MathDB

3

Part of 2001 Estonia National Olympiad

Problems(4)

square and a circle both tangent and intersecting

Source: 2001 Estonia National Olympiad Final Round grade 9 p3

3/12/2020
A circle of radius 1010 is tangent to two adjacent sides of a square and intersects its two remaining sides at the endpoints of a diameter of the circle. Find the side length of the square.
squarecirclegeometry
3 squares in a row, find sum of 2 angles

Source: 2001 Estonia National Olympiad Final Round grade 10 p3

3/12/2020
There are three squares in the picture. Find the sum of angles ADCADC and BDCBDC. https://cdn.artofproblemsolving.com/attachments/c/9/885a6c6253fca17e24528f8ba8a5d31a18c845.png
anglesSumgeometry
AO/OD= AE/EC+AF/FB

Source: 2001 Estonia National Olympiad Final Round grade 11 p3

3/12/2020
Points D,ED,E and FF are taken on the sides BC,CA,ABBC,CA,AB of a triangle ABCABC respectively so that the segments AD,BEAD, BE and CFCF intersect at point OO. Prove that AOOD=AEEC+AFFB\frac{AO}{OD}= \frac{AE}{EC}+\frac{AF}{FB} .
geometryratio
1/AE+1/BD=1/r , incricle related

Source: 2001 Estonia National Olympiad Final Round grade 12 p3

3/12/2020
A circle with center II and radius rr is inscribed in a triangle ABCABC with a right angle at CC. Rays AIAI and CICI meet the opposite sides at DD and EE respectively. Prove that 1AE+1BD=1r\frac{1}{AE}+\frac{1}{BD}=\frac{1}{r}
geometryincircleinradiusratio