2

Part of 2008 All-Russian Olympiad

Problems(3)

Cubic polynomial has root in segment [0,2]

Source: All-Russian Olympiad 2008 9.2,11.1

a,b,c

are such that the equation x^3 \plus{} ax^2 \plus{} bx \plus{} c has three real roots.Prove that if \minus{} 2\leq a \plus{} b \plus{} c\leq 0,then at least one of these roots belongs to the segment

[0,2]

algebrapolynomialalgebra proposed

arrangement of good cells

Source: All Russian 2008, Grade 10, Problem 2, Day 1

The columns of an

n\times n

board are labeled

1

n

1,2,...,n

are arranged in the board so that the numbers in each row and column are pairwise different. We call a cell "good" if the number in it is greater than the label of its column. For which

n

is there an arrangement in which each row contains equally many good cells?

modular arithmeticgraph theorycombinatorics unsolvedcombinatorics

Find the smallest possible value

Source: All Russian Mathematical Olympiad 2008. 11.2

Petya and Vasya are given equal sets of

N

weights, in which the masses of any two weights are in ratio at most

1.25

. Petya succeeded to divide his set into

10

groups of equal masses, while Vasya succeeded to divide his set into

11

groups of equal masses. Find the smallest possible

N

ratioalgebra proposedalgebra