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Difference between revisions of "1962 IMO Problems"

(→‎Day I)
(→‎Day II)
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== Day I ==
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== Day II ==
  
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=== Problem 1 ===
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=== Problem 4 ===
  
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Find the smallest natural number <math>n</math> which has the following properties:
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Solve the equation <math>cos^2{x}+cos^2{2x}+cos^3{3x}=1</math>.
  
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(a) Its decimal representation has 6 as the last digit.
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[[1962 IMO Problems/Problem 4 | Solution]]
  
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(b) If the last digit 6 is erased and placed in front of the remaining digits, the
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=== Problem 5 ===
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resulting number is four times as large as the original number <math>n</math>.
 
  
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[[1962 IMO Problems/Problem 1 | Solution]]
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On the circle <math>K</math> there are given three distinct points <math>A,B,C</math>. Construct
  
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=== Problem 2 ===
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(using only straightedge and compasses) a fourth point <math>D</math> on <math>K</math> such that
  
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Determine all real numbers <math>x</math> which satisfy the inequality:
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a circle can be inscribed in the quadrilateral thus obtained.
  
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<center>
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[[1962 IMO Problems/Problem 5 | Solution]]
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<math>\sqrt{\sqrt{3-x}-\sqrt{x+1}}>\dfrac{1}{2}</math>
 
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</center>
 
  
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[[1962 IMO Problems/Problem 2 | Solution]]
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=== Problem 6 ===
  
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=== Problem 3 ===
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Consider an isosceles triangle. Let <math>r</math> be the radius of its circumscribed
  
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Consider the cube <math>ABCDA'B'C'D'</math>(<math>ABCD</math> and <math>A'B'C'D'</math> are the upper and lower bases, respectively, and edges <math>AA'</math>, <math>BB'</math>, <math>CC'</math>, <math>DD'</math> are parallel). The point <math>X</math> moves at constant speed along the perimeter of the square <math>ABCD</math> in the direction <math>ABCDA</math>, and the point <math>Y</math> moves at the same rate along the perimeter of the square <math>B'C'CB</math> in the direction <math>B'C'CBB'</math>. Points <math>X</math> and <math>Y</math> begin their motion at the same instant from the starting positions <math>A</math> and <math>B'</math>, respectively. Determine and draw the locus of the midpoints of the segments <math>XY</math>.
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circle and <math>rho</math> the radius of its inscribed circle. Prove that the  
  
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[[1962 IMO Problems/Problem 3 | Solution]]
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distance <math>d</math> between the centers of these two circles is <math>d=\sqrt{r(r-
 +
 
 +
rho)}</math>
 +
 
 +
[[1962 IMO Problems/Problem 6 | Solution]]
 +
 
 +
=== Problem 7 ===
 +
 
 +
The tetrahedron <math>SABC</math> has the following property: there exist five
 +
 
 +
spheres, each tangent to the edges <math>SA, SB, SC, BCCA, AB,</math> or to their
 +
 
 +
extensions.
 +
 
 +
(a) Prove that the tetrahedron <math>SABC</math> is regular.
 +
 
 +
(b) Prove conversely that for every regular tetrahedron five such spheres
 +
 
 +
exist.
 +
 
 +
[[1962 IMO Problems/Problem 7 | Solution]]

Revision as of 13:59, 29 November 2007

Day II

Problem 4

Solve the equation $cos^2{x}+cos^2{2x}+cos^3{3x}=1$.

Solution

Problem 5

On the circle $K$ there are given three distinct points $A,B,C$. Construct

(using only straightedge and compasses) a fourth point $D$ on $K$ such that

a circle can be inscribed in the quadrilateral thus obtained.

Solution

Problem 6

Consider an isosceles triangle. Let $r$ be the radius of its circumscribed

circle and $rho$ the radius of its inscribed circle. Prove that the

distance $d$ between the centers of these two circles is $d=\sqrt{r(r-

rho)}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 7

The tetrahedron $SABC$ has the following property: there exist five

spheres, each tangent to the edges $SA, SB, SC, BCCA, AB,$ or to their

extensions.

(a) Prove that the tetrahedron $SABC$ is regular.

(b) Prove conversely that for every regular tetrahedron five such spheres

exist.

Solution

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