MathDB

2

Part of 2020 Tournament Of Towns

Problems(5)

TOT O level Problem 2

Source:

6/3/2020
What is the maximum number of distinct integers in a row such that the sum of any 11 consequent integers is either 100 or 101 ? What~ is~ the~ maximum~ number~ of~ distinct~ integers~ in~ a~ row~ such~ that~ the~sum~ of~ any~ 11~ consequent~ integers~ is~ either~ 100~ or~ 101~?
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ToT
3 legendary knights against a multiheaded dragon with 41! heads

Source: Tournament of Towns, Junior A-Level Paper, Spring 2020 , p2

6/10/2020
Three legendary knights are fighting against a multiheaded dragon. Whenever the first knight attacks, he cuts off half of the current number of heads plus one more. Whenever the second knight attacks, he cuts off one third of the current number of heads plus two more. Whenever the third knight attacks, he cuts off one fourth of the current number of heads plus three more. They repeatedly attack in an arbitrary order so that at each step an integer number of heads is being cut off. If all the knights cannot attack as the number of heads would become non-integer, the dragon eats them. Will the knights be able to cut off all the dragon’s heads if it has 41!41! heads?
Alexey Zaslavsky
number theoryfactorialcombinatoricsgame
minimal number of places where Alice has spent the New Year

Source: Tournament of Towns, Senior Ο-Level Paper, Spring 2020 , p2

6/4/2020
Alice asserts that after her recent visit to Addis-Ababa she now has spent the New Year inside every possible hemisphere of Earth except one. What is the minimal number of places where Alice has spent the New Year?
Note: we consider places of spending the New Year to be points on the sphere. A point on the border of a hemisphere does not lie inside the hemisphere.
Ilya Dumansky, Roman Krutovsky
combinatorics
lcm 2 player game a = lcm (x, y), b = lcm(x, z), c = lcm(y, z)

Source: Tournament of Towns, Senior A-Level Paper, Spring 2020 , p2

6/3/2020
Alice had picked positive integers a,b,ca, b, c and then tried to find positive integers x,y,zx, y, z such that a=lcm(x,y)a = lcm (x, y), b=lcm(x,z)b = lcm(x, z), c=lcm(y,z)c = lcm(y, z). It so happened that such x,y,zx, y, z existed and were unique. Alice told this fact to Bob and also told him the numbers aa and bb. Prove that Bob can find cc. (Note: lcm = least common multiple.)
Boris Frenkin
LCMnumber theoryleast common multiple
circumcenter of A_1B_1C_1 is incenter of ABC, AA_1 = BB_1 = CC_1 = R,

Source: Tournament of Towns 2020 oral p2 (15 March 2020)

5/17/2020
At heights AA0,BB0,CC0AA_0, BB_0, CC_0 of an acute-angled non-equilateral triangle ABCABC, points A1,B1,C1A_1, B_1, C_1 were marked, respectively, so that AA1=BB1=CC1=RAA_1 = BB_1 = CC_1 = R, where RR is the radius of the circumscribed circle of triangle ABCABC. Prove that the center of the circumscribed circle of the triangle A1B1C1A_1B_1C_1 coincides with the center of the inscribed circle of triangle ABCABC.
E. Bakaev
circumcirclegeometryincentercircumradiusaltitudes