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Difference between revisions of "2014 Canadian MO Problems"

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== Problem 1 ==
 
== Problem 1 ==
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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)\cdo...</math> is greater than or equal to <math>\frac{2^n-1}{2^n}</math>.     
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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>.     
 
   
 
   
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== Solution ==
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[[2014 Canadian MO Problems/Problem 1|Solution]]
  
 
== Problem 2==
 
== Problem 2==
 
Let <math>m</math> and <math>n</math> be odd positive integers. Each square of an <math>m</math> by <math>n</math> 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 <math>m</math> and <math>n</math>.     
 
Let <math>m</math> and <math>n</math> be odd positive integers. Each square of an <math>m</math> by <math>n</math> 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 <math>m</math> and <math>n</math>.     
 
    
 
    
 +
[[2014 Canadian MO Problems/Problem 2|Solution]]
 +
 
== Problem 3==
 
== Problem 3==
 
Let <math>p</math> be a fixed odd prime. A <math>p</math>-tuple <math>(a_1,a_2,a_3,\ldots,a_p)</math> of integers is said to be good if  
 
Let <math>p</math> be a fixed odd prime. A <math>p</math>-tuple <math>(a_1,a_2,a_3,\ldots,a_p)</math> of integers is said to be good if  
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Determine the number of good <math>p</math>-tuples.     
 
Determine the number of good <math>p</math>-tuples.     
 
    
 
    
 +
[[2014 Canadian MO Problems/Problem 3|Solution]]
 +
 
== Problem 4==
 
== Problem 4==
 
The quadrilateral <math>ABCD</math> is inscribed in a circle. The point <math>P</math> lies in the interior of <math>ABCD</math>, and <math>\angle P AB = \angle P BC = \angle P CD = \angle P DA</math>. The lines <math>AD</math> and <math>BC</math> meet at <math>Q</math>, and the lines <math>AB</math> and <math>CD</math> meet at <math>R</math>. Prove that the lines <math>PQ</math> and <math>PR</math> form the same angle as the diagonals of <math>ABCD</math>.     
 
The quadrilateral <math>ABCD</math> is inscribed in a circle. The point <math>P</math> lies in the interior of <math>ABCD</math>, and <math>\angle P AB = \angle P BC = \angle P CD = \angle P DA</math>. The lines <math>AD</math> and <math>BC</math> meet at <math>Q</math>, and the lines <math>AB</math> and <math>CD</math> meet at <math>R</math>. Prove that the lines <math>PQ</math> and <math>PR</math> form the same angle as the diagonals of <math>ABCD</math>.     
 
    
 
    
 +
[[2014 Canadian MO Problems/Problem 4|Solution]]
 +
 
== Problem 5==
 
== Problem 5==
 
Fix positive integers <math>n</math> and <math>k\ge 2</math>. 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 <math>1</math> to all of them or subtract <math>1</math> 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 <math>n-k+2</math> of the numbers on the blackboard are all simultaneously divisible by <math>k</math>.
 
Fix positive integers <math>n</math> and <math>k\ge 2</math>. 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 <math>1</math> to all of them or subtract <math>1</math> 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 <math>n-k+2</math> of the numbers on the blackboard are all simultaneously divisible by <math>k</math>.
 +
 +
[[2014 Canadian MO Problems/Problem 5|Solution]]

Latest revision as of 18:07, 22 January 2017

Problem 1

Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum $\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}$ is greater than or equal to $\frac{2^n-1}{2^n}$.

Solution

Problem 2

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ 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 $m$ and $n$.

Solution

Problem 3

Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be good if

(i) $0\le a_i\le p-1$ for all $I$, and (ii) $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and (iii) $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.

Determine the number of good $p$-tuples.

Solution

Problem 4

The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $PQ$ and $PR$ form the same angle as the diagonals of $ABCD$.

Solution

Problem 5

Fix positive integers $n$ and $k\ge 2$. 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 $1$ to all of them or subtract $1$ 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 $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

Solution

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