MathDB

3

Part of 2003 Romania National Olympiad

Problems(6)

Positive integers

Source: RMO 2003, Grade 7, Problem 3

10/23/2008
For every positive integer n n consider A_n\equal{}\sqrt{49n^2\plus{}0,35n}. (a) Find the first three digits after decimal point of A1 A_1. (b) Prove that the first three digits after decimal point of An A_n and A1 A_1 are the same, for every n n.
floor function
Real numbers

Source: RMO 2003, Grade 8, Problem 3

10/23/2008
The real numbers a,b a,b fulfil the conditions (i) 0b b has the first 12 digits after the decimal point equal to 9. Mircea Fianu
Midpoints of altitudes

Source: RMO 2003, Grade 9, Problem 3

10/23/2008
Prove that the midpoints of the altitudes of a triangle are collinear if and only if the triangle is right. Dorin Popovici
Area of a triangle given some condition upon complex numbers

Source: Romanian National Olympiad 2003, grade x, p. 3

8/27/2019
Let be a circumcircle of radius 1 1 of a triangle whose centered representation in the complex plane is given by the affixes of a,b,c, a,b,c, and for which the equation a+bcos⁡x+csin⁡x=0 a+b\cos x +c\sin x=0 has a real root in (0,π2). \left( 0,\frac{\pi }{2} \right) . prove that the area of the triangle is a real number from the interval (1,1+22]. \left( 1,\frac{1+\sqrt 2}{2} \right] .
Gheorghe Iurea
geometryComplex Geometrycomplex numbers
Limit of function in function of limit of sequence of values of function

Source: RomNO 2003, grade xi. p. 3

8/27/2019
Let be two functions f,g:R≥0⟶R f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} having that properties that f f is continuous, g g is nondecreasing and unbounded, and for any sequence of rational numbers (xn)n≥1 \left( x_n \right)_{n\ge 1} that diverges to ∞, \infty , we have 1=lim⁡n→∞f(xn)g(xn). 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . Prove that 1=lim⁡x→∞f(x)g(x).1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) .
Radu Gologan
functioncontinuitylimitsreal analysisromania
Integral problem using periodicity

Source: RomNO 2003, grade xii, p.3

8/27/2019
Let be a continuous function f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} that has the property that xf(x)≥∫0xf(t)dt, xf(x)\ge \int_0^x f(t)dt , for all real numbers x. x. Prove that
a) the mapping x↦1x∫0xf(t)dt x\mapsto \frac{1}{x}\int_0^x f(t) dt is nondecreasing on the restrictions R<0 \mathbb{R}_{<0 } and R>0. \mathbb{R}_{>0 } .
b) if ∫xx+1f(t)dt=∫x−1xf(t)dt, \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , for any real number x, x, then f f is constant.
Mihai Piticari
functioncalculusintegration