MathDB

4

Part of 2000 Tournament Of Towns

Problems(8)

TOT 2000 Spring AJ4 100 coins game

Source:

5/10/2020
Give and Take divide 100100 coins between themselves as follows. In each step, Give chooses a handful of coins from the heap, and Take decides who gets this handful. This is repeated until all coins have been taken, or one of them has 99 handfuls. In the latter case, the other gets all the remaining coins. What is the largest number of coins that Give can be sure of getting no matter what Take does?
(A Shapovalov)
combinatoricsgamegame strategy
pos. integers at all vertices of a cube such neighboor vertices 1 divides other

Source: Tournament Of Towns Spring 2000 Junior 0 Level p4

4/22/2020
Can one place positive integers at all vertices of a cube in such a way that for every pair of numbers connected by an edge, one will be divisible by the other , and there are no other pairs of numbers with this property?
(A Shapovalov)
cubecombinatoricsDivisibility
TOT 2000 Spring OS4 sum of first 10n + 1 successive numbers is negative

Source:

5/11/2020
(a) Does there exist an infinite sequence of real numbers such that the sum of every ten successive numbers is positive, while for every nn the sum of the first 10n+110n + 1 successive numbers is negative? (b) Does there exist an infinite sequence of integers with the same properties?
(AK Tolpygo)
SequenceSumalgebra
TOT 2000 Spring AS4 convex polygon on lattice points

Source:

5/11/2020
Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form x=mx = m, mm an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form y=ny = n, nn an integer, that lie within the polygon.
(G Galperin)
convex polygonlattice pointsSumcombinatorial geometrycombinatorics
TOT 2000 Autumn OJ4 32 coins , 30 real and 2 fake

Source:

5/10/2020
Among a set of 3232 coins , all identical in appearance, 3030 are real and 22 are fake. Any two real coins have the same weight . The fake coins have the same weight , which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most 44 times?
(A Shapovalov)
weighingscombinatorics
TOT 2000 Autumn AJ4 mark 31 on 8x8 chessboard

Source:

5/10/2020
In how many ways can 3131 squares be marked on an 8×88 \times 8 chessboard so that no two of the marked squares have a common side?
(R Zhenodarov)
combinatoricsChessboardtable
TOT 2000 Autumn OS4 2N coins, 2N-2 real, 2 fake

Source:

5/11/2020
Among a set of 2N2N coins, all identical in appearance, 2N−22N - 2 are real and 22 are fake. Any two real coins have the same weight . The fake coins have the same weight, which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most 44 times, if (a) N=16N = 16, ( b ) N=11N = 11 ?
(A Shapovalov)
weighingscombinatorics
TOT 2000 Autumn AS4 a_1 + 1/(a_2+1/(a_3+ ...))) = x

Source:

5/11/2020
Let a1,a2,...,ana_1 , a_2 , ..., a_n be non-zero integers that satisfy the equation a1+1a2+1a3+...1an+1x=xa_1 +\dfrac{1}{a_2+\dfrac{1}{a_3+ ... \dfrac{1}{a_n+\dfrac{1}{x}} } } = x for all values of xx for which the lefthand side of the equation makes sense. (a) Prove that nn is even. (b) What is the smallest n for which such numbers a1,a2,...,ana_1 , a_2 , ..., a_n exist?
(M Skopenko)
Evenalgebraequation