Difference between revisions of "2014 Canadian MO Problems"
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== Problem 1 == | == Problem 1 == | ||
− | Let <math>a_1,a_2,\dots,a_n</math> be positive real numbers whose product is <math>1</math>. Show that the sum <math>\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\ | + | Let <math>a_1,a_2,\dots,a_n</math> be positive real numbers whose product is <math>1</math>. Show that the sum <math>\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}</math> is greater than or equal to <math>\frac{2^n-1}{2^n}</math>. |
[[2014 Canadian MO Problems/Problem 1|Solution]] | [[2014 Canadian MO Problems/Problem 1|Solution]] |
Latest revision as of 19:07, 22 January 2017
Problem 1
Let be positive real numbers whose product is
. Show that the sum
is greater than or equal to
.
Problem 2
Let and
be odd positive integers. Each square of an
by
board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of
and
.
Problem 3
Let be a fixed odd prime. A
-tuple
of integers is said to be good if
(i) for all
, and
(ii)
is not divisible by
, and
(iii)
is divisible by
.
Determine the number of good -tuples.
Problem 4
The quadrilateral is inscribed in a circle. The point
lies in the interior of
, and
. The lines
and
meet at
, and the lines
and
meet at
. Prove that the lines
and
form the same angle as the diagonals of
.
Problem 5
Fix positive integers and
. A list of n integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add
to all of them or subtract
from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least
of the numbers on the blackboard are all simultaneously divisible by
.