MathDB

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Part of 2023 Durer Math Competition (First Round)

Problems(2)

group of students counting from 00-99 taking turns

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E1

5/25/2024
A group of students play the following game: they are counting one by one from 0000 to 9999 taking turns, but instead of every number they only say one of its digits. (The numbers in order are 0000, 0101, 0202, ......., meaning that one-digit numbers are regarded as two-digit numbers with a first digit 00.) One way of starting the counting could be for example 00, 11, 22, 00, 44, 00, 66, 77, 88, 9,9, 11, 11, 22, 11, 11, 55, 66, 11, 88, 11, 00, 22 etc. When they reach 9999, the counting restarts from 0000. At some point Csongor enters the room and after listening to the counting for a while, he discovers that he is able to tell what number the counting is at. How many digits has Csongor heard at least?
number theory
[ \sqrt{n} ] + [ n / [ \sqrt{n} ] ] > 2 \sqrt{n}

Source: (2022 -) 2023 XVI Dürer Math Competition Regional E+1

5/25/2024
Find all positive integers nn such that ⌊n⌋+⌊n⌊n⌋⌋>2n.\lfloor \sqrt{n} \rfloor + \left\lfloor \frac{n}{\lfloor \sqrt{n} \rfloor} \right \rfloor> 2\sqrt{n}. If kk is a real number, then ⌊k⌋\lfloor k \rfloor means the floor of kk, this is the greatest integer less than or equal to kk.
algebrainequalitiesfloor function