1970 IMO Problems

Problems of the 12th IMO 1970 Hungary.

Day 1

Problem 1

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

$\displaystyle \frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}$.

Solution

Problem 2

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

$\displaystyle A_{n} = x_{n}x_{n-1}\cdots x_{0}, A_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}$,

$\displaystyle B_{n} = x_{n}x_{n-1}\cdots x_{0}, B_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}$,

$\displaystyle x_{n} \neq 0, x_{n-1} \neq 0$.

Prove:

$\frac{A_{n-1}}{A_{n}} < \frac{B_{n-1}}{B_{n}}$ if and only if $\displaystyle a > b$.

Solution

Problem 3

The real numbers $\displaystyle a_0, a_1, \ldots, a_n, \ldots$ satisfy the condition:

$\displaystyle 1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots$.

The numbers $\displaystyle b_{1}, b_{2}, \ldots, b_n, \ldots$ are defined by

$b_n = \sum_{k=1}^{n} \left( 1 - \frac{a_{k-1}}{a_{k}} \right)$

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

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

Solution

Day 2

Problem 4

Find the set of all positive integers $\displaystyle n$ with the property that the set $\displaystyle \{ n, n+1, n+2, n+3, n+4, n+5 \}$ 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.

Solution

Problem 5

In the tetrahedron $\displaystyle ABCD$, angle $\displaystyle BDC$ is a right angle. Suppose that the foot $\displaystyle H$ of the perpendicular from $\displaystyle D$ to the plane $\displaystyle ABC$ in the tetrahedron is the intersection of the altitudes of $\displaystyle \triangle ABC$. Prove that

$\displaystyle ( AB+BC+CA )^2 \leq 6( AD^2 + BD^2 + CD^2 )$.

For what tetrahedra does equality hold?

Solution

Problem 6

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

Solution

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