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Difference between revisions of "1962 IMO Problems"

(→‎Day I)
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=== Problem 3 ===
 
=== Problem 3 ===
  
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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  
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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>.
  
−
lower bases, respectively, and edges <math>AA'</math>, <math>BB'</math>, <math>CC'</math>, <math>DD'</math> are
+
[[1962 IMO Problems/Problem 3 | Solution]]
 +
 
 +
== Day II ==
 +
 
 +
=== Problem 4 ===
 +
Solve the equation <math>cos^2{x}+cos^2{2x}+cos^2{3x}=1</math>.
  
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parallel). The point <math>X</math> moves at constant speed along the perimeter of the
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[[1962 IMO Problems/Problem 4 | Solution]]
  
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square <math>ABCD</math> in the direction <math>ABCDA</math>, and the point <math>Y</math> moves at the same
+
=== 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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rate along the perimeter of the square <math>B'C'CB</math> in the direction
+
[[1962 IMO Problems/Problem 5 | Solution]]
  
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<math>B'C'CBB'</math>. Points <math>X</math> and <math>Y</math> begin their motion at the same instant from
+
=== Problem 6 ===
 +
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>
  
−
the starting positions <math>A</math> and <math>B'</math>, respectively. Determine and draw the
+
[[1962 IMO Problems/Problem 6 | Solution]]
  
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locus of the midpoints of the segments <math>XY</math>.
+
=== Problem 7 ===
 +
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.
  
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[[1962 IMO Problems/Problem 3 | Solution]]
+
(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]]
 +
 
 +
== Resources ==
 +
* [[1962 IMO]]
 +
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1962 IMO 1962 Problems on the Resources page]

Revision as of 16:55, 30 December 2008

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

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