4
Part of 2007 Tournament Of Towns
Problems(8)
Three numbers as roots of polynomials
Source: Tournament of Towns 2007 - Spring - Junior O-Level - P4
9/2/2011
Three nonzero real numbers are given. If they are written in any order as coefficients of a quadratic trinomial, then each of these trinomials has a real root. Does it follow that each of these trinomials has a positive root?
algebrapolynomialquadraticsalgebra unsolved
Maximum number of the triangles (ToT 2007-Spring-A4)
Source:
9/2/2011
Several diagonals (possibly intersecting each other) are drawn in a convex -gon in such a way that no three diagonals intersect in one point. If the -gon is cut into triangles, what is the maximum possible number of these triangles?
Prove that the sequence is not periodic
Source: Tournament of Towns 2007 - Spring - Senior O-Level - P4
9/2/2011
A binary sequence is constructed as follows. If the sum of the digits of the positive integer is even, the -th term of the sequence is . Otherwise, it is . Prove that this sequence is not periodic.
number theory unsolvednumber theory
On a 29 × 29 board with numbers
Source: Tournament of Towns 2007 - Fall - Junior O-Level - P4
9/3/2011
Each cell of a table contains one of the integers , and each of these integers appears times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
combinatorics proposedcombinatorics
Entering symbols on a 1 × n table
Source:
9/3/2011
Two players take turns entering a symbol in an empty cell of a chessboard, where is an integer greater than . Aaron always enters the symbol and Betty always enters the symbol . Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes first, which player has a winning strategy?
Two Numbers Sum to 65
Source: Fall 2007 Tournament of Towns Junior P-Level #4
3/31/2015
From the first 64 positive integers are chosen two subsets with 16 numbers in each. The first subset contains only odd numbers while the second one contains only even numbers. Total sums of both subsets are the same. Prove that among all the chosen numbers there are two whose sum equals 65.(3 points)
Choosing two of twenty-nine cards numbered from 1 to 29
Source: Tournament of Towns 2007 - Fall - Senior O-Level - P4
9/4/2011
The audience chooses two of twenty-nine cards, numbered from to respectively. The assistant of a magician chooses two of the remaining twenty-seven cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in an arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.
combinatorics unsolvedcombinatorics
105 Piles of Pebbles
Source: Fall 2007 Tournament of Towns Senior P-Level #4
3/31/2015
There three piles of pebbles, containing 5, 49, and 51 pebbles respectively. It is allowed to combine any two piles into a new one or to split any pile consisting of even number of pebbles into two equal piles. Is it possible to have 105 piles with one pebble in each in the end?(3 points)
number theorycombinatoricsinvariant