MathDB

island inhabited by dragons, snakes and crocodiles

Source: 1997 Estonia National Olympiad Final Round grade 9 p5

3/13/2020

In the creation of the world there is a lonely island inhabited by dragons, snakes and crocodiles. Every inhabitant eats once a day: every snake eats one dragon for breakfast, every dragon eats one crocodile for lunch and every crocodile eats a snake for dinner. Find the total number of dragons, snakes and crocodiles on the island immediately after the creation of the world (at the beginning of the first day), when, at the end of the sixth day, there is only one inhabitant alive on the island, only one crocodile and during these six days none of the inhabitants of the island considered any to give up their meals due to lack of food.

algebracombinatorics

6 small circles tangent to a lerger circle, ractangle related

Source: 1997 Estonia National Olympiad Final Round grade 11 p5

3/11/2020

Six small circles of radius

1

are drawn so that they are all tangent to a larger circle, and two of them are tangent to sides

BC

and

AD

of a rectangle

ABCD

at their midpoints

K

and

L

. Each of the remaining four small circles is tangent to two sides of the rectangle. The large circle is tangent to sides

AB

and

CD

of the rectangle and cuts the other two sides. Find the radius of the large circle. https://cdn.artofproblemsolving.com/attachments/b/4/a134da78d709fe7162c48d6b5c40bd1016c355.png

geometryrectanglecirclestangent circles

7 tangent circles and a rectangle

Source: 1997 Estonia National Olympiad Final Round grade 12 p5

3/11/2020

Find the length of the longer side of the rectangle on the picture, if the shorter side has length

1

and the circles touch each other and the sides of the rectangle as shown.
https://cdn.artofproblemsolving.com/attachments/b/8/3986683247293bd089d8e83911309308ce0c3a.png

geometryrectanglecircles

7 tangent circles and an equilateral triangle

Source: 1997 Estonia National Olympiad Final Round grade 10

3/11/2020

There are six small circles in the figure with a radius of

1

and tangent to a large circle and the sides of the

ABC

of an equilateral triangle, where touch points are

K, L

and

M

respectively with the midpoints of sides

AB, BC

and

AC

. Find the radius of the large circle and the side of the triangle

ABC

. https://cdn.artofproblemsolving.com/attachments/3/0/f858dcc5840759993ea2722fd9b9b15c18f491.png

geometrytangent circlescirclesEquilateral

5

Problems(4)