IMO 1962 (IV International Olympiad)

Minimo n che termina con 6 e diventa quadruplo spostando il 6

Trovare il più piccolo numero naturale $n$ con le seguenti proprietà: (a) la sua rappresentazione decimale ha 6 come ultima cifra; (b) se l’ultima cifra 6 viene cancellata e messa davanti alle cifre rimanenti, il numero risultante è quattro volte il numero originale $n$.

Topic: Teoria dei Numeri Metodo: congruenze Abilita: Manipolazione algebrica Area: Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Minimum n that ends with 6 and becomes quadrupled by moving 6

Find the smallest natural number $n$ with the following properties: (a) its decimal representation has 6 as the last digit; (b) if the last digit 6 is deleted and placed before the remaining digits, the resulting number is four times the original number $n$.

src_imho_1962__Q01

Reali x con sqrt(3-x)-sqrt(x+1) > 1/2

1962/3. Consider the cube ABCDA′B′C′D′ (ABCD and A′B′C′D′ are the upper and lower bases, respectively, and edges AA′, BB′, CC′, DD′ are parallel). The point X moves at constant speed along the perimeter of the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B′C′CB in the direction B′C′CBB′. Points X and Y begin their motion at the same instant from the starting positions A and B′, respectively. Determine and draw the locus of the midpoints of the segments XY. 1962/4. Solve the equation cos2 x + cos2 2x + cos2 3x = 1. 1962/5. On the circle K there are given three distinct points A, B, C. Construct (using only straightedge and compasses) a fourth point D on K such that a circle can be inscribed in the quadrilateral thus obtained. 1962/6. Consider an isosceles triangle. Let r be the radius of its circumscribed circle and ρ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is d = q r(r −2ρ). 1962/7.

The tetrahedron SABC has the following property: there exist five spheres, each tangent to the edges SA, SB, SC, BCCA, AB, or to their extensions. (a) Prove that the tetrahedron SABC is regular. (b) Prove conversely that for every regular tetrahedron five such spheres exist.

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

*Real x with square x-x-x-sqrt x+1) > 1/2 *

1962/3. Consider the ABCDA′B′C′D′ (ABCD and A′B′C′D′ are the upper and lower bases, respectively, and edges AA′, BB′, CC′, DD′ are parallel). The point X moves at constant speed along the perimeter of the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B′C′CB in the direction B′C′CBB′. Points X and Y begin their motion at the same instant from the starting positions A and B′, respectively. Determine and draw the locus of the midpoints of the XY segments. 1962/4. Solve the equation cos2 x + cos2 2x + cos2 3x = 1. 1962/5. On the circle K there are given three distinct points A, B, C. Construct (using only straightedge and compasses) a fourth point D on K such that a circle can be inscribed in the quadrilateral thus obtained. 1962/6. Consider an isosceles triangle. Let r be the radius of its circumscribed circle and ρ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is d = q r(r −2ρ). 1962/7.

The tetrahedron SABC has the following property: there exist five spheres, each tangent to the edges SA, SB, SC, BCCA, AB, or to their extensions. Prove that the tetrahedron SABC is regular. (b) Prove conversely that for every regular tetrahedron five such spheres exist.

src_imho_1962__Q02

Luogo dei medi di XY in moto sul cubo

Si consideri il cubo $ABCD{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ (la faccia $ABCD$ è la base inferiore, la faccia $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ è quella superiore, $A{A}^{\prime },B{B}^{\prime },C{C}^{\prime },D{D}^{\prime }$ sono gli spigoli laterali). Il punto $X$ si muove a velocità costante lungo il perimetro del quadrato $ABCD$, e il punto $Y$ si muove alla stessa velocità lungo il perimetro del quadrato $B^{\prime }{C}^{\prime }CB$. Inizialmente $X$ è in $A$ e $Y$ è in $B^{\prime }$. Determinare e tracciare il luogo geometrico del punto medio di $XY$.

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

Place of moving XY media on the cube

Consider the $ABCD{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ cube (the face $ABCD$ is the bottom base, the face $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ is the top, $A{A}^{\prime },B{B}^{\prime },C{C}^{\prime },D{D}^{\prime }$ are the side tips). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$, and the point $Y$ moves at the same speed along the perimeter of the square $B^{\prime }{C}^{\prime }CB$. Initially $X$ is in $A$ and $Y$ is in $B^{\prime }$. Determine and map the geometric location of the midpoint of $XY$.

src_imho_1962__Q03

Risolvere cos^2 x + cos^2 2x + cos^2 3x = 1

Risolvere l’equazione ${\cos }^{2}x+{\cos }^{2}2x+{\cos }^{2}3x=1$.

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

Solve cos^2 x + cos^2 2x + cos^2 3x = 1

Solve the equation ${\cos }^{2}x+{\cos }^{2}2x+{\cos }^{2}3x=1$.

src_imho_1962__Q04

Costruire D sul cerchio per quadrilatero circoscrittibile

Sul cerchio $K$ sono dati tre punti distinti $A,B,C$. Costruire (con riga e compasso) un punto $D$ su $K$ tale che nel quadrilatero $ABCD$ così formato si possa inscrivere un cerchio.

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

Building D on the circle by circumcirculating quadrilateral

Three distinct points $A,B,C$ are given on the $K$ circle. Build (with line and compass) a point $D$ on $K$ such that a circle can be inscribed in the $ABCD$ quadrilateral thus formed.

src_imho_1962__Q05

Provare d=sqrt(r(r-2rho)) (formula di Eulero)

Si consideri un triangolo isoscele. Sia $r$ il raggio del suo cerchio circoscritto e $\rho$ il raggio del suo cerchio inscritto. Dimostrare che la distanza $d$ tra i centri di questi due cerchi è $d=\sqrt{r(r-2\rho )}$.

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

*Try d=sqrt(r(r-2rho)) (Euler formula) *

Consider yourself an isosceles triangle. either $r$ the radius of its circumscribed circle and $\rho$ the radius of its inscribed circle. Show that the $d$ distance between the centers of these two circles is $d=\sqrt{r(r-2\rho )}$.

src_imho_1962__Q06

Tetraedro con cinque sfere tangenti agli spigoli e regolarita

Il tetraedro $SABC$ ha la seguente proprietà: esistono cinque sfere, ciascuna tangente agli spigoli $SA,SB,SC,BC,CA,AB$ o alle loro estensioni. (a) Dimostrare che il tetraedro $SABC$ è isoscele, cioè $SA=BC$, $SB=CA$, $SC=AB$. (b) Dimostrare che le cinque sfere sono congruenti.

Topic: Geometria solida Abilita: Ragionamento geometrico, generalizzazione Area: Geometria Fonte: apri PDF p.2

Tetrahedron with five spheres tangent to the spines and regulated

The tetrahedron $SABC$ has the following property: there are five spheres, each tangent to the $SA,SB,SC,BC,CA,AB$ beams or their extensions. (a) Demonstrate that the tetrahedron $SABC$ is isosceles, i.e. $SA=BC$, $SB=CA$, $SC=AB$. (b) Demonstrate that the five spheres are congruent.

src_imho_1962__Q07