IMO 1965 (Seventh International Olympiad)

Solve trig inequality with abs value on [0,2pi]

Determinare tutti i valori di $x$ nell’intervallo $0\le x\le 2\pi$ che soddisfano la disuguaglianza $2\cos x\le \left|\sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}\right|\le \sqrt{2}.$

Topic: Trigonometria, Disuguaglianze Metodo: Disuguaglianze classiche, Tecniche trigonometriche Abilita: Manipolazione algebrica Area: Algebra e Analisi, Geometria Fonte: apri PDF p.1

Solves trig inequality with abs value on [0,2pi]

Determine all values of $x$ in the $0\le x\le 2\pi$ range that satisfy the inequality $2\cos x\le \left|\sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}\right|\le \sqrt{2}.$

src_imo_1965_all__Q01

Prove diagonally-dominant 3x3 system has only zero solution

1965/2. Consider the system of equations a11x1 + a12x2 + a13x3

0 a21x1 + a22x2 + a23x3

0 a31x1 + a32x2 + a33x3

0 with unknowns x1, x2, x3. The coefficients satisfy the conditions: (a) a11, a22, a33 are positive numbers; (b) the remaining coefficients are negative numbers; (c) in each equation, the sum of the coefficients is positive. Prove that the given system has only the solution x1 = x2 = x3 = 0. 1965/3. Given the tetrahedron ABCD whose edges AB and CD have lengths a and b respectively. The distance between the skew lines AB and CD is d, and the angle between them is ω. Tetrahedron ABCD is divided into two solids by plane ε, parallel to lines AB and CD. The ratio of the distances of ε from AB and CD is equal to k. Compute the ratio of the volumes of the two solids obtained. 1965/4. Find all sets of four real numbers x1, x2, x3, x4 such that the sum of any one and the product of the other three is equal to 2. 1965/5. Consider ∆OAB with acute angle AOB. Through a point M ̸= O perpendiculars are drawn to OA and OB, the feet of which are P and Q respectively. The point of intersection of the altitudes of ∆OPQ is H. What is the locus of H if M is permitted to range over (a) the side AB, (b) the interior of ∆OAB?

1965/6. In a plane a set of n points (n ≥3) is given. Each pair of points is connected by a segment. Let d be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length d. Prove that the number of diameters of the given set is at most n.

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

Prove diagonally-dominant 3x3 system has only zero solution

1965/2. Consider the system of equations a11x1 + a12x2 + a13x3 = 0 a21x1 + a22x2 + a23x3 = 0 a31x1 + a32x2 + a33x3 = 0 with unknowns x1, x2, x3. The coefficients satisfy the conditions: (a) a11, a22, a33 are positive numbers; (b) the remaining coefficients are negative numbers; (c) in each equation, the sum of the coefficients is positive. Prove that the given system has only the solution x1 = x2 = x3 = 0. 1965/3. Given the tetrahedron ABCD whose edges AB and CD have lengths a and b respectively. The distance between the skew lines AB and CD is d, and the angle between them is ω. Tetrahedron ABCD is divided into two solids by plane ε, parallel to lines AB and CD. The ratio of the distances of ε from AB and CD is equal to k. Calculate the ratio of the volumes of the two solids obtained. 1965/4. Find all sets of four real numbers x1, x2, x3, x4 such that the sum of any one and the product of the other three is equal to 2. 1965/5. Consider ∆OAB with acute angle AOB. Through a point M = O perpendiculars are drawn to OA and OB, the feet of which are P and Q respectively. The point of intersection of the altitudes of ∆OPQ is H. What is the locus of H if M is allowed to range over (a) the side AB, (b) the interior of ∆OAB?

1965/6. In a plane a set of n points (n ≥3) is given. Each pair of points is connected by a segment. Let d be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length d. Prove that the number of diameters of the given set is at most n.

src_imo_1965_all__Q02

Volume ratio of tetrahedron cut by plane parallel to skew edges

Dato il tetraedro $ABCD$ i cui spigoli $AB$ e $CD$ hanno lunghezze rispettivamente $a$ e $b$. La distanza tra le rette sghembe $AB$ e $CD$ è $d$, e l’angolo tra esse è $\omega$. Il tetraedro $ABCD$ è diviso in due solidi dal piano $\epsilon$, parallelo alle rette $AB$ e $CD$. Il rapporto delle distanze di $\epsilon$ da $AB$ e da $CD$ è uguale a $k$. Calcolare il rapporto dei volumi dei due solidi ottenuti.

Topic: Geometria solida Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Volume ratio of tetrahedron cut by plane parallel to skew edges

Given the tetrahedron $ABCD$ whose beams $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the straight lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. The $ABCD$ tetrahedron is divided into two solids by the $\epsilon$ plane, parallel to the $AB$ and $CD$ lines. The ratio of $\epsilon$ to $AB$ and $CD$ is equal to $k$. Calculate the ratio of the volumes of the two solids obtained.

src_imo_1965_all__Q03

Find four reals where each plus product of other three is 2

Trovare tutti gli insiemi di quattro numeri reali $x_1,x_2,x_3,x_4$ tali che la somma di ciascuno di essi e il prodotto degli altri tre sia uguale a $2$.

Topic: Algebra Metodo: Sfruttamento della simmetria Area: Algebra e Analisi Fonte: apri PDF p.1

Find four real where each plus product of other three is 2

Find all sets of four real numbers $x_1,x_2,x_3,x_4$ such that the sum of each of them and the product of the other three is $2$.

src_imo_1965_all__Q04

Locus of orthocenter H of pedal triangle OPQ

Considerare il triangolo $OAB$ con angolo acuto $AOB$. Per un punto $M\ne O$ si tracciano le perpendicolari a $OA$ e a $OB$, i cui piedi sono rispettivamente $P$ e $Q$. Il punto di intersezione delle altezze del triangolo $OPQ$ è $H$. Qual è il luogo geometrico di $H$ se $M$ varia su: (a) il lato $AB$; (b) l’interno del triangolo $OAB$?

Topic: Geometria piana Metodo: Metodo delle coordinate Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Locus of orthocenter H of pedal triangle OPQ

Consider the $OAB$ triangle with a sharp angle $AOB$. For a point $M\ne O$ the perpendiculars to $OA$ and to $OB$ shall be drawn, the feet of which are $P$ and $Q$ respectively. The height intersection point of the $OPQ$ triangle is $H$. What is the geometric location of $H$ if $M$ varies on: (a) the side $AB$; (b) the inside of the $OAB$ triangle?

src_imo_1965_all__Q05

Prove number of diameters of n-point set is at most n

In un piano è dato un insieme di $n$ punti ($n\ge 3$). Ogni coppia di punti è collegata da un segmento. Sia $d$ la lunghezza del più lungo di questi segmenti. Si definisce diametro dell’insieme qualsiasi segmento di collegamento di lunghezza $d$. Dimostrare che il numero dei diametri dell’insieme dato è al più $n$.

Topic: Combinatoria, Geometria piana Metodo: Principio di estremalita Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.2

Prove number of diameters of n-point set is at most n

A set of $n$ points ($n\ge 3$) is given in a plane. Each pair of points is connected by a segment. Whether $d$ the length of the longest of these segments. The diameter of the assembly of any length $d$ link segment is defined. Demonstrate that the number of diameters of the given set is at most $n$.

src_imo_1965_all__Q06