IMO 1995 (36th IMO, Toronto)

Prove AM, DN, XY concurrent in two-diameter-circles config

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 . The line XY meets BC at Z. 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 the lines AM, DN, XY are concurrent.

Topic: Geometria piana Metodo: Sfruttamento della simmetria Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Prove AM, DN, XY concurrent in two-diameter-circles config

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 . The line XY meets BC at Z. 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 the lines AM, DN, XY are concurrent.

src_imo_1995__Q01

Prove sum of 1/(a^3(b+c)) at least 3/2 for abc=1

Let a, b, c be positive real numbers such that abc = 1. Prove that 1 a3(b + c) + 1 b3(c + a) + 1 c3(a + b) ≥3 2.

Topic: Disuguaglianze Metodo: Disuguaglianze classiche Area: Algebra e Analisi Fonte: apri PDF p.1

Prove sum of 1/(a^3(b+c)) at least 3/2 for abc=1

Let a, b, c be positive real numbers such that abc = 1. Prove that 1 a3(b + c) + 1 b3(c + a) + 1 c3(a + b) ≥3 2.

src_imo_1995__Q02

Determine n with point areas equal to ri+rj+rk

Determine all integers n > 3 for which there exist n points A1, … , An in the plane, no three collinear, and real numbers r1, … , rn such that for 1 ≤i < j < k ≤n, the area of △AiAjAk is ri + rj + rk. 36th International Mathematical Olympiad Second Day - Toronto - July 20, 1995 Time Limit: 41 2 hours

Topic: Geometria piana, Combinatoria Metodo: Metodo delle coordinate Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Determinate n with point areas equal to ri+rj+rk

Determine all integers n > 3 for which there exist n points A1, … , An in the plane, no three collinear, and real numbers r1, … , rn such that for 1 ≤i < j < k ≤n, the area of △AiAjAk is ri + rj + rk. 36th International Mathematical Olympiad Second Day - Toronto - July 20, 1995 Time limit: 41 2 hours

src_imo_1995__Q03

Maximum x0 for cyclic positive-real recurrence sequence

Trova il valore massimo di $x_0$ per cui esiste una sequenza $x_0,x_1,\dots ,x_{1995}$ di numeri reali positivi con $x_0=x_{1995}$, tale che per $i=1,\dots ,1995$: $x_{i-1}+\frac{2}{x_{i-1}}=2x_i+\frac{1}{x_i}.$

Topic: successioni Metodo: Disuguaglianze classiche, Ricorsione Area: Algebra e Analisi Fonte: apri PDF p.1

Maximum x0 for cyclic positive-real recurrence sequence

Find the maximum value of $x_0$ for which there exists a sequence $x_0,x_1,\dots ,x_{1995}$ of positive real numbers 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_imo_1995__Q04

Hexagon inequality AG+GB+GH+DH+HE>=CF

Sia $ABCDEF$ un esagono convesso con $AB=BC=CD$ e $DE=EF=FA$, tale che $\angle BCD=\angle EFA=\pi /3$. Supponiamo che $G$ e $H$ siano punti nell’interno dell’esagono tali che $\angle AGB=\angle DHE=2\pi /3$. Dimostrare che $AG+GB+GH+DH+HE\ge CF.$

Topic: Geometria piana, Disuguaglianze Metodo: Disuguaglianze classiche Abilita: Ragionamento geometrico Area: Algebra e Analisi, Geometria Fonte: apri PDF p.1

In the case of the equation, the following equation is used: Hexagon inequality AG+GB+GH+DH+HE>=CF

Whether $ABCDEF$ is a convex hexagon with $AB=BC=CD$ and $DE=EF=FA$, such as $\angle BCD=\angle EFA=\pi /3$. Suppose $G$ and $H$ are points within the hexagon such as $\angle AGB=\angle DHE=2\pi /3$. Show that $AG+GB+GH+DH+HE\ge CF.$

src_imo_1995__Q05

Count p-element subsets of 1..2p with p-divisible sum

Sia $p$ un numero primo dispari. Quanti sottoinsiemi $A$ di $p$ elementi di $\{1,2,\dots ,2p\}$ esistono tali che la somma degli elementi di $A$ sia divisibile per $p$?

Topic: Combinatoria, Teoria dei Numeri Metodo: congruenze, Conteggio combinatorio Area: Aritmetica e Teoria dei Numeri, Combinatoria, Logica e Probabilita Fonte: apri PDF p.1

Count p-element subsets of 1..2p with p-divisible sum

Whether $p$ is an odd prime number. How many $A$ subsets of $p$ elements of $\{1,2,\dots ,2p\}$ exist that the sum of the elements of $A$ is divisible by $p$?

src_imo_1995__Q06