MathDB

4

Part of 2019 Durer Math Competition Finals

Problems(3)

sum of ratios wanted, similar isosceles triangles on sides of a triangle

Source: 2019 Dürer Math Competition Finals E1.4

11/28/2020
Let ABCABC be an acute-angled triangle having angles α,β,γ\alpha,\beta,\gamma at vertices A,B,CA, B, C respectively. Let isosceles triangles BCA1,CAB1,ABC1BCA_1, CAB_1, ABC_1 be erected outwards on its sides, with apex angles 2α,2β,2γ2\alpha ,2\beta ,2\gamma respectively. Let A2A_2 be the intersection point of lines AA1AA_1 and B1C1B_1C_1 and let us define points B2B_2 and C2C_2 analogously. Find the exact value of the expression AA1A2A1+BB1B2B1+CC1C2C1\frac{AA_1}{A_2A_1}+\frac{BB_1}{B_2B_1}+\frac{CC_1}{C_2C_1}
ratiogeometryisosceles
2 tram lines in Miskolc

Source: 2019 Dürer Math Competition Finals Day2 E4 https://artofproblemsolving.com/community/c1621835_2019_

1/5/2022
In Miskolc there are two tram lines: line 11 runs between Tiszai railway station and UpperMajláth, while line 22 runs between Tiszai railway station and the Ironworks. The timetable for trams leaving Tiszai railway station is as follows: tram 1 1 leaves at every minute ending in a 00 or 66, and tram 22 leaves at every minute ending in a 33. There are three types of passengers waiting for the trams: those who will take tram 1 1 only, those who will take tram 22 only and those who will take any tram. Every minute there is a constant number of passengers of each type arriving at the station. (This number is not necessarily the same for the different types.) Also, every tram departs with an equal number of passengers from Tiszai railway station. How many passengers are there on a departing tram, if we know that every minute there are 33 passengers arriving at the station who will take tram 22 only?
combinatorics
7 numbers are drawn out of 55 in the Intergalactic Lottery

Source: 2019 Dürer Math Competition Finals E+1.4

11/28/2020
In the Intergalactic Lottery, 77 numbers are drawn out of 5555. R2-D2 and C-3PO decide that they want to win this lottery, so they fill out lottery tickets separately such that for each possible draw one of them does have a winning ticket for that draw. Prove that one of them has 77 tickets with all different numbers.
combinatorics