IMO 1970

Problem 1. Let $M$ be a point on the side $A B$ of $Δ A B C$ . Let $r_{1}, r_{2}$ and $r$ be the radii of the inscribed circles of triangles $A M C, B M C$ and $A B C$ . Let $q_{1}, q_{2}$ and $q$ be the radii of the escribed circles of the same triangles that lie in the angle $A C B$ . Prove that

$\frac{r _{1}}{q _{1}} \cdot \frac{r _{2}}{q _{2}} = \frac{r}{q} .$

Problem 2. Let $a, b$ and $n$ be integers greater than $1$ , and let $a$ and $b$ be the bases of two number systems. $A_{n - 1}$ and $A_{n}$ are numbers in the system with base $a$ , and $B_{n - 1}$ and $B_{n}$ are numbers in the system with base $b;$ these are related as follows:

A_{n} = x_{n} x_{n - 1} \dots x_{0}, A_{n - 1} = x_{n - 1} x_{n - 2} \dots x_{0}, B_{n} = x_{n} x_{n - 1} \dots x_{0}, B_{n - 1} = x_{n - 1} x_{n - 2} \dots x_{0}, x_{n} \neq = 0, x_{n - 1} \neq = 0.

Prove:

$\frac{A _{n - 1}}{A _{n}} < \frac{B _{n - 1}}{B _{n}} if and only if a > b .$

Problem 3. The real numbers $a_{0}, a_{1}, ..., a_{n}, ..$ . satisfy the condition:

$1 = a_{0} \leq a_{1} \leq a_{2} \leq \dots \leq a_{n} \leq \dots .$

The numbers $b_{1}, b_{2}, ..., b_{n}, ..$ . are defined by

b_{n} = k = 1 \sum n (1 - \frac{a _{k - 1}}{a _{k}}) \frac{1}{a _{k}} .

(a) Prove that $0 \leq b_{n} < 2$ for all $n$ .

(b) Given $c$ with $0 \leq c < 2$ , prove that there exist numbers $a_{0}, a_{1}, ...$ with the above properties such that $b_{n} > c$ for large enough $n$ .

Problem 4. Find the set of all positive integers $n$ with the property that the set ${n, n + 1, n + 2, n + 3, n + 4, n + 5}$ can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

Problem 5. In the tetrahedron $A B C D$ , angle $B D C$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $A B C$ is the intersection of the altitudes of $Δ A B C$ . Prove that

$(A B + B C + C A)^{2} \leq 6 (A D^{2} + B D^{2} + C D^{2}) .$

For what tetrahedra does equality hold?

Problem 6. In a plane there are $100$ points, no three of which are collinear. Consider all possible triangles having these points as vertices. Prove that no more than $70%$ of these triangles are acute-angled.

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