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Difference between revisions of "2013 Canadian MO Problems"

(Created page with "==Problem 1== Determine all polynomials P(x) with real coefficients such that (x+1)P(x-1)-(x-1)P(x) is a constant polynomial. [[2013 Canadian MO Problems/Problem 1|Solution...")
 
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==Problem 1==
 
==Problem 1==
−
Determine all polynomials P(x) with real coefficients such that
+
Determine all polynomials <math>P(x)</math> with real coefficients such that
−
(x+1)P(x-1)-(x-1)P(x)
+
<math>(x+1)P(x-1)-(x-1)P(x)</math>
 
is a constant polynomial.     
 
is a constant polynomial.     
  
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==Problem 2==
 
==Problem 2==
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The sequence a_1, a_2, \dots, a_n consists of the numbers 1, 2, \dots, n in some order. For which positive integers n is it possible that the n+1 numbers 0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n all have di fferent remainders when divided by n + 1?     
+
The sequence <math>a_1, a_2, \dots, a_n</math> consists of the numbers <math>1, 2, \dots, n</math> in some order. For which positive integers <math>n</math> is it possible that the <math>n+1</math> numbers <math>0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n</math> all have di fferent remainders when divided by <math>n + 1</math>?     
  
  
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==Problem 3==
 
==Problem 3==
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Let G be the centroid of a right-angled triangle ABC with \angle BCA = 90^\circ. Let P be the point on ray AG such that \angle CPA = \angle CAB, and let Q be the point on ray BG such that \angle CQB = \angle ABC. Prove that the circumcircles of triangles AQG and BPG meet at a point on side AB.     
+
Let <math>G</math> be the centroid of a right-angled triangle <math>ABC</math> with <math>\angle BCA = 90^\circ</math>. Let <math>P</math> be the point on ray <math>AG</math> such that <math>\angle CPA = \angle CAB</math>, and let <math>Q</math> be the point on ray <math>BG</math> such that <math>\angle CQB = \angle ABC</math>. Prove that the circumcircles of triangles <math>AQG</math> and <math>BPG</math> meet at a point on side <math>AB</math>.     
  
  
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==Problem 4==
 
==Problem 4==
−
Let n be a positive integer. For any positive integer j and positive real number r, define f_j(r) and g_j(r) by
+
Let <math>n</math> be a positive integer. For any positive integer <math>j</math> and positive real number <math>r</math>, define  
−
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\l...
+
<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 \lceil x\rceil denotes the smallest integer greater than or equal to x. Prove that
+
where <math>\lceil x\rceil</math> denotes the smallest integer greater than or equal to <math>x</math>. Prove that
−
\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)
+
<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 r.     
+
for all positive real numbers <math>r</math>.     
  
  
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==Problem 5==
 
==Problem 5==
−
Let O denote the circumcentre of an acute-angled triangle ABC. Let point P on side AB be such that \angle BOP = \angle ABC, and let point Q on side AC be such that \angle COQ = \angle ACB. Prove that the reflection of BC in the line PQ is tangent to the circumcircle of triangle APQ.   
+
Let <math>O</math> denote the circumcentre of an acute-angled triangle <math>ABC</math>. Let point <math>P</math> on side <math>AB</math> be such that <math>\angle BOP = \angle ABC</math>, and let point <math>Q</math> on side <math>AC</math> be such that <math>\angle COQ = \angle ACB</math>. Prove that the reflection of <math>BC</math> in the line <math>PQ</math> is tangent to the circumcircle of triangle <math>APQ</math>.   
  
  
 
[[2013 Canadian MO Problems/Problem 5|Solution]]
 
[[2013 Canadian MO Problems/Problem 5|Solution]]

Latest revision as of 12:45, 8 October 2014

Problem 1

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.

Solution

Problem 2

The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have di fferent remainders when divided by $n + 1$?


Solution

Problem 3

Let $G$ be the centroid of a right-angled triangle $ABC$ with $\angle BCA = 90^\circ$. Let $P$ be the point on ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circumcircles of triangles $AQG$ and $BPG$ meet at a point on side $AB$.


Solution

Problem 4

Let $n$ be a positive integer. For any positive integer $j$ and positive real number $r$, define \[f_j(r) =\min (jr, n)+\min\left(\frac{j}{r}, n\right),\text{ and }g_j(r) =\min (\lceil jr\rceil, n)+\min\left(\left\lceil\frac{j}{r}\right\rceil, n\right),\] where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that \[\sum_{j=1}^n f_j(r)\leq n^2+n\leq \sum_{j=1}^n g_j(r)\] for all positive real numbers $r$.


Solution

Problem 5

Let $O$ denote the circumcentre of an acute-angled triangle $ABC$. Let point $P$ on side $AB$ be such that $\angle BOP = \angle ABC$, and let point $Q$ on side $AC$ be such that $\angle COQ = \angle ACB$. Prove that the reflection of $BC$ in the line $PQ$ is tangent to the circumcircle of triangle $APQ$.


Solution

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