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Difference between revisions of "2007 IMO Problems"

m (Created page with '<Strong>Problem 1</Strong> <hr> Real numbers <math>a_1, a_2, \dots , a_n</math> are given. For each <math>i</math> (<math>1\le i\le n</math>) define <cmath>d_i=\max\{a_j:1\le …')
 
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<Strong>Problem 1</Strong>
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==Problem 1==
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<hr>
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Real numbers <math>a_1, a_2, \dots , a_n</math> are given. For each <math>i</math> (<math>1\le i\le n</math>) define
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<cmath>d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}</cmath>
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and let
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<cmath>d=\max\{d_i:1\le i\le n\}.</cmath>
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(a) Prove that, for any real numbers <math>x_1\le x_2\le \cdots\le x_n</math>,
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<cmath>\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2}  \qquad (*)</cmath>
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 +
(b) Show that there are real numbers <math>x_1\le x_2\le x_n</math> such that equality holds in (*)
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 +
 
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[[2007 IMO Problems/Problem 1 | Solution]]
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==Problem 2==
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Consider five points <math>A,B,C,D</math>, and <math>E</math> such that <math>ABCD</math> is a parallelogram and <math>BCED</math> is a cyclic quadrilateral.
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Let <math>\ell</math> be a line passing through <math>A</math>. Suppose that <math>\ell</math> intersects the interior of the segment <math>DC</math> at <math>F</math> and intersects
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line <math>BC</math> at <math>G</math>. Suppose also that <math>EF=EG=EC</math>. Prove that <math>\ell</math> is the bisector of <math>\angle DAB</math>.
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[[2007 IMO Problems/Problem 2 | Solution]]
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==Problem 3==
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In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.
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[[2007 IMO Problems/Problem 3 | Solution]]
  
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<hr>
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==Problem 4==
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In <math>\triangle ABC</math> the bisector of <math>\angle{BCA}</math> intersects the circumcircle again at <math>R</math>, the perpendicular bisector of <math>BC</math> at <math>P</math>, and the perpendicular bisector of <math>AC</math> at <math>Q</math>. The midpoint of <math>BC</math> is <math>K</math> and the midpoint of <math>AC</math> is <math>L</math>. Prove that the triangles <math>RPK</math> and <math>RQL</math> have the same area.
  
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Real numbers <math>a_1, a_2, \dots , a_n</math> are given. For each <math>i</math> (<math>1\le i\le n</math>) define
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[[2007 IMO Problems/Problem 4 | Solution]]
  
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<cmath>d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}</cmath>
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==Problem 5==
  
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and let
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(''Kevin Buzzard and Edward Crane, United Kingdom'')
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Let <math>a</math> and <math>b</math> be positive integers.  Show that if <math>4ab-1</math> divides <math>(4a^2-1)^2</math>, then <math>a=b</math>.
  
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<cmath>d=\max\{d_i:1\le i\le n\}</cmath>.
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[[2007 IMO Problems/Problem 5 | Solution]]
  
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(a) Prove that, for any real numbers <math>x_1\le x_2\le \cdots\le x_n</math>,
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==Problem 6==
  
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<cmath>\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2}  (*)</cmath>  
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Let <math>n</math> be a positive integer. Consider
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<cmath>S=\{(x,y,z)~:~x,y,z\in \{0,1,\ldots,n \},~x+y+z>0\}</cmath>
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as a set of <math>(n+1)^3-1</math> points in three-dimensional space.
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Determine the smallest possible number of planes, the union of which contain <math>S</math> but does not include <math>(0,0,0)</math>.
  
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(b) Show that there are real numbers <math>x_1\le x_2\le x_n</math> such that equality holds in (*)
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[[2007 IMO Problems/Problem 6 | Solution]]

Revision as of 23:27, 8 October 2014

Problem 1

Real numbers $a_1, a_2, \dots , a_n$ are given. For each $i$ ($1\le i\le n$) define \[d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}\] and let \[d=\max\{d_i:1\le i\le n\}.\]

(a) Prove that, for any real numbers $x_1\le x_2\le \cdots\le x_n$, \[\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2} \qquad (*)\]

(b) Show that there are real numbers $x_1\le x_2\le x_n$ such that equality holds in (*)


Solution

Problem 2

Consider five points $A,B,C,D$, and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$. Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$. Suppose also that $EF=EG=EC$. Prove that $\ell$ is the bisector of $\angle DAB$.

Solution

Problem 3

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.

Solution

Problem 4

In $\triangle ABC$ the bisector of $\angle{BCA}$ intersects the circumcircle again at $R$, the perpendicular bisector of $BC$ at $P$, and the perpendicular bisector of $AC$ at $Q$. The midpoint of $BC$ is $K$ and the midpoint of $AC$ is $L$. Prove that the triangles $RPK$ and $RQL$ have the same area.

Solution

Problem 5

(Kevin Buzzard and Edward Crane, United Kingdom) Let $a$ and $b$ be positive integers. Show that if $4ab-1$ divides $(4a^2-1)^2$, then $a=b$.

Solution

Problem 6

Let $n$ be a positive integer. Consider \[S=\{(x,y,z)~:~x,y,z\in \{0,1,\ldots,n \},~x+y+z>0\}\] as a set of $(n+1)^3-1$ points in three-dimensional space. Determine the smallest possible number of planes, the union of which contain $S$ but does not include $(0,0,0)$.

Solution

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