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Difference between revisions of "1970 IMO Problems"

 
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=== Problem 1 ===
 
=== Problem 1 ===
  
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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 <math>\angle ACB</math>.  Prove that
  
 
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<center>
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<math>\displaystyle \frac{r_1}{q_1} \cdot \frac{r_2}{q_2} = \frac{r}{q}</math>.
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<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 ===
  
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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:
  
 
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<center>
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<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>,
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<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>,
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<math>B_{n} = x_{n}x_{n-1}\cdots x_{0}, B_{n-1} = x_{n-1}x_{n-2}\cdots x_{0}</math>,
  
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<math>\displaystyle x_{n} \neq 0, x_{n-1} \neq 0</math>.
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<math>x_{n} \neq 0, x_{n-1} \neq 0</math>.
 
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<center>
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<math> \frac{A_{n-1}}{A_{n}} < \frac{B_{n-1}}{B_{n}}</math> if and only if <math>\displaystyle a > b</math>.
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<math> \frac{A_{n-1}}{A_{n}} < \frac{B_{n-1}}{B_{n}}</math> if and only if <math>a > b</math>.
 
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</center>
  
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=== Problem 3 ===
 
=== Problem 3 ===
  
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The real numbers <math>\displaystyle a_0, a_1, \ldots, a_n, \ldots</math> satisfy the condition:
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The real numbers <math>a_0, a_1, \ldots, a_n, \ldots</math> satisfy the condition:
  
 
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<center>
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<math>\displaystyle 1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots</math>.
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<math>1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots</math>.
 
</center>
 
</center>
  
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The numbers <math>\displaystyle b_{1}, b_{2}, \ldots, b_n, \ldots</math> are defined by
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The numbers <math>b_{1}, b_{2}, \ldots, b_n, \ldots</math> are defined by
  
 
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</center>
 
</center>
  
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(a) Prove that <math>\displaystyle 0 \leq b_n < 2</math> for all <math>\displaystyle n</math>.
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(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 ===
  
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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 ===
  
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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
  
 
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<center>
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<math>\displaystyle ( AB+BC+CA )^2 \leq 6( AD^2 + BD^2 + CD^2 )</math>.
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<math>( AB+BC+CA )^2 \leq 6( AD^2 + BD^2 + CD^2 )</math>.
 
</center>
 
</center>
  
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=== Problem 6 ===
 
=== Problem 6 ===
  
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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.
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In a plane there are <math>100</math> points, no three of which are collinear.  Consider all possible triangles having these point as vertices.  Prove that no more than <math>70 \%</math> of these triangles are acute-angled.
  
 
[[1970 IMO Problems/Problem 6 | Solution]]
 
[[1970 IMO Problems/Problem 6 | Solution]]
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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]]}}

Latest revision as of 23:17, 9 December 2022

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

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

(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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