MathDB

4

Part of 2007 Nicolae Coculescu

Problems(6)

Point chosen from a locus of points

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12/13/2019
Let M M be a point in the interior of a triangle ABC, ABC, let D D be the intersection of AM AM with BC, BC, let E E be the intersection of M M with AC, let F F be the intersection of CM CM with AB. AB. Knowing that the expression MAMD⋅MBME⋅MCMF \frac{MA}{MD}\cdot \frac{MB}{ME}\cdot \frac{MC}{MF} is minimized, describe the point M. M.
Locusgeometryanalytic geometryinfimum
Unique partition under certain conditions of the first n naturals

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12/13/2019
Let be a natural number n≥2. n\ge 2. Prove that there exists an unique bipartition (A,B) \left( A,B \right) of the set {1,2…,n} \{ 1,2\ldots ,n \} such that ⌊x⌋≠y, \lfloor \sqrt x \rfloor\neq y , for any x,y∈A, x,y\in A , and ⌊z⌋≠t, \lfloor \sqrt z \rfloor\neq t , for any z,t∈B. z,t\in B.
Costin Bădică
floor functionnumber theoryset
Interesting Cauchy function with period on a one-variable restriction

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12/13/2019
Prove that there exists a nonconstant function f:R2⟶R f:\mathbb{R}^2\longrightarrow\mathbb{R} verifying the following system of relations: {f(x,x+y)=f(x,y),emsp;∀x,y∈Rf(x,y+z)=f(x,y)+f(x,z),emsp;∀x,y∈R \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,&   \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,&   \forall x,y\in\mathbb{R} \end{matrix} \right.
functionalgebraFind all functions
Shur-like inequality

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12/13/2019
Let be three nonnegative integers m,n,p m,n,p and three real numbers x,y,z x,y,z such that 2mx+2ny+2pz≥0. 2^mx+2^ny+2^pz\ge 0. Prove: 2m(2x−1)+2n(2y−1)+2p(2z−1)≥0 2^m\left( 2^x-1 \right)+2^n\left( 2^y-1 \right)+2^p\left( 2^z-1 \right)\ge 0
Cristinel Mortici
inequalities
Nicolae Coculescu 2007

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2/24/2008
Let n∈N∗ n\in{N^*},n≥3 n\ge{3} and a1,a2,...,an∈R∗ a_1,a_2,...,a_n\in{R^*}, so that ∣ai∣≠∣aj∣ |a_i|\neq{|a_j|}, for every i,j∈{1,2,...,n},i≠j i,j\in{\{1,2,...,n\}}, i\neq{j}. Find p∈Sn p\in{S_n} with the property: a_ia_j < \equal{} a_{p(i)}a_{p(j)}, for every i,j∈{1,2,....n} i,j\in{\{1,2,....n\}},i≠j i\neq{j} (Teodor Radu)
linear algebralinear algebra unsolved
Totient divisibility

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12/13/2019
Prove that p p divides φ(1+ap), \varphi (1+a^p) , where a≥2 a\ge 2 is a natural number, p p is a prime, and φ \varphi is Euler's totient.
Cristinel Mortici
TotientEulernumber theoryDivisibility