IMO 1963 (V International Olympiad)

Radici reali di sqrt(x^2-p)+2 sqrt(x^2-1)=x

Find all real roots of the equation $\sqrt{x^2-p+2\sqrt{x^2-1}}=x,$ dove $p$ è un parametro reale.

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

*Real roots of square x^2-p) + 2 square x^2-1) = x *

Find all the real roots of the equation $\sqrt{x^2-p+2\sqrt{x^2-1}}=x,$ where $p$ is a real parameter.

src_imho_1963__Q01

Luogo di vertici di angoli retti con un lato per A

1963/6. Five students, A, B, C, D, E, took part in a contest. One prediction was that the contestants would finish in the order ABCDE. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order DAECB. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

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

Place of vertices of right angles with one side for A

1963/6. Five students, A, B, C, D, E, took part in a contest. One prediction was that the contestants would finish in the ABCDE order. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the DAECB order. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

src_imho_1963__Q02

provare equilatero

In an $n$-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation $a_1\ge a_2\ge \cdots \ge a_n.$ Prove that $a_1=a_2=\cdots =a_n$.

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

try to equilibrate

In an $n$-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation $a_1\ge a_2\ge \cdots \ge a_n.$ Prove that $a_1=a_2=\cdots =a_n$.

src_imho_1963__Q03

Sistema ciclico di 5 equazioni con parametro y

Find all solutions $x_1,x_2,x_3,x_4,x_5$ of the system $\{\begin{array}{l}{x}_5+x_2=y{x}_1\\ x_1+x_3=y{x}_2\\ x_2+x_4=y{x}_3\\ x_3+x_5=y{x}_4\\ x_4+x_1=y{x}_5,\end{array}$ where $y$ is a parameter.

Topic: Algebra, successioni Metodo: Sfruttamento della simmetria Abilita: Manipolazione algebrica Area: Algebra e Analisi Fonte: apri PDF p.1

Cyclic system of 5 equations with parameter y

Find all solutions $x_1,x_2,x_3,x_4,x_5$ of the system $\{\begin{array}{l}{x}_5+x_2=y{x}_1\\ x_1+x_3=y{x}_2\\ x_2+x_4=y{x}_3\\ x_3+x_5=y{x}_4\\ x_4+x_1=y{x}_5,\end{array}$ where $y$ is a parameter.

src_imho_1963__Q04

Provare cos(pi/7)-cos(2pi/7)+cos(3pi/7)=1/2

Prove that $\cos \frac{\pi }{7}-\cos \frac{2\pi }{7}+\cos \frac{3\pi }{7}=\frac{1}{2}$.

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

This is the total number of samples taken from the sample.

Prove that $\cos \frac{\pi }{7}-\cos \frac{2\pi }{7}+\cos \frac{3\pi }{7}=\frac{1}{2}$.

src_imho_1963__Q05

Determinare l’ordine di arrivo dei cinque studenti

Five students, $A$, $B$, $C$, $D$, $E$, took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

Topic: Logica, giochi, strategie, Combinatoria Metodo: Analisi per casi Abilita: Casework accurato Area: Combinatoria, Logica e Probabilita Fonte: apri PDF p.2

Determining the order of arrival of the five students

Five students, $A$, $B$, $C$, $D$, $E$, took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

src_imho_1963__Q06