MathDB

1

max number of equal spheres touching a fixed equal one, no common points

Show that the maximum number of spheres of radius

1

that can be placed touching a fixed sphere of radius

1

so that no pair of spheres has an interior point in common is between

12

and

14

.

5

1

smallest k such that 1 \le n ^{1/n} \le k for all positive integers n

Show that

1 \le n^{1/n} \le 2

for all positive integers

n

. Find the smallest

k

such that

1 \le n ^{1/n} \le k

for all positive integers

n

.

4

1

coloring points of circle so that no righttriangle inscribed has same color

Show that it is possible to color each point of a circle red or blue so that no right-angled triangle inscribed in the circle has its vertices all the same color.

3

1

1 + 1/2 + 1/3 + ... + 1/n not an integer for n > 1

Show that

1 + 1/2 + 1/3 + ... + 1/n

is not an integer for

n > 1

.

2

1

some vertices of regular pyramids form a regular hexagon

An equilateral triangle

ABC

has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side

a

is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done, the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.

1

finite solutions of 1/a + 1/b + 1/c = 1/1983 in positive integers

Show that there are only finitely many solutions to

1/a + 1/b + 1/c = 1/1983

in positive integers.

1983 Brazil National Olympiad

Subcontests

max number of equal spheres touching a fixed equal one, no common points

smallest k such that 1 \le n ^{1/n} \le k for all positive integers n

coloring points of circle so that no righttriangle inscribed has same color

1 + 1/2 + 1/3 + ... + 1/n not an integer for n > 1

some vertices of regular pyramids form a regular hexagon

finite solutions of 1/a + 1/b + 1/c = 1/1983 in positive integers

1983 Brazil National Olympiad

Subcontests

max number of equal spheres touching a fixed equal one, no common points

smallest k such that 1 \le n ^{1/n} \le k for all positive integers n

coloring points of circle so that no righttriangle inscribed has same color

1 + 1/2 + 1/3 + ... + 1/n not an integer for n &gt; 1

some vertices of regular pyramids form a regular hexagon

finite solutions of 1/a + 1/b + 1/c = 1/1983 in positive integers

1 + 1/2 + 1/3 + ... + 1/n not an integer for n > 1