MathDB

4

Part of 2011 Tuymaada Olympiad

Problems(4)

In a 100-digit n, the decimal period length of 1/n > 2011

Source: XVIII Tuymaada Mathematical Olympiad (2011), Junior Level

7/29/2011
Prove that, among 100000100000 consecutive 100100-digit positive integers, there is an integer nn such that the length of the period of the decimal expansion of 1n\frac1n is greater than 20112011.
Eulerquadraticsnumber theory unsolvednumber theory
Are the squares closest to a point on a chessboard connected

Source: XVIII Tuymaada Mathematical Olympiad (2011), Junior Level

7/29/2011
The Duke of Squares left to his three sons a square estate, 100×100100\times 100 square miles, made up of ten thousand 1×11\times 1 square mile square plots. The whole estate was divided among his sons as follows. Each son was assigned a point inside the estate. A 1×11\times 1 square plot was bequeathed to the son whose assigned point was closest to the center of this square plot. Is it true that, irrespective of the choice of assigned points, each of the regions bequeathed to the sons is connected (that is, there is a path between every two of its points, never leaving the region)?
Dukecollegegeometry unsolvedgeometry
100 cubes & 10 fourths powers implies 2000 squares

Source: XVIII Tuymaada Mathematical Olympiad (2011), Senior Level

7/29/2011
In a set of consecutive positive integers, there are exactly 100100 perfect cubes and 1010 perfect fourth powers. Prove that there are at least 20002000 perfect squares in the set.
geometry3D geometrynumber theory unsolvednumber theory
Quadratic P with divisors of P(1), P(2),... increasing

Source: XVIII Tuymaada Mathematical Olympiad (2011), Senior Level

7/29/2011
Let P(n)P(n) be a quadratic trinomial with integer coefficients. For each positive integer nn, the number P(n)P(n) has a proper divisor dnd_n, i.e., 1<dn<P(n)1<d_n<P(n), such that the sequence d1,d2,d3,…d_1,d_2,d_3,\ldots is increasing. Prove that either P(n)P(n) is the product of two linear polynomials with integer coefficients or all the values of P(n)P(n), for positive integers nn, are divisible by the same integer m>1m>1.
quadraticsalgebrapolynomialAnalytic Number TheoryInteger sequenceInteger Polynomial