MathDB

1

Part of 2001 All-Russian Olympiad

Problems(5)

Partitioning integers from 1 to 999999

Source: All-Russian MO 2001 Grade 9 #1; Grade 10 #1

1/2/2012
The integers from 11 to 999999999999 are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even. In which group is the sum of the elements greater?
ceiling functionfloor functionnumber theory unsolvednumber theory
2001 Kopeyka Coins

Source: All-Russian MO 2001 Grade 9 #5

1/2/2012
Yura put 20012001 coins of 11, 22 or 33 kopeykas in a row. It turned out that between any two 11-kopeyka coins there is at least one coin; between any two 22-kopeykas coins there are at least two coins; and between any two 33-kopeykas coins there are at least 33 coins. How many 33-koyepkas coins could Yura put?
combinatorics unsolvedcombinatorics
Polynomial Relationships

Source: MOP 2002

7/12/2008
The polynomial P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d has three distinct real roots. The polynomial P(Q(x)) P(Q(x)), where Q(x)\equal{}x^2\plus{}x\plus{}2001, has no real roots. Prove that P(2001)>164 P(2001)>\frac{1}{64}.
algebrapolynomialquadraticsquadratic formula
100 given weights

Source: All-Russian MO 2001 Grade 11 #1

1/3/2012
The total mass of 100100 given weights with positive masses equals 2S2S. A natural number kk is called middle if some kk of the given weights have the total mass SS. Find the maximum possible number of middle numbers.
inductioncombinatorics unsolvedcombinatorics
Two monic trinomials

Source: All-Russian MO 2001 Grade 11 #5

1/3/2012
Two monic quadratic trinomials f(x)f(x) and g(x)g(x) take negative values on disjoint intervals. Prove that there exist positive numbers α\alpha and β\beta such that αf(x)+βg(x)>0\alpha f(x) + \beta g(x) > 0 for all real xx.
quadraticsalgebrapolynomialalgebra unsolved