10
Problems(2)
area of triangle, medians
Source: 2021 Dürer Math Competition Finals Day2 E10 https://artofproblemsolving.com/community/c2749870_
1/9/2022
A triangle is given. Its side a is of length cm, and its area is cm. It is also known that one of the angles lying on side a is twice as large as the other one. We cut the triangle into two parts at the median belonging to side a. Then we move the so-obtained two parts towards each other, such that the two segments of side a remain on the same line (i.e., the line initially occupied by side a). We move the two parts towards each other until we first reach a moment when the common part of the two segments is of length cm. What is the area of the so-obtained shape in cm? The so-obtained shape is the union of the two parts, which is a heptagon.
https://cdn.artofproblemsolving.com/attachments/3/0/3d45e2df6a0043dfa4fe5ccf64865da8879b42.png
geometryareas
max no r of horses Billy can own standing on lattice points
Source: 2021 Dürer Math Competition Finals Day2 E+10 https://artofproblemsolving.com/community/c2749870_
1/9/2022
Billy owns some really energetic horses. They are hopping around on points of the plane having two integer coordinates. Each horse can make the following types of jumps. They can hop to a point obtained from their current position via a translation by vector , , or .
The horses are now standing on lattice points such that no two can meet by making jumps as above. What is the maximal possible number of horses Billy can own?
combinatorics