IMO 1961 (III International Olympiad)

Sistema x+y+z=a, x^2+y^2+z^2=b^2, xy=z^2 con condizioni

Solve the system of equations: $x+y+z=a$ $x^2+y^2+z^2=b^2$ $xy=z^2$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ (the solutions of the system) are distinct positive numbers.

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

System x+y+z=a, x^2+y^2+z^2=b^2, xy=z^2 with conditions

Solve the system of equations: $x+y+z=a$ $x^2+y^2+z^2=b^2$ $xy=z^2$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x,y,z$ (the solutions of the system) are distinct positive numbers.

src_imho_1961__Q01

Provare a^2+b^2+c^2 >= 4 sqrt3 T (disuguaglianza di Weitzenbock)

Let $a,b,c$ be the sides of a triangle, and $T$ its area. Prove: $a^2+b^2+c^2\ge 4\sqrt{3}T.$ In what case does equality hold?

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

*Try a^2+b^2+c^2 >= 4 squared 3 T (Weitzenbock inequality) *

Let $a,b,c$ be the sides of a triangle, and $T$ its area. In what case does equality hold?

src_imho_1961__Q02

Risolvere cos^n x - sin^n x = 1

Solve the equation ${\cos }^{n}x-{\sin }^{n}x=1$, where $n$ is a natural number.

Topic: Trigonometria Metodo: Analisi per casi, Tecniche trigonometriche Abilita: Manipolazione algebrica Area: Geometria Fonte: apri PDF p.1

Resolving cos^n x - sin^n x = 1

Solve the equation ${\cos }^{n}x-{\sin }^{n}x=1$, where $n$ is a natural number.

src_imho_1961__Q03

almeno un rapporto <=2 e uno >=2

Consider triangle $P_1{P}_2{P}_3$ and a point $P$ within the triangle. Lines $P_{1}P$, $P_{2}P$, $P_{3}P$ intersect the opposite sides in points $Q_1$, $Q_2$, $Q_3$ respectively. Prove that, of the numbers $\frac{P_{1}P}{P{Q}_1},\frac{P_{2}P}{P{Q}_2},\frac{P_{3}P}{P{Q}_3}$ at least one is $\le 2$ and at least one is $\ge 2$.

Topic: Geometria piana Metodo: Principio di estremalita Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

at least one ratio <=2 and one >=2

Consider triangle $P_1{P}_2{P}_3$ and a point $P$ within the triangle. Lines $P_{1}P$, $P_{2}P$, $P_{3}P$ intersect the opposite sides in points $Q_1$, $Q_2$, $Q_3$ respectively. Prove that, of the numbers $\frac{P_{1}P}{P{Q}_1},\frac{P_{2}P}{P{Q}_2},\frac{P_{3}P}{P{Q}_3}$ at least one is $\le 2$ and at least one is $\ge 2$.

src_imho_1961__Q04

Costruire triangolo dati b, c e angolo AMB con condizione

Construct triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=\omega$, where $M$ is the midpoint of segment $BC$ and $\omega <90^{\circ }$. Prove that a solution exists if and only if $b\tan \frac{\omega }{2}\le c<b.$ In what case does equality hold?

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

Build data triangle b, c and AMB with condition

Construct triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=\omega$, where $M$ is the midpoint of segment $BC$ and $\omega <90^{\circ }$. Prove that a solution exists if and only if $b\tan \frac{\omega }{2}\le c<b.$ In what case does equality hold?

src_imho_1961__Q05

Luogo del baricentro G dei medi al variare di A’,B’,C’

Consider a plane $\epsilon$ and three non-collinear points $A$, $B$, $C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A^{\prime }$, $B^{\prime }$, $C^{\prime }$. Let $L$, $M$, $N$ be the midpoints of segments $A{A}^{\prime }$, $B{B}^{\prime }$, $C{C}^{\prime }$; let $G$ be the centroid of triangle $LMN$. (We will not consider positions of the points $A^{\prime }$, $B^{\prime }$, $C^{\prime }$ such that the points $L$, $M$, $N$ do not form a triangle.) What is the locus of point $G$ as $A^{\prime }$, $B^{\prime }$, $C^{\prime }$ range independently over the plane $\epsilon$?

Topic: Geometria solida, Geometria analitica Metodo: Metodo delle coordinate Abilita: Ragionamento geometrico Area: Geometria Fonte: apri PDF p.2

Place of the G-barycentre of the media at A’,B’,C’

Consider a plane $\epsilon$ and three non-collinear points $A$, $B$, $C$ on the same side of $\epsilon$; assume the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A^{\prime }$, $B^{\prime }$, $C^{\prime }$. Let $L$, $M$, $N$ be the midpoints of segments $A{A}^{\prime }$, $B{B}^{\prime }$, $C{C}^{\prime }$; let $G$ be the centroid of triangle $LMN$. (We will not consider positions of the points $A^{\prime }$, $B^{\prime }$, $C^{\prime }$ such that the points $L$, $M$, $N$ do not form a triangle.)

src_imho_1961__Q06