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>.) | |
[[1973 Canadian MO Problems/Problem 2 | Solution]] | [[1973 Canadian MO Problems/Problem 2 | Solution]] | ||
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==Problem 3== | ==Problem 3== | ||
| − | + | Prove that if <math>p</math> and <math>p+2</math> are prime integers greater than <math>3</math>, then <math>6</math> is a factor of <math>p+1</math>. | |
[[1973 Canadian MO Problems/Problem 3 | Solution]] | [[1973 Canadian MO Problems/Problem 3 | Solution]] | ||
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==Problem 4== | ==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: <math>P_{0}P_{1}P_{3}, P_{0}P_{3}P_{6}, P_{0}P_{6}P_{7}, P_{0}P_{7}P_{8}, P_{1}P_{2}P_{3}, P_{3}P_{4}P_{6}, P_{4}P_{5}P_{6}</math>. In how many ways can these triangles be labeled with the names <math>\triangle_{1}, \triangle_{2}, \triangle_{3}, \triangle_{4}, \triangle_{5}, \triangle_{6}, \triangle_{7}</math> so that <math>P_{i}</math> is a vertex of triangle <math>\triangle_{i}</math> for <math>i = 1, 2, 3, 4, 5, 6, 7</math>? Justify your answer. | ||
[[1973 Canadian MO Problems/Problem 4 | Solution]] | [[1973 Canadian MO Problems/Problem 4 | Solution]] | ||
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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 <math>n=2,3,4,\ldots n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n)</math>. | |
[[1973 Canadian MO Problems/Problem 5 | Solution]] | [[1973 Canadian MO Problems/Problem 5 | Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
| − | + | If <math>A</math> and <math>B</math> are fixed points on a given circle not collinear with center <math>O</math> of the circle, and if <math>XY</math> is a variable diameter, find the locus of <math>P</math> (the intersection of the line through <math>A</math> and <math>X</math> and the line through <math>B</math> and <math>Y</math>). | |
[[1973 Canadian MO Problems/Problem 6 | Solution]] | [[1973 Canadian MO Problems/Problem 6 | Solution]] | ||
Revision as of 16:42, 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
