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1970 IMO Problems: Difference between revisions

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=== Problem 1 ===
=== Problem 1 ===


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


<center>
<center>
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=== Problem 3 ===
=== Problem 3 ===


The real numbers <math>a_0, a_1, \ldots, a_n, \ldots</math> satisfy the condition:
The real numbers <imath>a_0, a_1, \ldots, a_n, \ldots</imath> satisfy the condition:


<center>
<center>
<math>1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots</math>.
<imath>1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots</imath>.
</center>
</center>


The numbers <math>b_{1}, b_{2}, \ldots, b_n, \ldots</math> are defined by
The numbers <imath>b_{1}, b_{2}, \ldots, b_n, \ldots</imath> are defined by


<center>
<center>
<math>b_n = \sum_{k=1}^{n} \left( 1 - \frac{a_{k-1}}{a_{k}} \right)</math>
<imath>b_n = \sum_{k=1}^{n} \left( 1 - \frac{a_{k-1}}{a_{k}} \right)\dfrac{1}{\sqrt{a_k}}</imath>
</center>
</center>


(a) Prove that <math>0 \leq b_n < 2</math> for all <math>n</math>.
(a) Prove that <imath>0 \leq b_n < 2</imath> for all <imath>n</imath>.


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


[[1970 IMO Problems/Problem 3 | Solution]]
[[1970 IMO Problems/Problem 3 | Solution]]

Latest revision as of 00:10, 12 November 2025

Problems of the 12th IMO 1970 Hungary.

Day 1

Problem 1

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

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

Solution

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:

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

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

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

Prove:

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

Solution

Problem 3

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

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

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

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

(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 $n$.

Solution

Day 2

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.

Solution

Problem 5

In the tetrahedron $ABCD$, angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ in the tetrahedron 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?

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

Resources

1970 IMO (Problems) • Resources
Preceded by
1969 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1971 IMO
All IMO Problems and Solutions
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