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Difference between revisions of "2017 EGMO Problems"

(Created page with "==Day 1== ===Problem 1=== Let <math>ABCD</math> be a convex quadrilateral with <math>\angle DAB=\angle BCD=90^{\circ}</math> and <math>\angle ABC> \angle CDA</math>. Let <math...")
 
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Find the smallest positive integer <math>k</math> for which there exists a colouring of the positive integers <math>\mathbb{Z}_{>0}</math> with <math>k</math> colours and a function <math>f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}</math> with the following two properties:
 
Find the smallest positive integer <math>k</math> for which there exists a colouring of the positive integers <math>\mathbb{Z}_{>0}</math> with <math>k</math> colours and a function <math>f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}</math> with the following two properties:
  
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<math>(i)</math> For all positive integers <math>m,n</math> of the same colour, <math>f(m+n)=f(m)+f(n).</math>
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(i) For all positive integers <math>m,n</math> of the same colour, <math>f(m+n)=f(m)+f(n).</math>
  
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<math>(ii)</math> There are positive integers <math>m,n</math> such that <math>f(m+n)\ne f(m)+f(n).</math>
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(ii) There are positive integers <math>m,n</math> such that <math>f(m+n)\ne f(m)+f(n).</math>
  
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[i]In a colouring of <math>\mathbb{Z}_{>0}</math> with <math>k</math> colours, every integer is coloured in exactly one of the <math>k</math> colours. In both <math>(i)</math> and <math>(ii)</math> the positive integers <math>m,n</math> are not necessarily distinct.[/i]
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In a colouring of <math>\mathbb{Z}_{>0}</math> with <math>k</math> colours, every integer is coloured in exactly one of the <math>k</math> colours. In both <math>(i)</math> and <math>(ii)</math> the positive integers <math>m,n</math> are not necessarily distinct.
  
 
[[2016 EGMO Problems/Problem 2|Solution]]
 
[[2016 EGMO Problems/Problem 2|Solution]]
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b) Prove that for every odd positive integer <math>m</math> there exists an integer <math>n\geq2</math> such that <math>m</math> belongs to an expensive <math>n</math>-tuple.
 
b) Prove that for every odd positive integer <math>m</math> there exists an integer <math>n\geq2</math> such that <math>m</math> belongs to an expensive <math>n</math>-tuple.
  
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[i]There are exactly <math>n</math> factors in the product on the left hand side.[/i]
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There are exactly <math>n</math> factors in the product on the left hand side.
  
 
[[2016 EGMO Problems/Problem 5|Solution]]
 
[[2016 EGMO Problems/Problem 5|Solution]]
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Let <math>ABC</math> be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid <math>G</math> and the circumcentre <math>O</math> of <math>ABC</math> in its sides <math>BC,CA,AB</math> are denoted by <math>G_1,G_2,G_3</math> and <math>O_1,O_2,O_3</math>, respectively. Show that the circumcircles of triangles <math>G_1G_2C</math>, <math>G_1G_3B</math>, <math>G_2G_3A</math>, <math>O_1O_2C</math>, <math>O_1O_3B</math>, <math>O_2O_3A</math> and <math>ABC</math> have a common point.
 
Let <math>ABC</math> be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid <math>G</math> and the circumcentre <math>O</math> of <math>ABC</math> in its sides <math>BC,CA,AB</math> are denoted by <math>G_1,G_2,G_3</math> and <math>O_1,O_2,O_3</math>, respectively. Show that the circumcircles of triangles <math>G_1G_2C</math>, <math>G_1G_3B</math>, <math>G_2G_3A</math>, <math>O_1O_2C</math>, <math>O_1O_3B</math>, <math>O_2O_3A</math> and <math>ABC</math> have a common point.
  
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[i]The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.[/i]
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The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.
  
 
[[2016 EGMO Problems/Problem 6|Solution]]
 
[[2016 EGMO Problems/Problem 6|Solution]]

Revision as of 12:52, 24 December 2022

Day 1

Problem 1

Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.

Solution

Problem 2

Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:

(i) For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$

(ii) There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$

In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.

Solution

Problem 3

There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?

Solution

Day 2

Problem 4

Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:

(i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$.

(ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.

Solution

Problem 5

Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is expensive if there exists a positive integer $k$ such that\[(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.\]a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.

b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.

There are exactly $n$ factors in the product on the left hand side.

Solution

Problem 6

Let $ABC$ be an acute-angled triangle in which no two sides have the same length. The reflections of the centroid $G$ and the circumcentre $O$ of $ABC$ in its sides $BC,CA,AB$ are denoted by $G_1,G_2,G_3$ and $O_1,O_2,O_3$, respectively. Show that the circumcircles of triangles $G_1G_2C$, $G_1G_3B$, $G_2G_3A$, $O_1O_2C$, $O_1O_3B$, $O_2O_3A$ and $ABC$ have a common point.

The centroid of a triangle is the intersection point of the three medians. A median is a line connecting a vertex of the triangle to the midpoint of the opposite side.

Solution

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