MathDB

3

Part of 1999 Estonia National Olympiad

Problems(4)

computational, EG | : | GC | : | CE | inside a square, midpoints related

Source: 1999 Estonia National Olympiad Final Round grade 9

3/11/2020
Let EE and FF be the midpoints of the lines ABAB and DADA of a square ABCDABCD, respectively and let GG be the intersection of DEDE with CFCF. Find the aspect ratio of sidelengths of the triangle EGCEGC, ∣EG∣:∣GC∣:∣CE∣| EG | : | GC | : | CE |.
geometrysquareratio
cover sides wall of staircase of n steps with L-trominos

Source: 1999 Estonia National Olympiad Final Round grade 11 p4

3/11/2020
For which values of nn it is possible to cover the side wall of staircase of n steps (for n=6n = 6 in the figure) with plates of shown shape? The width and height of each step is 11 dm, the dimensions of plate are 2×22 \times 2 dm and from the corner there is cut out a piece with dimensions 1×11\times 1 dm. https://cdn.artofproblemsolving.com/attachments/e/e/ac7a52f3dd40480f82024794708c5a449e0c2b.png
combinatoricscombinatorial geometry
area between APIQB and CPIQ is equal to area of CLIM, incircle related

Source: 1999 Estonia National Olympiad Final Round grade 10 p3

3/11/2020
The incircle of the triangle ABCABC, with the center II , touches the sides AB,ACAB, AC and BCBC in the points K,LK, L and MM respectively. Points PP and QQ are taken on the sides ACAC and BCBC respectively, such that ∣AP∣=∣CL∣|AP| = |CL| and ∣BQ∣=∣CM∣|BQ| = |CM|. Prove that the difference of areas of the figures APIQBAPIQB and CPIQCPIQ is equal to the area of the quadrangle CLIMCLIM
geometryareasincircle
euler line // AB iff tan A tanB = 3 in acute ABC

Source: 1999 Estonia National Olympiad Final Round grade 12 p3

3/11/2020
Prove that the line segment, joining the orthocenter and the intersection point of the medians of the acute-angled triangle ABCABC is parallel to the side ABAB iff tan⁡∠A⋅tan⁡∠B=3\tan \angle A \cdot \tan \angle B = 3.
Euler LineCentroidorthocenterparallelgeometrytrigonometry