MathDB

P1

Part of 2022/2023 Tournament of Towns

Problems(8)

Large root for quadratic polynomial

Source: 44th International Tournament of Towns, Senior A-Level P1, Fall 2022

2/16/2023
What is the largest possible rational root of the equation ax2+bx+c=0ax^2 + bx + c = 0{} where a,ba, b and cc{} are positive integers that do not exceed 100100{}?
algebraTournament of Towns
One kilometer is approximatively 0,62 miles

Source: 44th International Tournament of Towns, Junior A-Level P1, Fall 2022

2/16/2023
One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is 1000 km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? (Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).
combinatoricsgeometryTournament of Towns
Filling a 6x6 table

Source: 44th International Tournament of Towns, Junior O-Level P1, Fall 2022

2/16/2023
Is it possible to arrange 3636 distinct numbers in the cells of a 6×66 \times 6 table, so that in each 1×51\times 5 rectangle (both vertical and horizontal) the sum of the numbers equals 20222022 or 20232023?
Tournament of Townscombinatorics
Factorial times factorial equals factorial

Source: 44th International Tournament of Towns, Senior O-Level P1, Fall 2022

2/16/2023
Find the maximum integer mm such that m!⋅2022!m! \cdot 2022! is a factorial of an integer.
number theoryfactorialTournament of Towns
Sequences of letters

Source: 44th International Tournament of Towns, Senior A-Level P1, Spring 2023

4/4/2023
There are two letter sequences AA and BB, both with length 100100 letters. In one move you can insert in any place of sequence ( possibly to start or to end) any number of same letters or remove any number of consecutive same letters. Prove that it is possible to make second sequence from first sequence using not more than 100100 moves.
combinatorics
Length of bisectors

Source: 44th International Tournament of Towns, Junior A-Level P1, Spring 2023

1/9/2024
A right-angled triangle has an angle equal to 30∘.30^\circ. Prove that one of the bisectors of the triangle is twice as short as another one.
Egor Bakaev
geometryangle bisector
Moving dice around

Source: 44th International Tournament of Towns, Senior O-Level P1 and Junior O-Level P3, Spring 2023

1/9/2024
There are 2023 dice on the table. For 1 dollar, one can pick any dice and put it back on any of its four (other than top or bottom) side faces. How many dollars at a minimum will guarantee that all the dice have been repositioned to show equal number of dots on top faces?
Egor Bakaev
combinatoricsDice
Mess-loving clerks have lunch

Source: 44th International Tournament of Towns, Junior O-Level P1, Spring 2023

1/9/2024
There are NN{} mess-loving clerks in the office. Each of them has some rubbish on the desk. The mess-loving clerks leave the office for lunch one at a time (after return of the preceding one). At that moment all those remaining put half of rubbish from their desks on the desk of the one who left. Can it so happen that after all of them have had lunch the amount of rubbish at the desk of each one will be the same as before lunch if a) N=2N = 2{} and b) N=10N = 10?
Alexey Zaslavsky
combinatorics