MathDB

P3

Part of 2024 District Olympiad

Problems(4)

NT with quadratic equations

Source: Romanian District Olympiad 2024 9.3

3/10/2024
Let nn be a composite positive integer. Let 1=d1<d2<⋯<dk=n1=d_1<d_2<\cdots<d_k=n be the positive divisors of n.n.{} Assume that the equations di+2x2−2di+1x+di=0d_{i+2}x^2-2d_{i+1}x+d_i=0 for i=1,…,k−2i=1,\ldots,k-2 all have real solutions. Prove that n=pk−1n=p^{k-1} for some prime number p.p.{}
number theoryquadratic equation
C(omplex) numbers

Source: Romanian District Olympiad, grade 10, p3

3/10/2024
Let a,b,c∈C∖{0}a,b,c\in\mathbb{C}\setminus\left\{0\right\} such that ∣a∣=∣b∣=∣c∣|a|=|b|=|c| and A=a+b+cA=a+b+c respectively B=abcB=abc are both real numbers. Prove that Cn=an+bn+cn C_n=a^n+b^n+c^n is also a real number,, (∀)n∈N.(\forall)n\in\mathbb{N}.
algebracomplex numbers
Antisymmetric matrix problem

Source: Romanian District Olympiad 2024 11.3

3/10/2024
Let A∈Mn(C)A\in\mathcal{M}_n(\mathbb{C}) be an antisymmetric matrix, i.e. A=−At.A=-A^t. [*]Prove that if A∈Mn(R)A\in\mathcal{M}_n(\mathbb{R}) and A2=OnA^2=O_n then A=On.A=O_n. [*]Assume that nn{} is odd. Prove that if AA{} is the adjoint of another matrix B∈Mn(C)B\in\mathcal{M}_n(\mathbb{C}) then A2=On.A^2=O_n.
linear algebramatrix
Easy NT with a ring

Source: Romanian District Olympiad 2024 12.3

3/10/2024
Let kk be a positive integer. A ring (A,+,⋅)(A,+,\cdot) has property PkP_k if for any a,b∈Aa,b\in A there exists c∈Ac\in A such that ak=bk+ck.a^k=b^k+c^k. [*]Give an example of a finite ring (A,+,⋅)(A,+,\cdot) which does not have PkP_k for any k⩾2.k\geqslant 2. [*]Let n⩾3n\geqslant 3 be an integer and Mn={m∈N:(Zn,+,⋅) has Pm}.M_n=\{m\in\mathbb{N}:(\mathbb{Z}_n,+,\cdot)\text{ has }P_m\}. Prove that all the elements of MnM_n are odd integers and that (Mn,⋅)(M_n,\cdot) is a monoid.
number theoryRing Theory