Difference between revisions of "1973 Canadian MO Problems"
m (→Problem 2) |
(→Problem 4) |
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==Problem 4== | ==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); | ||
+ | MP("P_{"+string(k)+"}",A,11,A); | ||
+ | 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); | ||
+ | } | ||
+ | </asy> | ||
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. | 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. |
Revision as of 18:17, 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