MathDB

4

Part of 2015 Romania National Olympiad

Problems(4)

Asymmetrical acyclical inequality

Source: Romania NMO 2015,9th grade,Problem 4

4/25/2015
Let a,b,c,d≥0a,b,c,d \ge 0 real numbers so that a+b+c+d=1a+b+c+d=1.Prove that a+(b−c)26+(c−d)26+(d−b)26+b+c+d≤2.\sqrt{a+\frac{(b-c)^2}{6}+\frac{(c-d)^2}{6}+\frac{(d-b)^2}{6}} +\sqrt{b}+\sqrt{c}+\sqrt{d} \le 2.
inequalitiesalgebra
Inequality of cardinals

Source: Romania National Olympiad 2015, grade x, p.4

8/23/2019
Let be a finite set A A of real numbers, and define the sets S±={x±y∣x,y∈A}. S_{\pm }=\{ x\pm y| x,y\in A \} . Show that ∣A∣⋅∣S−∣≤∣S+∣2. \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 .
inequalitiesalgebracombinatoricsDiscrete
Strange problem about matrices

Source: Romanian National Olympiad 2015, grade xi, p.4

8/23/2019
Let be three natural numbers k,m,n k,m,n an m×n m\times n matrix A, A, an n×m n\times m matrix B, B, and k k complex numbers a0,a1,…,ak a_0,a_1,\ldots ,a_k such that the following conditions hold.
\text{(i)}  m\ge n\ge 2 \text{(ii)}  a_0I_m+a_1AB+a_2(AB)^2+\cdots +a_k(AB)^k=O_m \text{(iii)}  a_0I_m+a_1BA+a_2(BA)^2+\cdots +a_k(BA)^k\neq O_n
Prove that a0=0. a_0=0.
linear algebramatrixcomplex numbers
Pol. whose antider. of the recipr. of their assoc. f are frac. of rational pol.

Source: Romania National Olympiad 2015, grade xii, problem 4

8/23/2019
Find all non-constant polynoms f∈Q[X] f\in\mathbb{Q} [X] that don't have any real roots in the interval [0,1] [0,1] and for which there exists a function ξ:[0,1]⟶Q[X]×Q[X],ξ(x):=(gx,hx) \xi :[0,1]\longrightarrow\mathbb{Q} [X]\times\mathbb{Q} [X], \xi (x):=\left( g_x,h_x \right) such that hx(x)≠0 h_x(x)\neq 0 and ∫0xdtf(t)=gx(x)hx(x), \int_0^x \frac{dt}{f(t)} =\frac{g_x(x)}{h_x(x)} , for all x∈[0,1]. x\in [0,1] .
algebrapolynomialfunctioncalculusintegration