Difference between revisions of "1964 IMO Problems"
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== Day I == | == Day I == | ||
− | == Problem 1 == | + | === 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 == | + | === 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 == | + | === 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>). | ||
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=== Problem 6 === | === Problem 6 === | ||
− | In tetrahedron <math>ABCD</math>, vertex <math>D</math> is connected with <math>D_0</math>, the | + | In tetrahedron <math>ABCD</math>, vertex <math>D</math> is connected with <math>D_0</math>, the centroid of <math>\triangle ABC</math>. Lines parallel to <math>DD_0</math> are drawn through <math>A,B</math> and <math>C</math>. These lines intersect the planes <math>BCD, CAD</math> and <math>ABD</math> in points <math>A_1, B_1,</math> and <math>C_1</math>, respectively. Prove that the volume of <math>ABCD</math> is one third the volume of <math>A_1B_1C_1D_0</math>. Is the result true if point <math>D_o</math> is selected anywhere within <math>\triangle ABC</math>? |
[[1964 IMO Problems/Problem 6 | Solution]] | [[1964 IMO Problems/Problem 6 | Solution]] | ||
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* [[1964 IMO]] | * [[1964 IMO]] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1964 IMO 1964 Problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1964 IMO 1964 Problems on the Resources page] | ||
+ | |||
+ | |||
+ | * [[IMO Problems and Solutions, with authors]] | ||
+ | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{IMO box|year=1964|before=[[1963 IMO]]|after=[[1965 IMO]]}} |
Latest revision as of 12:38, 29 January 2021
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
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 centroid 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
?
Resources
1964 IMO (Problems) • Resources | ||
Preceded by 1963 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 1965 IMO |
All IMO Problems and Solutions |