Difference between revisions of "1975 Canadian MO Problems"
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== Problem 6 == | == Problem 6 == | ||
<div class=ol> | <div class=ol> | ||
− | <div class=li><span class=num>(i)</span><math>15</math> chairs are equally place around a circular table on which are name cards for <math>15</math> | + | <div class=li><span class=num>(i)</span><math>15</math> chairs are equally place around a circular table on which are name cards for <math>15</math> guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated. </div> |
<div class=li><span class=num>(ii)</span> Give an example of an arrangement in which just one of the 15 quests is correctly seated and for which no rotation correctly places more than one person.</div> | <div class=li><span class=num>(ii)</span> Give an example of an arrangement in which just one of the 15 quests is correctly seated and for which no rotation correctly places more than one person.</div> | ||
</div> | </div> | ||
[[1975 Canadian MO Problems/Problem 6 | Solution]] | [[1975 Canadian MO Problems/Problem 6 | Solution]] | ||
+ | |||
== Problem 7 == | == Problem 7 == | ||
A function <math>f(x)</math> is <math>\textit{periodic}</math> if there is a positive integer such that <math>f(x+p) = f(x)</math> for all <math>x</math>. For example, <math>\sin x</math> is periodic with period <math>2\pi</math>. Is the function <math>\sin(x^2)</math> periodic? Prove your assertion. | A function <math>f(x)</math> is <math>\textit{periodic}</math> if there is a positive integer such that <math>f(x+p) = f(x)</math> for all <math>x</math>. For example, <math>\sin x</math> is periodic with period <math>2\pi</math>. Is the function <math>\sin(x^2)</math> periodic? Prove your assertion. |
Latest revision as of 12:17, 11 August 2016
Contents
Problem 1
Simplify
.
Problem 2
A sequence of numbers satisfies
![$a_1 = \frac{1}{2}$](http://latex.artofproblemsolving.com/7/c/9/7c93752fc9b032db99ff38f9114dcc254b098bcf.png)
![$a_1+a_2+\cdots+a_n=n^2a_n\quad(n\ge1).$](http://latex.artofproblemsolving.com/a/f/2/af2aa146d3f29ed75142daa3c1a4819df46b0582.png)
Determine the value of
Problem 3
For each real number ,
denotes the largest integer less than or equal to
,
Indicate on the
-plane the set of all points
for which
.
Problem 4
For a positive number such as ,
is referred to as the integral part of the number and
as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
Problem 5
are four "consecutive" points on the circumference of a circle and
are points on the circumference which are respectively the midpoints of the arcs
Prove that
is perpendicular to
.
Problem 6
![$15$](http://latex.artofproblemsolving.com/f/1/9/f1965fae079a9ba2c0726c307070c2355dfcb213.png)
![$15$](http://latex.artofproblemsolving.com/f/1/9/f1965fae079a9ba2c0726c307070c2355dfcb213.png)
Problem 7
A function is
if there is a positive integer such that
for all
. For example,
is periodic with period
. Is the function
periodic? Prove your assertion.
Problem 8
Let be a positive integer. Find all polynomials
where the
are real, which satisfy the equation
.