MathDB

5

Part of 2021 Durer Math Competition Finals

Problems(4)

tiles : 2 1x4, 4 1x1, 6 1x1, 8 1x1

Source: 2021 Dürer Math Competition Finals Day 1 E5 , E+4

1/2/2022
A torpedo set consists of 22 pieces of 1×41 \times 4, 44 pieces of 1×31 \times 3, 66 pieces of 1×21 \times 2 and 8 8 pieces of 1×11 \times 1 ships. a) Can one put the whole set to a 10×1010 \times 10 table so that the ships do not even touch with corners? (The ships can be placed both horizontally and vertically.) b) Can we solve this problem if we change 44 pieces of 1×11 \times 1 ships to 33 pieces of 1×21 \times 2 ships? c) Can we solve the problem if we change the remaining 44 pieces of 1×11 \times 1 ships to one piece of 1×31 \times 3 ship and one piece of 1×21 \times 2 ship? (So the number of pieces are 2,5,10,02, 5, 10, 0.)
combinatoricstilesTiling
(2x^3 - 6x^2 - 3x -20)/5(x - 4) is an integer

Source: 2021 Dürer Math Competition Finals Day2 E5 https://artofproblemsolving.com/community/c2749870_

1/8/2022
How many integers 1≤x≤20211\le x \le 2021 make the value of the expression 2x3−6x2−3x−205(x−4)\frac{2x^3 - 6x^2 - 3x -20}{5(x - 4)} an integer?
number theoryInteger
every divisors of 2n^2 - 1 gives a different remainder after division by 2n

Source: 2021 Dürer Math Competition Finals Day 1 E+5

1/2/2022
Let nn be a positive integer. Show that every divisors of 2n2−12n^2 - 1 gives a different remainder after division by 2n2n.
number theoryremainderdivisor
special exam in 3 different areas for a sheriff

Source: 2021 Dürer Math Competition Finals Day2 E+5 https://artofproblemsolving.com/community/c2749870_

1/9/2022
Joe, who is already feared by all bandits in the Wild West, would like to officially become a sheriff. To do that, he has to take a special exam where he has to demonstrate his talent in three different areas: tracking, shooting and lasso throwing. He successfully completes each task with a given probability, independently of each other. He passes the exam if he can complete at least two of the tasks successfully. Joe calculated that in case he starts with tracking and completes it successfully, his chance of passing the exam is 32%32\%. If he starts with successful shooting, the chance of passing is 49%49\%, whereas if he starts with successful lasso throwing, he passes with probability 52%52\%. The overall probability of passing (calculated before the start of the exam) is X/1000X/1000 . What is the value of XX?
combinatoricsprobability