Difference between revisions of "1973 Canadian MO Problems"
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==Problem 2== | ==Problem 2== | ||
| − | Find all real numbers that satisfy the equation <math>|x+3|-|x-1|=x+1</math>. (Note: <math>|a| = a</math> if <math>a\ge 0; |a|=-a if a<0</math>.) | + | Find all real numbers that satisfy the equation <math>|x+3|-|x-1|=x+1</math>. (Note: <math>|a| = a</math> if <math>a\ge 0; |a|=-a</math> if <math>a<0</math>.) |
[[1973 Canadian MO Problems/Problem 2 | Solution]] | [[1973 Canadian MO Problems/Problem 2 | Solution]] | ||
Revision as of 16:50, 8 October 2014
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
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
