International Mathematical Olympiad 1970

Let $M$ be a point on the side $AB$ of $\triangle ABC$. Let $r_1, r_2$ and $r$ be the radii of the inscribed circles of triangles $AMC, BMC$ and $ABC$. Let $q_1, q_2$ and $q$ be the radii of the escribed circles of the same triangles that lie in the angle $ACB$. Prove that $$\frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}.$$

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: $$A_n = x_n x_{n-1} \cdots x_0, \quad A_{n-1} = x_{n-1} x_{n-2} \cdots x_0,$$ $$B_n = x_n x_{n-1} \cdots x_0, \quad B_{n-1} = x_{n-1} x_{n-2} \cdots x_0,$$ $$x_n \neq 0, \quad x_{n-1} \neq 0.$$

Prove: $$\frac{A_{n-1}}{A_n} < \frac{B_{n-1}}{B_n} \text{ if and only if } a > b.$$

The real numbers $a_0, a_1, \ldots, a_n, \ldots$ satisfy the condition: $$1 = a_0 \leq a_1 \leq a_2 \leq \cdots \leq a_n \leq \cdots.$$

The numbers $b_1, b_2, \ldots, b_n, \ldots$ are defined by $$b_n = \sum_{k=1}^{n} \left(1 - \frac{a_{k-1}}{a_k}\right) \frac{1}{\sqrt{a_k}}.$$

(a) Prove that $0 \leq b_n < 2$ for all $n$.

(b) Given $c$ with $0 \leq c < 2$, prove that there exist numbers $a_0, a_1, \ldots$ with the above properties such that $b_n > c$ for large enough $n$.

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.

In the tetrahedron $ABCD$, angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ is the intersection of the altitudes of $\triangle ABC$. Prove that $$(AB + BC + CA)^2 \leq 6(AD^2 + BD^2 + CD^2).$$

For what tetrahedra does equality hold?

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.