1973 Canadian MO Problems
Contents
Problem 1
Solve the simultaneous inequalities, 
and 
; i.e. find a single inequality equivalent to the two simultaneous inequalities.
What is the greatest integer that satisfies both inequalities 
and 
.
Give a rational number between 
and 
.
Express 
as a product of two integers neither of which is an integral multiple of 
.
Without the use of logarithm tables evaluate 
.
Problem 2
Find all real numbers that satisfy the equation 
. (Note: 
if 
if 
.)
Problem 3
Prove that if 
and 
are prime integers greater than 
, then 
is a factor of 
.
Problem 4
![[asy] size(200); pair A=dir(120), B=dir(80); for(int k=0;k<9;++k) { pair C=dir(120-(40)*(k+2)); D(A--B); A=B;B=C; } for(int k=0;k<3;++k) { pair A1=dir(120-(40)*(3*k)); pair B1=dir(120-(40)*(3*k+2)); pair C1=dir(120-(40)*(3*k+3)); D(A1--B1); D(A1--C1); } for(int k=0;k<9;++k) { pair A=dir(120+(40)*(k)); MP("P_{"+string(k)+"}",A,11,A); } [/asy]](http://latex.artofproblemsolving.com/8/3/4/8344776ea853dd08ddb05d7aea1fc48e2f92214a.png)
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: 
. In how many ways can these triangles be labeled with the names 
so that 
is a vertex of triangle 
for 
? Justify your answer.
Problem 5
For every positive integer 
, let 
.
For example, 
.
Prove that 
for 
Problem 6
If 
and 
are fixed points on a given circle not collinear with center 
of the circle, and if 
is a variable diameter, find the locus of 
(the intersection of the line through and 
and the line through and 
).
Problem 7
Observe that $\frac{1}{1}= \frac{1}{2}+\frac{1}{2};\quad \frac{1}{2}=\frac{1}{3}+\frac{1}{6};\quad \frac{1}{3}=\frac{1}{4}+\frac{1}{12};\qu...$ (Error compiling LaTeX. Unknown error_msg) State a general law suggested by these examples, and prove it.
Prove that for any integer greater than 
there exist positive integers 
and 
such that
