MathDB

Problem 6

Part of 2002 Mongolian Mathematical Olympiad

Problems(2)

prove angles equal given side ratios

Source: Mongolian MO 2002 Grade 10 P6

4/9/2021
Let A1,B1,C1A_1,B_1,C_1 be the midpoints of the sides BC,CA,ABBC,CA,AB respectively of a triangle ABCABC. Points KK on segment C1A1C_1A_1 and LL on segment A1B1A_1B_1 are taken such that C1KKA1=BC+ACAC+ABandA1LLB1=AC+ABBC+AB.\frac{C_1K}{KA_1}=\frac{BC+AC}{AC+AB}\enspace\enspace\text{and}\enspace\enspace\frac{A_1L}{LB_1}=\frac{AC+AB}{BC+AB}.If BKBK and CLCL meet at SS, prove that ∠C1A1S=∠B1A1S\angle C_1A_1S=\angle B_1A_1S.
geometryratio
stingin two squares connected by a curve

Source: Mongolian MO 2002 Teachers P6

4/9/2021
Two squares of area 3838 are given. Each of the squares is divided into 3838 connected pieces of unit area by simple curves. Then the two squares are patched together. Show that one can sting the patched squares with 3838 needles so that every piece of each square is stung exactly once.
geometrygame