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Difference between revisions of "1964 IMO Problems"

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== Day I ==
 
== Day I ==
  
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== Problem 1 ==
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=== 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]]
  
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== Problem 2 ==
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=== 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]]
  
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== Problem 3 ==
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=== 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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== Day II ==
 
== Day II ==
  
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== Problem 4 ==
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=== Problem 4 ===
 
Seventeen people correspond by mail with one another - each one with all
 
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
 
the rest. In their letters only three different topics are discussed. Each pair
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[[1964 IMO Problems/Problem 4 | Solution]]
 
[[1964 IMO Problems/Problem 4 | Solution]]
  
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== Problem 5 ==
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=== Problem 5 ===
 
Suppose five points in a plane are situated so that no two of the straight lines
 
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
 
joining them are parallel, perpendicular, or coincident. From each point perpendiculars
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[[1964 IMO Problems/Problem 5 | Solution]]
 
[[1964 IMO Problems/Problem 5 | Solution]]
  
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== Problem 6 ==
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=== Problem 6 ===
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In tetrahedron <math>ABCD</math>, vertex <math>D</math> is connected with <math>D_0</math>, the centrod 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>
+
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]
 +
 +
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* [[IMO Problems and Solutions, with authors]]
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* [[Mathematics competition resources]]
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{{IMO box|year=1964|before=[[1963 IMO]]|after=[[1965 IMO]]}}

Latest revision as of 11:38, 29 January 2021

Problems of the 6th IMO 1964 in USSR.

Day I

Problem 1

(a) Find all positive integers $n$ for which $2^n-1$ is divisible by $7$.

(b) Prove that there is no positive integer $n$ for which $2^n+1$ is divisible by $7$.

Solution

Problem 2

Suppose $a, b, c$ are the sides of a triangle. Prove that

\[a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le{3abc}.\]

Solution

Problem 3

A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructed. Each of these tangents cuts off a triangle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).

Solution

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.

Solution

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.

Solution

Problem 6

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centroid of $\triangle ABC$. Lines parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_1, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result true if point $D_o$ is selected anywhere within $\triangle ABC$?

Solution

Resources


1964 IMO (Problems) • Resources
Preceded by
1963 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1965 IMO
All IMO Problems and Solutions
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