MathDB

3

Part of 2016 Romania National Olympiad

Problems(6)

4 digit no = 2^m3^n(m + n)

Source: 2016 Romanian NMO grade VII P3

9/3/2024
Find all the positive integers pp with the property that the sum of the first pp positive integers is a four-digit positive integer whose decomposition into prime factors is of the form 2m3n(m+n)2^m3^n(m + n), where m,n∈N∗m, n \in N^*.
number theoryprime factorization
3/2 <= sum (b + c) /( b + c + 2a) <= 5/3 for a,b,c triangle sidelenghts

Source: 2016 Romanian NMO grade VIII P3

9/3/2024
If a,ba, b and cc are the length of the sides of a triangle, show that 32≤b+cb+c+2a+a+ca+c+2b+a+ba+b+2c≤53.\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.
geometrygeometric inequalityGeometric Inequalities
A property of spheres defined on rational grids

Source: Romania National Olympiad 2016, grade ix, p. 3

8/25/2019
We say that a rational number is spheric if it is the sum of three squares of rational numbers (not necessarily distinct). Prove that:
a) 7 7 is not spheric. b) a rational spheric number raised to the power of any natural number greater than 1 1 is spheric.
number theorygeometry3D geometrysphere
Area of triangles, eventually represented by complex numbers

Source: Romanian National Olympiad 2016, grade x, p. 3

8/25/2019
a) Let be two nonzero complex numbers a,b. a,b. Show that the area of the triangle formed by the representations of the affixes 0,a,b 0,a,b in the complex plane is 14∣a‾b−ab‾∣. \frac{1}{4}\left| \overline{a} b-a\overline{b} \right| .
b) Let be an equilateral triangle ABC, ABC, its circumcircle C, \mathcal{C} , its circumcenter O, O, and two distinct points P1,P2 P_1,P_2 in the interior of C. \mathcal{C} . Prove that we can form two triangles with sides P1A,P1B,P1C, P_1A,P_1B,P_1C, respectively, P2A,P2B,P2C, P_2A,P_2B,P_2C, whose areas are equal if and only if OP1=OP2. OP_1=OP_2.
Complex Geometryanalytic geometrygeometryPure geometrycomplex numbers
Useful characterization that compares powers of the id. func with positive func.

Source: Romanian National Olympiad 2016, grade xi, p. 3

8/25/2019
Let be a real number a, a, and a function f:R>0⟶R>0. f:\mathbb{R}_{>0 }\longrightarrow\mathbb{R}_{>0 } . Show that the following relations are equivalent.
\text{(i)} \varepsilon\in\mathbb{R}_{>0 } \implies\left( \lim_{x\to\infty } \frac{f(x)}{x^{a+\varepsilon }} =0\wedge \lim_{x\to\infty } \frac{f(x)}{x^{a-\varepsilon }} =\infty \right) \text{(ii)} \lim_{x\to\infty } \frac{\ln f(x)}{\ln x } =a
real analysis
An ineq. involving sum of def. int. as a characterization of locally cont. func.

Source: Romanian National Olympiad 2016, grade xii, p.3

8/25/2019
Let be a real number a, a, and a nondecreasing function f:R⟶R. f:\mathbb{R}\longrightarrow\mathbb{R} . Prove that f f is continuous in a a if and only if there exists a sequence (an)n≥1 \left( a_n \right)_{n\ge 1} of real positive numbers such that ∫aa+anf(x)dx+∫aa−anf(x)dx≤ann, \int_a^{a+a_n} f(x)dx+\int_a^{a-a_n} f(x)dx\le\frac{a_n}{n} , for all natural numbers n. n.
Dan Marinescu
functioncalculusintegrationinequalities