IMO 1995 (XXXVI International Mathematical Olympiad)

Four collinear points, circle intersections, concurrency proof

Let $A$, $B$, $C$, $D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that $AM$, $DN$, $XY$ are concurrent.

Topic: Geometria piana Metodo: Coordinate Abilita: Ragionamento geometrico, Lettura attenta Area: Geometria Fonte: apri PDF p.1

Four collinear points, circle intersections, concurrency proof

Let $A$, $B$, $C$, $D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that $AM$, $DN$, $XY$ are concurrent.

src_imho_1995__Q01

Inequality for positive reals with abc=1

Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge \frac{3}{2}.$

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

Inequality for positive reals with abc=1*

Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)}\ge \frac{3}{2}.$

src_imho_1995__Q02

Determine all n>3 for which n points allow equal triangle perimeters

Determine all integers $n>3$ for which there exist $n$ points $A_1,\dots ,A_n$ in the plane, no three collinear, and real numbers $r_1,\dots ,r_n$ such that for $1\le i<j<k\le n$, the area of $△A_i{A}_j{A}_k$ is $r_i+r_j+r_k$.

Topic: Geometria piana, Combinatoria Metodo: Casework, Estremalità Abilita: Ragionamento geometrico, Astrazione, Lettura attenta Area: Geometria, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Determine all n>3 for which n points allow equal triangle perimeters

Determine all integers $n>3$ for which there exist $n$ points $A_1,\dots ,A_n$ in the plane, no three collinear, and real numbers $r_1,\dots ,r_n$ such that for $1\le i<j<k\le n$, the area of $△A_i{A}_j{A}_k$ is $r_i+r_j+r_k$.

src_imho_1995__Q03

Maximum value of x_0 in a sequence satisfying a recurrence

Find the maximum value of $x_0$ for which there exists a sequence $x_0,x_1,\dots ,x_{1995}$ of positive reals with $x_0=x_{1995}$, such that for $i=1,\dots ,1995$, $x_{i-1}+\frac{2}{x_{i-1}}=2x_i+\frac{1}{x_i}.$

Topic: Algebra Metodo: Ricorsione, Estremalità Abilita: Manipolazione algebrica, Riconoscimento di pattern, Lettura attenta Area: Algebra e Analisi Fonte: apri PDF p.1

Maximum value of x_0 in a sequence satisfying a recurrence

Find the maximum value of $x_0$ for which there exists a sequence $x_0,x_1,\dots ,x_{1995}$ of positive reals with $x_0=x_{1995}$, such that for $i=1,\dots ,1995$, $x_{i-1}+\frac{2}{x_{i-1}}=2x_i+\frac{1}{x_i}.$

src_imho_1995__Q04

Convex hexagon ABCDEF with equal sides, angle and area condition

Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$ and $DE=EF=FA$, such that $\angle BCD=\angle EFA=\pi /3$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB=\angle DHE=2\pi /3$. Prove that $AG+GB+GH+DH+HE\ge CF$.

Topic: Geometria piana, Disuguaglianze Metodo: Trigonometria, Disuguaglianze Abilita: Ragionamento geometrico, Manipolazione algebrica, Lettura attenta Area: Geometria, Algebra e Analisi Fonte: apri PDF p.1

Convex hexagon ABCDEF with equal sides, angle and area condition

Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$ and $DE=EF=FA$, such that $\angle BCD=\angle EFA=\pi /3$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB=\angle DHE=2\pi /3$. Prove that $AG+GB+GH+DH+HE\ge CF$.

src_imho_1995__Q05

p-element subsets of {1,…,2p} with sum divisible by p

Let $p$ be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\dots ,2p\}$ are there, the sum of whose elements is divisible by $p$?

Topic: Combinatoria, Teoria dei Numeri Metodo: Congruenze, Conteggio Abilita: Conteggio sistematico, Manipolazione algebrica, Lettura attenta Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

p-element subsets of {1,…,2p} with sum divisible by p

Let$p$be an odd prime number. How many $p$-element subsets $A$ of $\{1,2,\dots ,2p\}$ are there, the sum of whose elements is divisible by $p$?

src_imho_1995__Q06