MathDB

Wizard card game

Source: 43rd International Tournament of Towns, Senior A-Level P1, Fall 2021

2/18/2023

The wizards

A, B, C, D

know that the integers

1, 2, \ldots, 12

are written on 12 cards, one integer on each card, and that each wizard will get three cards and will see only his own cards. Having received the cards, the wizards made several statements in the following order.
[*]“One of my cards contains the number 8”. [*]“All my numbers are prime”. [*]“All my numbers are composite and they all have a common prime divisor”. [*]“Now I know all the cards of each wizard”.
What were the cards of

A{}

if everyone was right?
Mikhail Evdokimov

combinatoricscardsTournament of Towns

Alice and Bob write numbers

Source: 43rd International Tournament of Towns, Junior A-Level P1, Fall 2021

2/18/2023

Alice wrote a sequence of

n > 2

nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those n numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the same as in Alice’s sequence. What are the possible values of

n{}

?
Alexey Zaslavsky

combinatoricsalgebraTournament of Towns

Consecutive interesting integers

Source: 43rd International Tournament of Towns, Senior O-Level P1, Fall 2021

2/18/2023

Let us call a positive integer

k{}

interesting if the product of the first

k{}

primes is divisible by

k{}

. For example the product of the first two primes is

2\cdot3 = 6

, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers?
Boris Frenkin

number theoryprime numbersTournament of Towns

Edition number divides the year

Source: 43rd International Tournament of Towns, Junior O-Level P1, Fall 2021

2/18/2023

The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament:

2021\div 43 = 47

. How many times more will the humanity witness such a wonderful event?
Alexey Zaslavsky

number theoryTournament of Towns

2022 TT spring SA1

Source: 2022 Tournament of towns Spring

5/11/2022

For each of the

9

positive integers

n,2n,3n,\dots , 9n

Alice take the first decimal digit (from the left) and writes it onto a blackboard. She selected

n

so that among the nine digits on the blackboard there is the least possible number of different digits. What is this number of different digits equals to?

number theory

2022TTsJA1

Source: 2022 Tournament of towns Spring A level Junior

5/18/2022

Find the largest positive integer

n

such that for each prime

p

with

2<p<n

the difference

n-p

is also prime.

number theoryTournament of Towns

2022 TT spring SO1

Source: 2022 Tournament of towns Spring Senior O level

5/11/2022

Peter picked a positive integer, multiplied it by 5, multiplied the result by 5,then multiplied the result by 5 again and so on. Altogether

k

multiplications were made. It so happened that the decimal representations of the original number and of all

k

resulting numbers in this sequence do not contain digit

7

. Prove that there exists a positive integer such that it can be multiplied

k

times by

2

so that no number in this sequence contains digit

7

.

number theory

2022 TTsJO1

Source: 2022 Tournament of towns Spring Junior O level

5/14/2022

Two friends walked towards each other along a straight road. Each had a constant speed but one was faster than the other. At one moment each friend released his dog to run freely forward, the speed of each dog is the same and constant. Each dog reached the other person and then returned to its owner. Which dog returned to its owner the first, of the person who walks fast or who walks slow?

P1

Problems(8)