MathDB

4

Part of 2019 Teodor Topan

Problems(4)

[2019] Ordered Inequality of an Odd nr. of Variables

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12/16/2019
Let be an odd natural number n, n, and n n real numbers y1≤y2≤⋯≤yn y_1\le y_2\le\cdots\le y_n whose sum is 0. 0. Prove that (n+2)yn+122≤y12+y22+⋯+yn2, (n+2)y_{\frac{n+1}{2}}^2\le y_1^2+y_2^2+\cdots +y_n^2, and specify where equality is attained.
Nicolae Bourbăcuț
inequalitiesn-variable inequality
Nice game theory

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12/16/2019
Ana choses two real numbers y>0,x y>0,x and Bogdan repeatedly tries to guess these in the following manner: at step j j he choses a real number bj, b_j, asks her if bj=x+jy, b_j=x+jy, and she tells him the truth.
a) If x=0, x=0, can Bogdan find Ana's numbers in a finite number of steps? b) If x≠0, x\neq 0, can Bogdan find Ana's numbers in a finite number of steps?
Discretegames
Strange problem

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12/16/2019
Calculate the minimum value of tr(AtA), \text{tr} (A^tA) , where A A in the cases where is a matrix of pairwise distinct nonnegative integers and:
a) det⁡A≡1(mod2) \det A\equiv 1\pmod 2 b) det⁡A=0 \det A=0
Vlad Mihaly
linear algebramatrixtranspose
Characterization of finite groups with injective maps

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12/16/2019
Let S S be a finite [url=https://en.wikipedia.org/wiki/Cancellation_property]cancellative semigroup.
a) Prove that S S contains an idempotent element. b) Prove that S S is a group. c) Disprove subpoint b) in the case that S S would not be finite.
Vlad Mihaly
group theory