MathDB

3

Part of 2014 Romania National Olympiad

Problems(6)

17^n +9^{n^2} = 23^n +3^{n^2} 2014 Romania NMO VII p3

Source:

8/15/2024
Find all positive integers nn so that 17n+9n2=23n+3n2.17^n +9^{n^2} = 23^n +3^{n^2} .
number theory
a/c=b/d=c/e from {n, n +1, n +2,...,2n}

Source: 2014 Romania NMO VII p3

8/15/2024
Find the smallest integer nn for which the set A={n,n+1,n+2,...,2n}A = \{n, n +1, n +2,...,2n\} contains five elements a<b<c<d<ea<b<c<d<e so that ac=bd=ce\frac{a}{c}=\frac{b}{d}=\frac{c}{e}
number theory
Another center of mass problem

Source: Romanian National Olympiad 2014, Grade IX, Problem 3

3/2/2019
Let P,Q P,Q be the midpoints of the diagonals BD, BD, respectively, AC, AC, of the quadrilateral ABCD, ABCD, and points M,N,R,S M,N,R,S on the segments BC,CD,PQ, BC,CD,PQ, respectively AC, AC, except their extremities, such that BMMC=DNNC=PRRQ=ASSC. \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . Show that the center of mass of the triangle AMN AMN is situated on the segment RS. RS.
geometrycenter of massanalytic geometry
Number of natural-valued non-contracted Lipschitz functions

Source: Romanian National Olympiad 2014, Grade X, Problem 3

3/2/2019
Let n n be a natural number, and A A the set of the first n n natural numbers. Find the number of nondecreasing functions f:A⟶A f:A\longrightarrow A that have the property x,y∈A  ⟹  ∣f(x)−f(y)∣≤∣x−y∣. x,y\in A\implies |f(x)-f(y)|\le |x-y|.
functionalgebra
Matrix and determinant

Source:

6/30/2015
Let A,B∈Mn(C)A,B\in M_n(C) be two square matrices satisfying A2+B2=2ABA^2+B^2 = 2AB.
1.Prove that det⁡(AB−BA)=0\det(AB-BA)=0. 2.If rank(A−B)=1rank(A-B)=1, then prove that AB=BAAB=BA.
linear algebramatrix
Real analysis: limits of f/id, F/id and more

Source: Romania National Olympiad 2014, Grade XII, Problem 3

3/3/2019
Let f:[1,∞)⟶(0,∞) f:[1,\infty )\longrightarrow (0,\infty ) be a continuous function satisfying the following properties:
(i)∃lim⁡x→∞f(x)x∈R‾ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} (ii)∃lim⁡x→∞1x∫1xf(t)dt∈R. \text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}.
a) Show that lim⁡x→∞f(x)x=0. \lim_{x\to\infty } \frac{f(x)}{x}=0. b) Prove that lim⁡x→∞1x2∫1xf2(t)dt=0. \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0.
functionreal analysis