Difference between revisions of "1962 IMO Problems"
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(a) Its decimal representation has 6 as the last digit. | (a) Its decimal representation has 6 as the last digit. | ||
| − | (b) If the last digit 6 is erased and placed in front of the remaining | + | (b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number <math>n</math>. |
| − | |||
| − | digits, the | ||
| − | resulting number is four times as large as the original number <math>n</math>. | ||
[[1962 IMO Problems/Problem 1 | Solution]] | [[1962 IMO Problems/Problem 1 | Solution]] | ||
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* [[1962 IMO]] | * [[1962 IMO]] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1962 IMO 1962 Problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1962 IMO 1962 Problems on the Resources page] | ||
| + | * [[IMO Problems and Solutions, with authors]] | ||
| + | * [[Mathematics competition resources]] | ||
| + | |||
| + | {{IMO box|year=1962|before=[[1961 IMO]]|after=[[1963 IMO]]}} | ||
Latest revision as of 20:17, 20 August 2020
Contents
Day I
Problem 1
Find the smallest natural number 
which has the following properties:
(a) Its decimal representation has 6 as the last digit.
(b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number .
Problem 2
Determine all real numbers 
which satisfy the inequality:
Problem 3
Consider the cube 
(
and 
are the upper and lower bases, respectively, and edges 
, 
, 
, 
are parallel). The point 
moves at constant speed along the perimeter of the square in the direction 
, and the point 
moves at the same rate along the perimeter of the square 
in the direction 
. Points and begin their motion at the same instant from the starting positions 
and 
, respectively. Determine and draw the locus of the midpoints of the segments 
.
Day II
Problem 4
Solve the equation 
.
Problem 5
On the circle 
there are given three distinct points 
. Construct (using only straightedge and compass) a fourth point 
on such that a circle can be inscribed in the quadrilateral thus obtained.
Problem 6
Consider an isosceles triangle. Let 
be the radius of its circumscribed circle and 
the radius of its inscribed circle. Prove that the distance 
between the centers of these two circles is
.Problem 7
The tetrahedron 
has the following property: there exist five spheres, each tangent to the edges 
, or to their extensions.
(a) Prove that the tetrahedron is regular.
(b) Prove conversely that for every regular tetrahedron five such spheres exist.
Resources
- 1962 IMO
- IMO 1962 Problems on the Resources page
- IMO Problems and Solutions, with authors
- Mathematics competition resources
| 1962 IMO (Problems) • Resources | ||
| Preceded by 1961 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1963 IMO |
| All IMO Problems and Solutions | ||
