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Difference between revisions of "1962 IMO Problems"

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== Day II ==
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== Day I ==
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=== Problem 1 ===
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Find the smallest natural number <math>n</math> which has the following properties:
  
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=== Problem 4 ===
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(a) Its decimal representation has 6 as the last digit.
  
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Solve the equation <math>cos^2{x}+cos^2{2x}+cos^3{3x}=1</math>.
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(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>.
  
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[[1962 IMO Problems/Problem 4 | Solution]]
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[[1962 IMO Problems/Problem 1 | Solution]]
  
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=== Problem 5 ===
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=== Problem 2 ===
  
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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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Determine all real numbers <math>x</math> which satisfy the inequality:
  
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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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<center>
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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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a circle can be inscribed in the quadrilateral thus obtained.
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[[1962 IMO Problems/Problem 2 | Solution]]
  
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[[1962 IMO Problems/Problem 5 | Solution]]
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=== Problem 3 ===
  
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=== Problem 6 ===
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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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Consider an isosceles triangle. Let <math>r</math> be the radius of its circumscribed
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[[1962 IMO Problems/Problem 3 | Solution]]
  
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circle and <math>rho</math> the radius of its inscribed circle. Prove that the
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== Day II ==
  
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distance <math>d</math> between the centers of these two circles is <math>d=\sqrt{r(r-
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=== Problem 4 ===
 +
Solve the equation <math>cos^2{x}+cos^2{2x}+cos^2{3x}=1</math>.
  
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rho)}</math>
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[[1962 IMO Problems/Problem 4 | Solution]]
  
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[[1962 IMO Problems/Problem 6 | Solution]]
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=== Problem 5 ===
 +
On the circle <math>K</math> there are given three distinct points <math>A,B,C</math>. Construct (using only straightedge and compass) a fourth point <math>D</math> on <math>K</math> such that a circle can be inscribed in the quadrilateral thus obtained.
  
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=== Problem 7 ===
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[[1962 IMO Problems/Problem 5 | Solution]]
  
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The tetrahedron <math>SABC</math> has the following property: there exist five
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=== Problem 6 ===
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Consider an isosceles triangle. Let <math>r</math> be the radius of its circumscribed circle and <math>\rho</math> the radius of its inscribed circle. Prove that the distance <math>d</math> between the centers of these two circles is
 +
<center><math>d=\sqrt{r(r-2p)}</math>.</center>
  
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spheres, each tangent to the edges <math>SA, SB, SC, BCCA, AB,</math> or to their
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[[1962 IMO Problems/Problem 6 | Solution]]
  
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extensions.
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=== Problem 7 ===
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The tetrahedron <math>SABC</math> has the following property: there exist five spheres, each tangent to the edges <math>SA, SB, SC, BC, CA, AB</math>, or to their extensions.
  
 
(a) Prove that the tetrahedron <math>SABC</math> is regular.
 
(a) Prove that the tetrahedron <math>SABC</math> is regular.
  
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(b) Prove conversely that for every regular tetrahedron five such spheres  
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(b) Prove conversely that for every regular tetrahedron five such spheres exist.
  
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exist.
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[[1962 IMO Problems/Problem 7 | Solution]]
  
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[[1962 IMO Problems/Problem 7 | Solution]]
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== Resources ==
 +
* [[1962 IMO]]
 +
* [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

Day I

Problem 1

Find the smallest natural number $n$ 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 $n$.

Solution

Problem 2

Determine all real numbers $x$ which satisfy the inequality:

$\sqrt{\sqrt{3-x}-\sqrt{x+1}}>\dfrac{1}{2}$

Solution

Problem 3

Consider the cube $ABCDA'B'C'D'$($ABCD$ and $A'B'C'D'$ are the upper and lower bases, respectively, and edges $AA'$, $BB'$, $CC'$, $DD'$ are parallel). The point $X$ moves at constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimeter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpoints of the segments $XY$.

Solution

Day II

Problem 4

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

Solution

Problem 5

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only straightedge and compass) 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

$d=\sqrt{r(r-2p)}$.

Solution

Problem 7

The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB$, 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

Resources

1962 IMO (Problems) • Resources
Preceded by
1961 IMO 		1 • 2 • 3 • 4 • 5 • 6 Followed by
1963 IMO
All IMO Problems and Solutions
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