MathDB

7

Part of 2006 Tournament of Towns

Problems(4)

Numbers on a Table

Source: Spring 2006 Tournament of Towns Junior A-Level #7

4/15/2015
Anna and Boris have the same copy of 5×55\times5 table filled with 2525 distinct numbers. After choosing the maximal number in the table, Anna erases the row and the column that contain this number. Then she continue the same operations with a smaller table till it is possible.Boris basically does the same; however, each time choosing the minimal number in a table. Can it happen that the total sum of the numbers chosen by Boris a) is greater than the total sum of the numbers chosen by Anna? (6 points) b) is greater than the total sum of any 55 numbers of initial table given that no two of the numbers are in the same row or in the same column? (2 points)
TOT 2006 Spring - Senior A-Level p7 ant along dodecahedron

Source:

2/25/2020
An ant craws along a closed route along the edges of a dodecahedron, never going backwards. Each edge of the route is passed exactly twice. Prove that one of the edges is passed both times in the same direction. (Dodecahedron has 1212 faces in the shape of pentagon, 3030 edges and 2020 vertices; each vertex emitting 3 edges). (8)
combinatoricsgeometry3D geometrydodecahedron
TOT 2006 Fall - Junior A-Level p7 Magician has a deck of 52 cards

Source:

2/25/2020
A Magician has a deck of 5252 cards. Spectators want to know the order of cards in the deck(without specifying face-up or face-down). They are allowed to ask the questions “How many cards are there between such-and-such card and such-and-such card?” One of the spectators knows the card order. Find the minimal number of questions he needs to ask to be sure that the other spectators can learn the card order. (9)
combinatorics
TOT 2006 Fall - Senior A-Level p7 sum x_i^2< (sum x_i)/n, sum x_i<(sum x_i^3)/2

Source:

2/25/2020
Positive numbers x1,...,xkx_1,..., x_k satisfy the following inequalities: x12+...+xk2<x1+...+xk2  and  x1+...+xk<x13+...+xk32x_1^2+...+ x_k^2 <\frac{x_1+...+x_k}{2} \ \ and \ \ x_1+...+x_k < \frac{x_1^3+...+ x_k^3}{2} a) Show that k>50k > 50, (3) b) Give an example of such numbers for some value of kk (3) c) Find minimum kk, for which such an example exists. (3)
inequalitiesalgebraSum