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1970 IMO Problems: Difference between revisions

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


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


<center>
<center>
<math>\displaystyle \frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}</math>.
<math>\frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}</math>.
</center>
</center>


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=== Problem 2 ===
=== Problem 2 ===


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


<center>
<center>
<math>\displaystyle A_{n} = x_{n}x_{n-1}\cdots x_{0}, A_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}</math>,
<math>A_{n} = x_{n}x_{n-1}\cdots x_{0}, A_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}</math>,


<math>\displaystyle B_{n} = x_{n}x_{n-1}\cdots x_{0}, B_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}</math>,
<math>B_{n} = x_{n}x_{n-1}\cdots x_{0}, B_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}</math>,


<math>\displaystyle x_{n} \neq 0, x_{n-1} \neq 0</math>.
<math>x_{n} \neq 0, x_{n-1} \neq 0</math>.
</center>
</center>


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


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=== Problem 3 ===
=== Problem 3 ===


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


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


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


<center>
<center>
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</center>
</center>


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


(b) given <math>\displaystyle 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>\displaystyle b_n > c</math> for large enough <math>\displaystyle n</math>.
(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>.


[[1970 IMO Problems/Problem 3 | Solution]]
[[1970 IMO Problems/Problem 3 | Solution]]
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=== Problem 4 ===
=== Problem 4 ===


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


[[1970 IMO Problems/Problem 4 | Solution]]
[[1970 IMO Problems/Problem 4 | Solution]]
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=== Problem 5 ===
=== Problem 5 ===


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


<center>
<center>
<math>\displaystyle ( AB+BC+CA )^2 \leq 6( AD^2 + BD^2 + CD^2 )</math>.
<math>( AB+BC+CA )^2 \leq 6( AD^2 + BD^2 + CD^2 )</math>.
</center>
</center>


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* [[1970 IMO]]
* [[1970 IMO]]
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1970 1970 IMO Problems on the Resources Page]
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1970 1970 IMO Problems on the Resources Page]
* [[IMO Problems and Solutions, with authors]]
* [[Mathematics competition resources]] {{IMO box|year=1970|before=[[1969 IMO]]|after=[[1971 IMO]]}}

Revision as of 12:48, 29 January 2021

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 the 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

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

(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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