Difference between revisions of "1964 IMO Problems"
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== Day I == | == Day I == | ||
| − | + | == Problem 1 == | |
(a) Find all positive integers <math>n</math> for which <math>2^n-1</math> is divisible by <math>7</math>. | (a) Find all positive integers <math>n</math> for which <math>2^n-1</math> is divisible by <math>7</math>. | ||
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[[1964 IMO Problems/Problem 1 | Solution]] | [[1964 IMO Problems/Problem 1 | Solution]] | ||
| − | + | == Problem 2 == | |
Suppose <math>a, b, c</math> are the sides of a triangle. Prove that | Suppose <math>a, b, c</math> are the sides of a triangle. Prove that | ||
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[[1964 IMO Problems/Problem 2 | Solution]] | [[1964 IMO Problems/Problem 2 | Solution]] | ||
| − | + | == Problem 3 == | |
A circle is inscribed in a triangle <math>ABC</math> with sides <math>a,b,c</math>. Tangents to the circle parallel to the sides of the triangle are contructed. Each of these tangents cuts off a triangle from <math>\triangle ABC</math>. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of <math>a,b,c</math>). | A circle is inscribed in a triangle <math>ABC</math> with sides <math>a,b,c</math>. Tangents to the circle parallel to the sides of the triangle are contructed. Each of these tangents cuts off a triangle from <math>\triangle ABC</math>. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of <math>a,b,c</math>). | ||
Revision as of 18:29, 15 July 2009
Problems of the 6th IMO 1964 in USSR.
Contents
Day I
Problem 1
(a) Find all positive integers 
for which 
is divisible by 
.
(b) Prove that there is no positive integer for which 
is divisible by .
Problem 2
Suppose 
are the sides of a triangle. Prove that
![\[a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le{3abc}.\]](http://latex.artofproblemsolving.com/2/c/8/2c8afb405fe890dd89759ea3a2ed2e527038adfc.png)
Problem 3
A circle is inscribed in a triangle 
with sides 
. Tangents to the circle parallel to the sides of the triangle are contructed. Each of these tangents cuts off a triangle from 
. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of ).
Day II
Problem 4
Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.
Problem 5
Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.
Problem 6
In tetrahedron 
, vertex 
is connected with 
, the centrod of . Lines parallel to 
are drawn through 
and 
. These lines intersect the planes 
and 
in points 
and 
, respectively. Prove that the volume of is one third the volume of 
. Is the result true if point 
is selected anywhere within
