MathDB

p1. a) Given four points in the plane, no three of which lie on the same line, each subset of three points determines the vertices of a triangle. Can all these triangles have equal areas? If so, give an example of four points (in the plane) with this property, and then describe all arrangements of four joints (in the plane) which permit this. If no such arrangement exists, prove this. b) Repeat part a) with "five" replacing "four" throughout.
p2. Three people at the base of a long stairway begin a race up the stairs. Person A leaps five steps with each stride (landing on steps

5

10

15

, etc.). Person B leaps a little more slowly but covers six steps with each stride. Person C leaps seven steps with each stride. A picture taken near the end of the race shows all three landing simultaneously, with Person A twenty-one steps from the top, person B seven steps from the top, and Person C one step from the top. How many steps are there in the stairway? If you can find more than one answer, do so. Justify your answer.
p3. Let

S

denote the sum of an infinite geometric series. Suppose the sum of the squares of the terms is

2S

, and that df the cubes is

64S/13

. Find the first three terms of the original series.
p4.

A

B

and

C

are three equally spaced points on a circular hoop. Prove that as the hoop rolls along the horizontal line

\ell

, the sum of the distances of the points

A, B

, and

C

above line

\ell

is constant. https://cdn.artofproblemsolving.com/attachments/3/e/a1efd0975cf8ff3cf6acb1da56da1dce35d81e.png
p5. A set of

n

numbers

x_1,x_2,x_3,...,x_n

(where

n>1

) has the property that the

k^{th}

number (that is,

x_k

) is removed from the set, the remaining

(n-1)

numbers have a sum equal to

k

(the subscript o

x_k

), and this is true for each

k = 1,2,3,...,n

. a) SoIve for these

n

numbers b) Find whether at least one of these

n

numbers can be an integer.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

1975 MMPC

Subcontests

1975 MMPC , Part 2 = Michigan Mathematics Prize Competition