MathDB

Maximize sum over x_i*x_{i+2}

Source: IMO Shortlist 2010, Algebra 3

7/17/2011

Let

x_1, \ldots , x_{100}

be nonnegative real numbers such that

x_i + x_{i+1} + x_{i+2} \leq 1

for all

i = 1, \ldots , 100

(we put

x_{101 } = x_1, x_{102} = x_2).

Find the maximal possible value of the sum

S = \sum^{100}_{i=1} x_i x_{i+2}.

Proposed by Sergei Berlov, Ilya Bogdanov, Russia

inequalitiesIMO Shortlistn-variable inequality

2500 chess kings have to be placed on a chessboard

Source: IMO Shortlist 2010, Combinatorics 3

7/17/2011

2500 chess kings have to be placed on a

100 \times 100

chessboard so that
(i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); (ii) each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
Proposed by Sergei Berlov, Russia

matrixcombinatoricsChessboardcountingIMO Shortlist

IMO Shortlist 2010 - Problem G3

Source:

7/17/2011

Let

A_1A_2 \ldots A_n

be a convex polygon. Point

P

inside this polygon is chosen so that its projections

P_1, \ldots , P_n

onto lines

A_1A_2, \ldots , A_nA_1

respectively lie on the sides of the polygon. Prove that for arbitrary points

X_1, \ldots , X_n

on sides

A_1A_2, \ldots , A_nA_1

respectively,

\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.

Proposed by Nairi Sedrakyan, Armenia

geometrycircumcircleInequalitypolygonIMO Shortlist

IMO Shortlist 2010 - Problem N3

Source:

7/17/2011

Find the smallest number

n

such that there exist polynomials

f_1, f_2, \ldots , f_n

with rational coefficients satisfying

x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.

Proposed by Mariusz Skałba, Poland

algebrapolynomialnumber theorySum of SquaresIMO Shortlist

3

Problems(4)