Difference between revisions of "1973 Canadian MO Problems"
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==Problem 5== | ==Problem 5== | ||
− | For every positive integer <math>n</math>, let <math>h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}</math>. For example, <math>h(1) = 1, h(2) = 1+\frac{1}{2}, h(3) = 1+\frac{1}{2}+\frac{1}{3}</math>. Prove that | + | For every positive integer <math>n</math>, let <math>h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}</math>. |
+ | |||
+ | For example, <math>h(1) = 1, h(2) = 1+\frac{1}{2}, h(3) = 1+\frac{1}{2}+\frac{1}{3}</math>. | ||
+ | |||
+ | Prove that <math>n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n)\qquad</math> for <math>n=2,3,4,\ldots</math> | ||
[[1973 Canadian MO Problems/Problem 5 | Solution]] | [[1973 Canadian MO Problems/Problem 5 | Solution]] |
Revision as of 17:46, 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
.)
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