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Difference between revisions of "2017 JBMO Problems"
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Determine all the sets of six consecutive positive integers such that the product of some two of them, added to the product of some other two of them is equal to the product of the remaining two numbers.
 
Determine all the sets of six consecutive positive integers such that the product of some two of them, added to the product of some other two of them is equal to the product of the remaining two numbers.
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[[2017 JBMO Problems/Problem 1#Solution|Solution]]
  
 
==Problem 2==
 
==Problem 2==
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Let <math>x,y,z</math> be positive integers such that <math>x\neq y\neq z \neq x</math> .Prove that <cmath>(x+y+z)(xy+yz+zx-2)\geq 9xyz.</cmath>
 
Let <math>x,y,z</math> be positive integers such that <math>x\neq y\neq z \neq x</math> .Prove that <cmath>(x+y+z)(xy+yz+zx-2)\geq 9xyz.</cmath>
 
When does the equality hold?
 
When does the equality hold?
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[[2017 JBMO Problems/Problem 2#Solution|Solution]]
  
 
==Problem 3==
 
==Problem 3==
  
 
Let  <math>ABC </math> be an acute triangle such that <math>AB\neq AC</math> ,with circumcircle <math> \Gamma</math> and circumcenter <math>O</math>. Let <math>M</math> be the midpoint of <math>BC</math> and <math>D</math> be a point on <math> \Gamma</math> such that <math>AD \perp  BC</math>. let <math>T</math> be a point such that <math>BDCT</math> is a parallelogram  and <math>Q</math> a point on the same side of  <math>BC</math> as <math>A</math> such that <math>\angle{BQM}=\angle{BCA}</math> and  <math>\angle{CQM}=\angle{CBA}</math>. Let the line <math>AO</math> intersect <math> \Gamma</math> at <math>E</math> <math>(E\neq A)</math> and let the circumcircle of <math>\triangle ETQ</math>  intersect <math> \Gamma</math> at point <math>X\neq E</math>. Prove that the point <math>A,M</math> and <math>X</math> are collinear .
 
Let  <math>ABC </math> be an acute triangle such that <math>AB\neq AC</math> ,with circumcircle <math> \Gamma</math> and circumcenter <math>O</math>. Let <math>M</math> be the midpoint of <math>BC</math> and <math>D</math> be a point on <math> \Gamma</math> such that <math>AD \perp  BC</math>. let <math>T</math> be a point such that <math>BDCT</math> is a parallelogram  and <math>Q</math> a point on the same side of  <math>BC</math> as <math>A</math> such that <math>\angle{BQM}=\angle{BCA}</math> and  <math>\angle{CQM}=\angle{CBA}</math>. Let the line <math>AO</math> intersect <math> \Gamma</math> at <math>E</math> <math>(E\neq A)</math> and let the circumcircle of <math>\triangle ETQ</math>  intersect <math> \Gamma</math> at point <math>X\neq E</math>. Prove that the point <math>A,M</math> and <math>X</math> are collinear .
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 +
[[2017 JBMO Problems/Problem 3#Solution|Solution]]
  
 
==Problem 4==
 
==Problem 4==
  
 
Consider a regular 2n-gon <math> P</math>,<math>A_1,A_2,\cdots ,A_{2n}</math> in the plane ,where <math>n</math> is a positive integer . We say that a point <math>S</math> on one of the sides of  <math>P</math> can be seen from a point <math>E</math> that is external to <math>P</math> , if the line segment <math>SE</math> contains no other points that lie  on the sides of <math>P</math> except  <math>S</math> .We color the sides of <math>P</math> in 3 different colors (ignore the vertices of <math>P</math>,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover  ,from every point in the plane external to <math>P</math> , points of most 2  different colors on <math>P</math> can be seen .Find the number of distinct such colorings of <math>P</math> (two colorings are considered distinct if at least one of sides is colored differently).
 
Consider a regular 2n-gon <math> P</math>,<math>A_1,A_2,\cdots ,A_{2n}</math> in the plane ,where <math>n</math> is a positive integer . We say that a point <math>S</math> on one of the sides of  <math>P</math> can be seen from a point <math>E</math> that is external to <math>P</math> , if the line segment <math>SE</math> contains no other points that lie  on the sides of <math>P</math> except  <math>S</math> .We color the sides of <math>P</math> in 3 different colors (ignore the vertices of <math>P</math>,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover  ,from every point in the plane external to <math>P</math> , points of most 2  different colors on <math>P</math> can be seen .Find the number of distinct such colorings of <math>P</math> (two colorings are considered distinct if at least one of sides is colored differently).
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 +
[[2017 JBMO Problems/Problem 4#Solution|Solution]]
  
 
==See also==
 
==See also==
 
{{JBMO box|year=2017|before=[[2016 JBMO Problems]]|after=[[2018 JBMO Problems]]|five=}}
 
{{JBMO box|year=2017|before=[[2016 JBMO Problems]]|after=[[2018 JBMO Problems]]|five=}}

Latest revision as of 15:32, 16 September 2017

Problem 1

Determine all the sets of six consecutive positive integers such that the product of some two of them, added to the product of some other two of them is equal to the product of the remaining two numbers.

Solution

Problem 2

Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that \[(x+y+z)(xy+yz+zx-2)\geq 9xyz.\] When does the equality hold?

Solution

Problem 3

Let $ABC$ be an acute triangle such that $AB\neq AC$ ,with circumcircle $\Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $\Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $\Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $\Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear .

Solution

Problem 4

Consider a regular 2n-gon $P$,$A_1,A_2,\cdots ,A_{2n}$ in the plane ,where $n$ is a positive integer . We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$ .We color the sides of $P$ in 3 different colors (ignore the vertices of $P$,we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once . Moreover ,from every point in the plane external to $P$ , points of most 2 different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently).

Solution

See also

2017 JBMO (Problems • Resources)
Preceded by
2016 JBMO Problems 		Followed by
2018 JBMO Problems
1 • 2 • 3 • 4
All JBMO Problems and Solutions


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