IMO 1965 (VII International Mathematical Olympiad)

Find all x in [0,2π] satisfying a trigonometric inequality

Determine all values $x$ in the interval $0\le x\le 2\pi$ which satisfy the inequality $2\cos x\le \sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}\le \sqrt{2}.$

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

Find the x in [0,2π] satisfying a trigonometric inequality

Determine the values $x$ in the interval $0\le x\le 2\pi$ which satisfy the inequality $2\cos x\le \sqrt{1+\sin 2x}-\sqrt{1-\sin 2x}\le \sqrt{2}.$

src_imho_1965__Q01

Prove a homogeneous linear system has only the trivial solution

Consider the system of equations $a_{11}{x}_1+a_{12}{x}_2+a_{13}{x}_3=0$ $a_{21}{x}_1+a_{22}{x}_2+a_{23}{x}_3=0$ $a_{31}{x}_1+a_{32}{x}_2+a_{33}{x}_3=0$ with unknowns $x_1,x_2,x_3$. The coefficients satisfy the conditions: (a) $a_{11},a_{22},a_{33}$ 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 $x_1=x_2=x_3=0$.

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

Prove a homogeneous linear system has only the trivial solution

Consider the system of equations $a_{11}{x}_1+a_{12}{x}_2+a_{13}{x}_3=0$ $a_{21}{x}_1+a_{22}{x}_2+a_{23}{x}_3=0$ $a_{31}{x}_1+a_{32}{x}_2+a_{33}{x}_3=0$ with unknowns $x_1,x_2,x_3$. The coefficients satisfy the conditions: (a) $a_{11},a_{22},a_{33}$ 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 $x_1=x_2=x_3=0$.

src_imho_1965__Q02

Tetrahedron cut by plane: ratio of distances equals k

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 $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

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

Tetrahedron cut by plane: ratio of distances equals k

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 $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Calculate the ratio of the volumes of the two solids obtained.

src_imho_1965__Q03

Find all sets of four real numbers with pairwise-product sums equal to 2

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

Topic: Algebra Metodo: Simmetria, Casework Abilita: Manipolazione algebrica, Riconoscimento di pattern, Casework accurato Area: Algebra e Analisi Fonte: apri PDF p.1

Find all sets of four real numbers with pairwise-product sums equal to 2

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

src_imho_1965__Q04

Locus of orthocenter foot as point ranges over triangle region

Consider $△OAB$ with acute angle $AOB$. Through a point $M\ne O$ 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 $△OAB$?

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

Locus of orthocenter foot as point ranges over triangle region

Consider $△OAB$ with acute angle $AOB$. Through a point $M\ne O$ 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 $△OAB$?

src_imho_1965__Q05

Bound the number of diameters of a finite planar point set

In a plane a set of $n$ points ($n\ge 3$) 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: Combinatoria, Geometria piana Metodo: Estremalità, Doppio conteggio Abilita: Ragionamento geometrico, Astrazione, Modellizzazione Area: Combinatoria, Logica e Probabilita, Geometria Fonte: apri PDF p.1

Bound the number of diameters of a finite planar point set

In a plane a set of $n$ points ($n\ge 3$) 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_imho_1965__Q06