Difference between revisions of "2013 Canadian MO Problems"
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==Problem 4== | ==Problem 4== | ||
− | Let <math>n</math> be a positive integer. For any positive integer <math>j</math> and positive real number <math>r</math>, define | + | Let <math>n</math> be a positive integer. For any positive integer <math>j</math> and positive real number <math>r</math>, define |
− | < | + | <cmath> f_j(r) =\min (jr, n)+\min\left(\frac{j}{r}, n\right),\text{ and }g_j(r) =\min (\lceil jr\rceil, n)+\min\left(\left\lceil\frac{j}{r}\right\rceil, n\right),</cmath> |
where <math>\lceil x\rceil</math> denotes the smallest integer greater than or equal to <math>x</math>. Prove that | where <math>\lceil x\rceil</math> denotes the smallest integer greater than or equal to <math>x</math>. Prove that | ||
− | < | + | <cmath>\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)</cmath> |
for all positive real numbers <math>r</math>. | for all positive real numbers <math>r</math>. | ||
Latest revision as of 13:45, 8 October 2014
Problem 1
Determine all polynomials with real coefficients such that
is a constant polynomial.
Problem 2
The sequence consists of the numbers
in some order. For which positive integers
is it possible that the
numbers
all have di fferent remainders when divided by
?
Problem 3
Let be the centroid of a right-angled triangle
with
. Let
be the point on ray
such that
, and let
be the point on ray
such that
. Prove that the circumcircles of triangles
and
meet at a point on side
.
Problem 4
Let be a positive integer. For any positive integer
and positive real number
, define
where
denotes the smallest integer greater than or equal to
. Prove that
for all positive real numbers
.
Problem 5
Let denote the circumcentre of an acute-angled triangle
. Let point
on side
be such that
, and let point
on side
be such that
. Prove that the reflection of
in the line
is tangent to the circumcircle of triangle
.