Difference between revisions of "1973 Canadian MO Problems"
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==Problem 7== | ==Problem 7== | ||
| + | Observe that | ||
| + | <math>\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...</math> | ||
| + | State a general law suggested by these examples, and prove it. | ||
| + | Prove that for any integer <math>n</math> greater than <math>1</math> there exist positive integers <math>i</math> and <math>j</math> such that | ||
| + | <math>\frac{1}{n}= \frac{1}{i(i+1)}+\frac{1}{(i+1)(i+2)}+\frac{1}{(i+2)(i+3)}+\cdots+\frac{1}{j(j+1)}. </math> | ||
[[1973 Canadian MO Problems/Problem 7 | Solution]] | [[1973 Canadian MO Problems/Problem 7 | Solution]] | ||
Revision as of 21:55, 16 December 2011
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
Problem 3
Problem 4
Problem 5
Problem 6
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
