Read full issue - Canadian Mathematical Society

264
12 th Grade
1. Solve the equation cos x cos 2x cos 3x =1.
2. All faces of a convex polytope are triangles. What can be the number
of the faces?
3. Does there exist a polynomial P (x; y) in two variables such that
(a) P (x; y) > 0 for all x; y,
(b) for each c>0there exist such x and y that P (x; y) =c?
4. Let S(x) be the digital sum of natural number x. Prove that
S(2 n ) !1when n !1,nnatural.
5. The centres of four equal circles are the vertices of a square. How
must A, B, C, D be chosen so that each circle contains at least one of them
and the area of ABCD is as big as possible?
As a nal set of problems for you to puzzle over after the hiatus, we give
the problems of the Dutch Mathematical Olympiad, Second Round, written
16 September 1994. My thanks again go to Bill Sands for collecting these
problems for me while he was helping out at the IMO in Toronto.
DUTCH MATHEMATICAL OLYMPIAD
Second Round
16 September, 1994
1. A unit square is divided in two rectangles in such a way that the
smaller rectangle can be put on the greater rectangle with every vertex of the
smaller on exactly one of the edges of the greater.
Calculate the dimensions of the smaller rectangle.
2. Given is a sequence of numbers a1; a2;a3;::: with the property:
a1 =2; a2=3and
an+1 =2an,1 or
an+1 =3an,2an,1
for all n 2:
Prove that no number between 1600 and 2000 can be an element of the
sequence.