IMO 2007 (XLVIII International Mathematical Olympiad)

Inequality for real sequences with defined differences

Real numbers $a_1,a_2,\dots ,a_n$ are given. For each $i$ ($1\le i\le n$) define $d_i=\max \{a_j:1\le j\le i\}-\min \{a_j:1\le j\le i\}$ 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 \frac{d}{2}. \tag{*} (b) Show that there are real numbers $x_1\le x_2\le \cdots \le x_n$ such that equality holds in $(\ast )$.

Topic: Algebra, Disuguaglianze Metodo: Estremalità, Casework Abilita: Manipolazione algebrica, Lettura attenta, Stima Area: Algebra e Analisi Fonte: apri PDF p.1

Inequality for real sequences with defined differences

Real numbers $a_1,a_2,\dots ,a_n$ are given. For each $i$ ($1\le i\le n$) define $d_i=\max \{a_j:1\le j\le i\}-\min \{a_j:1\le j\le i\}$ 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 \frac{d}{2}. \tag{*} (b) Show that there are real numbers $x_1\le x_2\le \cdots \le x_n$ such that equality holds in $(\ast )$.

src_imho_2007__Q01

Line through vertex of parallelogram meets cyclic quadrilateral

Consider five points $A$, $B$, $C$, $D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Suppose that a line $\ell$ passing through $A$ 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 $DAB$.

Topic: Geometria piana Metodo: Trigonometria, Coordinate Abilita: Ragionamento geometrico, Manipolazione algebrica Area: Geometria Fonte: apri PDF p.1

Line through vertex of parallelogram meets cyclic quadrilateral

Consider five points $A$, $B$, $C$, $D$ and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Suppose that a line $\ell$ passing through $A$ 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 $DAB$.

src_imho_2007__Q02

Clique sizes in two-room partition of friendship graph

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.

Topic: Combinatoria Metodo: Grafi, Estremalità, Casework Abilita: Ragionamento geometrico, Modellizzazione, Astrazione, Casework accurato Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Click sizes in two-room partition of friendship graph

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each of them are friends. 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.

src_imho_2007__Q03

Angle bisector, circumcircle, perpendicular bisectors, equal areas

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.

Topic: Geometria piana Metodo: Trigonometria, Coordinate Abilita: Ragionamento geometrico, Manipolazione algebrica Area: Geometria Fonte: apri PDF p.1

Angle bisector, circumcircle, perpendicular bisectors, equal areas

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 $RPK$ and $RQL$ triangles have the same area.

src_imho_2007__Q04

Divisibility condition forces a equals b

Let $a$ and $b$ be positive integers. Show that if $4ab-1$ divides $(4a^2-1{)}^2$, then $a=b$.

Topic: Teoria dei Numeri Metodo: Congruenze, Fattorizzazione, Estremalità, Induzione Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Divisibility condition forces at equals b

Let $a$ and $b$ be positive integers. Show that if $4ab-1$ divides $(4a^2-1{)}^2$, then $a=b$.

src_imho_2007__Q05

Minimum planes covering lattice points with positive coordinate sum

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

Topic: Combinatoria, Geometria solida Metodo: Estremalità, Induzione, Doppio conteggio Abilita: Astrazione, Modellizzazione, Conteggio sistematico, Stima Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Minimum planes covering lattice points with positive coordinate sum

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

src_imho_2007__Q06