MathDB

4-digit number

Source: (2022-) 2023 XVI 16th Dürer Math Competition Finals Day 2 E3

5/25/2024

Which is the largest four-digit number that has all four of its digits among its divisors and its digits are all different?

number theoryDigits

4 merchants want to travel from Athens to Rome by cart.

Source: (2022 -) 2023 XVI Dürer Math Competition Finals Day 1 E3

5/25/2024

a) Four merchants want to travel from Athens to Rome by cart. On the same day, but different times they leave Athens and arrive on another day to Rome, but in reverse order. Every day, when the evening comes, each merchant enters the next inn on the way. When some merchants sleep in the same inn at night, then on the following day at dawn they leave in reverse order of arrival, because they can only park this way on the narrow streets next to the inns. They cannot overtake each other, their order only changes after a night spent together in the same inn. Eventually each merchant arrives in Rome while they sleep with every other merchant in the same inn exactly once. Is it possible, that the number of the inns they sleep in is even every night?
b) Is it possible if there are

8

merchants instead of

4

and every other condition is the same?

combinatorics

y = 1000/(x^2+100)

Source: (2022-) 2023 XVI 16th Dürer Math Competition Finals Day 2 E+3

5/25/2024

Hapi, the god of the annual flooding of the Nile is preparing for this year’s flooding. The shape of the channel of the Nile can be described by the function

y = \frac{-1000}{ x^2+100}

where the

x

and

y

coordinates are in metres. The depth of the river is

5

metres now. Hapi plans to increase the water level by

3

metres. How many metres wide will the river be after the flooding? The depth of the river is always measured at its deepest point. https://cdn.artofproblemsolving.com/attachments/8/3/4e1d277e5cacf64bf82c110d521747592b928e.png

algebra

3

Problems(3)