MathDB

2007 Nicolae Păun

Part of Nicolae Păun

Subcontests

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Hard base 10 number theory

Prove that for any natural number n, n, there exists a number having n+1 n+1 decimal digits, namely, k0,k1,k2,…,kn k_0,k_1,k_2,\ldots ,k_n , and a (n+1)-tuple, \text{(n+1)-tuple}, say (ϵ0,ϵ1,ϵ2…,ϵn)∈{−1,1}n+1,\left( \epsilon_0 ,\epsilon_1 ,\epsilon_2\ldots ,\epsilon_n \right)\in\{-1,1\}^{n+1} , that satisfies: 1≤∏j=0n(2+j)kj⋅ϵj≤210n−1 1\le \prod_{j=0}^n (2+j)^{k_j\cdot \epsilon_j}\le \sqrt[10^n-1]{2}
Sorin Rădulescu and Ion Savu

Beauatiful combinatorial geometry: 27 discs inserted in a circle

20 20 discs of radius 1 1 are bounded by a circle of radius 10. 10. Show that in the interior of this circle is sufficient space to insert 7 7 discs of radius 13 \frac{1}{3} that doesn't touch any other disc.
Flavian Georgescu

Create a f. locally Darboux and not monotone on open set, but monotone on dense

Construct a function f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} having the following properties:
(i)f \text{(i)} f is not monotonic on any real interval. (ii)f \text{(ii)} f has Darboux property (intermediate value property) on any real interval. \text{(iii)} f(x)\leqslant f\left( x+1/n \right) ,  \forall x\in\mathbb{R} ,  \forall n\in\mathbb{N}
Alexandru Cioba
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Base 10 polynom algebra

Let be nine nonzero decimal digits a1,a2,a3,b1,b2,b3,c1,c2,c3 a_1,a_2,a_3,b_1,b_2,b_3,c_1,c_2,c_3 chosen such that the polynom (100a1+10a2+a3)X2+(100b1+10b2+b3)X+100c1+10c2+c3 \left( 100a_1+10a_2+a_3 \right) X^2 +\left( 100b_1+10b_2+b_3 \right) X +100c_1+10c_2+c_3 admits at least a real solution. Prove that at least one of the polynoms a_iX^2+b_iX+c_i  (i\in\{1,2,3\}) admits at least a real solution.
Nicolae Mușuroia

matrix equations

Prove that ∃X,Y,Z∈Mn(C) \exists X,Y,Z\in \mathcal{M}_n(\mathbb{C}) such that a) X^2\plus{}Y^2\equal{}A b) X^3\plus{}Y^3\plus{}Z^3\equal{}A , where A∈Mn(C) A\in \mathcal{M}_n(\mathbb{C})

Group theory; function of sets

Consider a finite group G G and the sequence of functions (An)n≥1:G⟶P(G) \left( A_n \right)_{n\ge 1} :G\longrightarrow \mathcal{P} (G) defined as An(g)={x∈G∣xn=g}, A_n(g) = \left\{ x\in G|x^n=g \right\} , where P(G) \mathcal{P} (G) is the power of G. G.
a) Prove that if G G is commutative, then for any natural numbers n, n, either An(g)=∅, A_n(g) =\emptyset , or ∣An(g)∣=∣An(1)∣. \left| A_n(g) \right| =\left| A_n(1) \right| .
b) Provide an example of what G G could be in the case that there exists an element g0 g_0 of G G and a natural number n0 n_0 such that ∣An0(g0)∣>∣An0(1)∣. \left| A_{n_0}\left( g_0 \right) \right| >\left| A_{n_0}(1) \right| .
Sorin Rădulescu and Ion Savu