By Radmila Bulajich Manfrino

This e-book is meant for the Mathematical Olympiad scholars who desire to organize for the research of inequalities, an issue now of widespread use at a number of degrees of mathematical competitions. during this quantity we current either vintage inequalities and the extra valuable inequalities for confronting and fixing optimization difficulties. a big a part of this ebook offers with geometric inequalities and this truth makes a major distinction with admire to many of the books that take care of this subject within the mathematical olympiad. The e-book has been prepared in 4 chapters that have every one of them a unique personality. bankruptcy 1 is devoted to provide simple inequalities. so much of them are numerical inequalities more often than not missing any geometric that means. notwithstanding, the place it really is attainable to supply a geometrical interpretation, we comprise it as we cross alongside. We emphasize the significance of a few of those inequalities, reminiscent of the inequality among the mathematics suggest and the geometric suggest, the Cauchy-Schwarz inequality, the rearrangementinequality, the Jensen inequality, the Muirhead theorem, between others. For these kinds of, along with giving the facts, we current numerous examples that express the right way to use them in mathematical olympiad difficulties. We additionally emphasize how the substitution procedure is used to infer numerous inequalities.

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**Example text**

1] i=1 = ([n, 0, 0, . . , 0] − [n − 1, 1, 0, . . , 0]) + ([n − 1, 1, 0, . . , 0] − [n − 2, 1, 1, 0, . . , 0]) + ([n − 2, 1, 1, 0, . . , 0] − [n − 3, 1, 1, 1, 0, . . , 0]) + · · · + ([2, 1, 1, . . , 1] − [1, 1, . . 1]) 1 (xn−1 − xn−1 )(x1 − x2 ) = 2 ! 1 2n! (xn−2 − xn−2 )(x1 − x2 )x3 + 1 2 ! + ! (xn−3 − xn−3 )(x1 − x2 )x3 x4 + · · · . 1 2 Since (xνr − xνs )(xr − xs ) > 0, unless xr = xs , the inequality follows. 6. If a, b are positive real numbers, then a2 + b Setting x = √ √ b2 ≥ a + b.

Xn ). We will consider only the particular case F (x1 , . . , xn ) = xa1 1 xa2 2 · · · xann We write [a] = [a1 , a2 , . . , an ] = ables x, y, z > 0 we have that 1 n! [1, 1] = xy, [1, 1, 1] = xyz, [2, 1, 0] = with xi > 0, ai ≥ 0. xa1 1 xa2 2 · · · xann . For instance, for the vari1 2 [x (y + z) + y 2 (x + z) + z 2 (x + y)]. 3! 44 Numerical Inequalities It is clear that [a] is invariant under any permutation of the (a1 , a2 , . . , an ) and therefore two sets of a are the same if they only diﬀer in arrangement.

We see that the equality holds if and only if ay = bx, that is, if and only if xa = yb . Another form to prove the inequality is using the Cauchy-Schwarz inequality in the following way: (a + b)2 = a √ b √ √ x+ √ y x y 2 ≤ a2 b2 + x y (x + y). 11) for all real numbers a1 , a2 , . . , an and x1 , x2 , . . , xn > 0, with equality if and only if a1 a2 an = = ··· = . 11) is also called the Cauchy-Schwarz inequality in Engel form or Arthur Engel’s Minima Principle. As a ﬁrst application of this inequality, we will present another proof of the Cauchy-Schwarz inequality.