MathDB

2

Part of 2022 Yasinsky Geometry Olympiad

Problems(3)

cumputational geo with altitude and angle bisector, <C=1/4 <A=1/4 <B

Source: 2022 Yasinsky Geometry Olympiad VIII-IX p2, Ukraine

11/7/2022
In the triangle ABCABC, angle CC is four times smaller than each of the other two angle The altitude AKAK and the angle bisector ALAL are drawn from the vertex of the angle AA. It is known that the length of ALAL is equal to ℓ\ell. Find the length of the segment LKLK.
(Gryhoriy Filippovskyi)
geometryangle bisectorbisectorangles
perimeter of ABC wanted, BH+CH=4r

Source: 2022 Yasinsky Geometry Olympiad X-XI advanced p2 , Ukraine

11/8/2022
In the acute triangle ABCABC, the sum of the distances from the vertices BB and CC to of the orthocenter HH is equal to 4r,4r, where rr is the radius of the circle inscribed in this triangle. Find the perimeter of triangle ABCABC if it is known that BC=aBC=a.
(Gryhoriy Filippovskyi)
geometryperimeter
square inside a square AP/PB=BQ/QC=CR/RD=DT/TA=1/2

Source: 2022 Yasinsky Geometry Olympiad X-XI p2 , Ukraine

11/8/2022
On the sides ABAB, BCBC, CDCD, DADA of the square ABCDABCD points P,Q,R,TP, Q, R, T are chosen such that APPB=BQQC=CRRD=DTTA=12.\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12. The segments ARAR, BTBT, CPCP, DQDQ in the intersection form the quadrilateral KLMNKLMN (see figure). https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.png
a) Prove that KLMNKLMN is a square. b) Find the ratio of the areas of the squares KLMNKLMN and ABCDABCD.
(Alexander Shkolny)
geometrySquaressquareratioareas