MathDB

2

Part of 2012 Tournament of Towns

Problems(8)

2012 ToT Spring Junior O p2 odd no of even positive divisors

Source:

3/4/2020
The number 44 has an odd number of odd positive divisors, namely 11, and an even number of even positive divisors, namely 22 and 44. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?
number theoryDivisors
2012 ToT Spring Junior A p2 100 points in the plane, no 3 collinear

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3/4/2020
One hundred points are marked in the plane, with no three in a line. Is it always possible to connect the points in pairs such that all fi fty segments intersect one another?
pointscombinatorial geometrycombinatorics
2012 ToT Spring Senior O p2 no 1,2,...,, n, -n,..., -2, -1 in a 1x 2n board

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3/5/2020
The cells of a 1\times 2n board are labelled 1,2,...,,n,−n,...,−2,−11,2,...,, n, -n,..., -2, -1 from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then 2n+12n + 1 is prime.
primecombinatoricstablerectangle table
2012 ToT Spring Senior A p2 100 points inside a circle, no 3 collinear

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3/22/2020
One hundred points are marked inside a circle, with no three in a line. Prove that it is possible to connect the points in pairs such that all fifty lines intersect one another inside the circle.
pointscirclecombinatoricscombinatorial geometry
2012 ToT Fall Junior A p2 Chip and Dale play a game with 222 nuts

Source:

3/22/2020
Chip and Dale play the following game. Chip starts by splitting 222222 nuts between two piles, so Dale can see it. In response, Dale chooses some number NN from 11 to 222222. Then Chip moves nuts from the piles he prepared to a new (third) pile until there will be exactly NN nuts in any one or two piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).
combinatoricsgame
2012 ToT Fall Junior O p2 subset with number of prime divisors

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3/22/2020
Let C(n)C(n) be the number of prime divisors of a positive integer n. (For example, C(10)=2,C(11)=1,C(12)=2C(10) = 2,C(11) = 1, C(12) = 2). Consider set S of all pairs of positive integers (a,b)(a, b) such that a≠ba\ne b and C(a+b)=C(a)+C(b)C(a + b) = C(a) + C(b). Is set SS finite or infinite?
number theoryprimeDivisors
2012 ToT Fall Senior A p2 1001 nuts into 3 piles game

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3/22/2020
Chip and Dale play the following game. Chip starts by splitting 10011001 nuts between three piles, so Dale can see it. In response, Dale chooses some number NN from 11 to 10011001. Then Chip moves nuts from the piles he prepared to a new (fourth) pile until there will be exactly NN nuts in any one or more piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).
game strategygamecombinatorics
2012 ToT Fall Senior O p2 all faces of polyhedron are congruent / regular

Source:

3/22/2020
Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are (a) congruent polygons? (b) regular polygons?
convex polyhedronregular polygoncongruentcombinatorial geometrygeometry