MathDB

3

Part of 2000 Tournament Of Towns

Problems(8)

coloring 2n vertices of a prism with base n-gon, into 3 colours

Source: Tournament Of Towns Spring 2000 Junior 0 Level p3

4/22/2020
The base of a prism is an nn-gon. We wish to colour its 2n2n vertices in three colours in such a way that every vertex is connected by edges to vertices of all three colours. (a) Prove that if nn is divisible by 33, then the task is possible. {b) Prove that if the task is possible, then nn is divisible by 33.
(A Shapovalov)
prismcombinatorial geometryColoringcombinatorics
TOT 2000 Spring AJ3 locus, rectangle and circle related

Source:

5/10/2020
AA is a fixed point inside a given circle. Determine the locus of points CC such that ABCDABCD is a rectangle with BB and DD on the circumference of the given circle.
(M Panov)
geometryrectangleLocuscirclefixed
TOT 2000 Spring OS3 1^k+2^k+...+n^k \le (n^{2k}-(n-1)^k)/(n^k-(n-1)^k)

Source:

5/11/2020
Prove the inequality 1k+2k+...+nk≤n2k−(n−1)knk−(n−1)k 1^k+2^k+...+n^k \le \frac{n^{2k}-(n-1)^k}{n^k-(n-1)^k} (L Emelianov)
inequalitiesalgebraSum of powers
TOT 2000 Spring AS3 Peter always loses in a solitaire game

Source:

5/11/2020
Peter plays a solitaire game with a deck of cards, some of which are face-up while the others are face-down. Peter loses if all the cards are face-down. As long as at least one card is face up, Peter must choose a stack of consecutive cards from the deck, so that the top and the bottom cards of the stack are face-up. They may be the same card. Then Peter turns the whole stack over and puts it back into the deck in exactly the same place as before. Prove that Peter always loses.
(A Shapovalov)
combinatoricsgamegame strategy
TOT 2000 Autumn OJ3 100 numbers on a blackboard

Source:

5/10/2020
(a) On a blackboard are written 100100 different numbers. Prove that you can choose 88 of them so that their average value is not equal to that of any 99 of the numbers on the blackboard. (b) On a blackboard are written 100100 integers. For any 88 of them, you can find 99 numbers on the blackboard so that the average value of the 88 numbers is equal to that of the 99. Prove that all the numbers on the blackboard are equal.
(A Shapovalov)
combinatoricsAverage
TOT 2000 Autumn AJ3 lcm (a,b,c,d)=a+b+c+d

Source:

5/10/2020
The least common multiple of positive integers a,b,ca, b, c and dd is equal to a+b+c+da + b + c + d. Prove that abcdabcd is divisible by at least one of 33 and 55.
( V Senderov)
least common multipleLCMnumber theorySumProductdivisibledivides
TOT 2000 Autumn OS3 angle on lateral face of a pentagonal prism

Source:

5/11/2020
In each lateral face of a pentagonal prism at least one of the four angles is equal to ff. Find all possible values of ff.
(A Shapovalov)
3D geometryprismgeometryangles
TOT 2000 Autumn AS3 ratio of segments wanted

Source:

5/11/2020
In a triangle ABC,AB=c,BC=a,CA=bABC, AB = c, BC = a, CA = b, and a<b<ca < b < c. Points B′B' and A′A' are chosen on the rays BCBC and ACAC respectively so that BB′=AA′=cBB'= AA'= c. Points C′′C'' and B′′B'' are chosen on the rays CACA and BABA so that CC′′=BB′′=aCC'' = BB'' = a. Find the ratio of the segment A′B′A'B' to the segment C′′B′′C'' B''.
(R Zhenodarov)
ratioequal segmentsgeometry