IMO 1966 (VIII International Mathematical Olympiad)

Count students who solved only problem B

In a mathematical contest, three problems, $A$ , $B$ , $C$ were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$ , the number who solved $B$ was twice the number who solved $C$ . The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$ . How many students solved only problem $B$ ?

Topic: Combinatoria, Teoria dei Numeri Metodo: Casework, Inclusione-esclusione Abilita: Lettura attenta, Modellizzazione, Manipolazione algebrica Area: Combinatoria, Logica e Probabilita, Aritmetica e Teoria dei Numeri Fonte: apri PDF p.1

Count students who solved only problem B

In a mathematical contest, three problems, $A$ , $B$ , $C$ were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$ , the number who solved $B$ was twice the number who solved $C$ . The number of students who solved only problem was one more than the number of students who solved problem and at least one other problem. Of all the students who solved just one problem, half didn’t solve problem. How many students solved only problem?

src_imho_1966__Q01

Prove triangle is isosceles given a trigonometric side condition

Let $a$ , $b$ , $c$ be the lengths of the sides of a triangle, and $α$ , $β$ , $γ$ , respectively, the angles opposite these sides. Prove that if $a + b = tan \frac{γ}{2} (a tan α + b tan β),$ the triangle is isosceles.

Topic: Trigonometria, Geometria piana Metodo: Trigonometria, skill_manipolazione_algebrica Abilita: Manipolazione algebrica, Ragionamento geometrico Area: Geometria Fonte: apri PDF p.1

Prove triangle is isosceles given a trigonometric side condition

Let $a$ , $b$ , $c$ be the lengths of the sides of a triangle, and $α$ , $β$ , $γ$ , respectively, the angles opposite these sides. Prove that if the triangle is isosceles.

src_imho_1966__Q02

Distances from tetrahedron vertices to circumsphere center inequality

Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

Topic: Geometria solida, Disuguaglianze Metodo: Disuguaglianze, Estremalità Abilita: Ragionamento geometrico, Modellizzazione, Stima Area: Geometria, Algebra e Analisi Fonte: apri PDF p.1

Distance from tetrahedron vertices to circumsphere center inequality

Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.

src_imho_1966__Q03

Telescoping sum of cosecants equals cotangent identity

Prove that for every natural number $n$ , and for every real number $x \neq = \frac{k π}{2 ^{t}}$ ( $t = 0, 1, \dots, n$ ; $k$ any integer) $\frac{1}{s i n 2 x} + \frac{1}{s i n 4 x} + \dots + \frac{1}{s i n 2 ^{n} x} = cot x - cot 2^{n} x .$

Topic: Trigonometria Metodo: Induzione, Telescoping Abilita: Manipolazione algebrica, Riconoscimento di pattern Area: Geometria Fonte: apri PDF p.1

Telescoping sum of cosecants equals cotangent identity

Prove that for every natural number $n$ , and for every real number $x \neq = \frac{k π}{2 ^{t}}$

src_imho_1966__Q04

Solve a symmetric system of four linear equations

Solve the system of equations $∣ a_{2} - a_{1} ∣ x_{1} + ∣ a_{3} - a_{1} ∣ x_{2} + ∣ a_{4} - a_{1} ∣ x_{3} = 1$ $∣ a_{1} - a_{2} ∣ x_{1} + ∣ a_{3} - a_{2} ∣ x_{2} + ∣ a_{4} - a_{2} ∣ x_{3} = 1$ $∣ a_{1} - a_{3} ∣ x_{1} + ∣ a_{2} - a_{3} ∣ x_{2} + ∣ a_{4} - a_{3} ∣ x_{3} = 1$ $∣ a_{1} - a_{4} ∣ x_{1} + ∣ a_{2} - a_{4} ∣ x_{2} + ∣ a_{3} - a_{4} ∣ x_{3} = 1$ where $a_{1}, a_{2}, a_{3}, a_{4}$ are four different real numbers.

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

Solves a symmetric system of four linear equations

Solve the system of equations $∣ a_{2} - a_{1} ∣ x_{1} + ∣ a_{3} - a_{1} ∣ x_{2} + ∣ a_{4} - a_{1} ∣ x_{3} = 1$ $∣ a_{1} - a_{2} ∣ x_{1} + ∣ a_{3} - a_{2} ∣ x_{2} + ∣ a_{4} - a_{2} ∣ x_{3} = 1$ $∣ a_{1} - a_{3} ∣ x_{1} + ∣ a_{2} - a_{3} ∣ x_{2} + ∣ a_{4} - a_{3} ∣ x_{3} = 1$ $∣ a_{1} - a_{4} ∣ x_{1} + ∣ a_{2} - a_{4} ∣ x_{2} + ∣ a_{3} - a_{4} ∣ x_{3} = 1$ where $a_{1}, a_{2}, a_{3}, a_{4}$ are four different real numbers.

src_imho_1966__Q05

Interior points on triangle sides: one sub-triangle has area ≤ quarter

In the interior of sides $B C$ , $C A$ , $A B$ of triangle $A B C$ , any points $K$ , $L$ , $M$ , respectively, are selected. Prove that the area of at least one of the triangles $A M L$ , $B K M$ , $C L K$ is less than or equal to one quarter of the area of triangle $A B C$ .

Topic: Geometria piana, Disuguaglianze Metodo: Estremalità, Casework Abilita: Ragionamento geometrico, Stima, Modellizzazione Area: Geometria, Algebra e Analisi Fonte: apri PDF p.1

Interior points on triangle sides: one sub-triangle has area ≤ quarter

In the interior of sides $B C$ , $C A$ , $A B$ of triangle $A B C$ , any points $K$ , $L$ , $M$ , respectively, are selected. Prove that the area of at least one of the $A M L$ , $B K M$ , $C L K$ is less than or equal to one quarter of the area of the triangle $A B C$ .

src_imho_1966__Q06

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IMO 1966 (VIII International Mathematical Olympiad)

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