Difference between revisions of "1962 IMO Problems"

Revision as of 14:00, 29 November 2007

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

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

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Difference between revisions of "1962 IMO Problems"

Revision as of 14:00, 29 November 2007

Contents

Day I

Problem 1

Problem 2

Problem 3