MathDB

p1. A man is walking due east at

2

m.p.h. and to him the wind appears to be blowing from the north. On doubling his speed to

4

m.p.h. and still walking due east, the wind appears to be blowing from the nortl^eas^. What is the speed of the wind (assumed to have a constant velocity)?
p2. Prove that any triangle can be cut into three pieces which can be rearranged to form a rectangle of the same area.
p3. An increasing sequence of integers starting with

1

has the property that if

n

is any member of the sequence, then so also are

3n

and

n + 7

. Also, all the members of the sequence are solely generated from the first nummber

1

; thus the sequence starts with

1,3,8,9,10, ...

and

2,4,5,6,7...

are not members of this sequence. Determine all the other positive integers which are not members of the sequence.
p4. Three prime numbers, each greater than

3

, are in arithmetic progression. Show that their common difference is a multiple of

6

.
p5. Prove that if

S

is a set of at least

7

distinct points, no four in a plane, the volumes of all the tetrahedra with vertices in

S

are not all equal.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

1968 MMPC

Subcontests

1968 MMPC , Part 2 = Michigan Mathematics Prize Competition