Difference between revisions of "1970 IMO Problems"
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=== Problem 1 === | === Problem 1 === | ||
− | Let <math> | + | 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> | ||
− | <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> | + | 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> | + | <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> | + | <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> | + | <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> | + | <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> | + | The real numbers <math>a_0, a_1, \ldots, a_n, \ldots</math> satisfy the condition: |
<center> | <center> | ||
− | <math> | + | <math>1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots</math>. |
</center> | </center> | ||
− | The numbers <math> | + | 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> | + | (a) Prove that <math>0 \leq b_n < 2</math> for all <math>n</math>. |
− | (b) given <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> | + | 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> | + | 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> | + | <math>( AB+BC+CA )^2 \leq 6( AD^2 + BD^2 + CD^2 )</math>. |
</center> | </center> | ||
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=== Problem 6 === | === 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. | + | 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 00:17, 10 December 2022
Problems of the 12th IMO 1970 Hungary.
Contents
Day 1
Problem 1
Let be a point on the side
of
. Let
, and
be the inscribed circles of triangles
, and
. Let
, and
be the radii of the exscribed circles of the same triangles that lie in
. Prove that
.
Problem 2
Let , and
be integers greater than 1, and let
and
be the bases of two number systems.
and
are numbers in the system with base
and
and
are numbers in the system with base
; these are related as follows:
,
,
.
Prove:
if and only if
.
Problem 3
The real numbers satisfy the condition:
.
The numbers are defined by
(a) Prove that for all
.
(b) given with
, prove that there exist numbers
with the above properties such that
for large enough
.
Day 2
Problem 4
Find the set of all positive integers with the property that the set
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.
Problem 5
In the tetrahedron , angle
is a right angle. Suppose that the foot
of the perpendicular from
to the plane
in the tetrahedron is the intersection of the altitudes of
. Prove that
.
For what tetrahedra does equality hold?
Problem 6
In a plane there are points, no three of which are collinear. Consider all possible triangles having these point as vertices. Prove that no more than
of these triangles are acute-angled.
Resources
- 1970 IMO
- 1970 IMO Problems on the Resources Page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
1970 IMO (Problems) • Resources | ||
Preceded by 1969 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1971 IMO |
All IMO Problems and Solutions |