequence are alternately added and subtracted, the result is an element of the sequence or the negative of an element of the sequence. For example, when n = 4 we have
1-2+3-5 = -3,
and 3 is an element of the Fibonacci sequence.
There are many sets of two different positive integers a and b, both less than 50, such that a2 and b2 end in the same last two digits. For example, 352 = 1225 and 452 = 2025 both end in 25. What are all possible values for the average of a and b?
For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider 22 to end with the digits 04, not 4.
Seven congruent line segments are connected together at their endpoints as shown in the figure below at the left. By raising point E the linkage can be made taller, as shown in the figure below and to the right.
Continuing to raise E in this manner, it is possible to use the linkage to make A, C, F, and E collinear, while simultaneously making B, G, D, and E collinear, thereby constructing a new triangle ABE.
Prove that a regular polygon with center E can be formed from a number of copies of this new triangle ABE, joined together at point E, and without overlapping interiors. Also find the number of sides of this polygon and justify your answer.