MathDB

4

Part of 2012 Tuymaada Olympiad

Problems(4)

Divisibility of a sum by a prime p

Source: Tuymaada 2012, Problem 4, Day 1, Seniors

7/20/2012
Let p=4k+3p=4k+3 be a prime. Prove that if 102+1+112+1+⋯+1(p−1)2+1=mn\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n} (where the fraction mn\dfrac {m} {n} is in reduced terms), then p∣2m−np \mid 2m-n.
Proposed by A. Golovanov
quadraticsmodular arithmeticcalculusalgebrapolynomial
Edge labeling on oriented graph (Tutte related)

Source: Tuymaada 2012, Problem 8, Day 2, Seniors

7/21/2012
Integers not divisible by 20122012 are arranged on the arcs of an oriented graph. We call the weight of a vertex the difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by 20122012. Prove that non-zero integers with absolute values not exceeding 20122012 can be arranged on the arcs of this graph, so that the weight of each vertex is zero.
Proposed by W. Tutte
abstract algebragraph theorycombinatorics proposedcombinatorics
Divisibility of a sum by a prime p

Source: Tuymaada 2012, Problem 4, Day 1, Juniors

7/21/2012
Let p=1601p=1601. Prove that if 102+1+112+1+⋯+1(p−1)2+1=mn,\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n}, where we only sum over terms with denominators not divisible by pp (and the fraction mn\dfrac {m} {n} is in reduced terms) then p∣2m+np \mid 2m+n.
Proposed by A. Golovanov
quadraticsmodular arithmeticnumber theorynumber theory proposed
Coloured balloons and pots

Source: Tuymaada 2012, Problem 8, Day 2, Juniors

7/21/2012
2525 little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that successive donkeys receive balloons of different colours, and so that at least one balloon of each colour is given to some donkey. Eeyore wants to give to each of the 2424 remaining donkeys a pot of one of six colours of the rainbow (except red), so that at least one pot of each colour is given to some donkey (but successive donkeys can receive pots of the same colour). Which of the two friends has more ways to get his plan implemented, and how many times more?
Eeyore is a character in the Winnie-the-Pooh books by A. A. Milne. He is generally depicted as a pessimistic, gloomy, depressed, old grey stuffed donkey, who is a friend of the title character, Winnie-the-Pooh. His name is an onomatopoeic representation of the braying sound made by a normal donkey. Of course, Winnie-the-Pooh is a fictional anthropomorphic bear.
Proposed by F. Petrov
combinatorics proposedcombinatorics