MathDB

2

Part of 2011 Romania National Olympiad

Problems(6)

ab <= (x + z) (y +t) , 2011 Romania NMO VII p2

Source:

8/15/2024
The numbers x,y,z,t,ax, y, z, t, a and bb are positive integers, so that xt−yz=1xt-yz = 1 and xy≥ab≥zt.\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .Prove that ab≤(x+z)(y+t)ab \le (x + z) (y +t)
algebrainequalities
sum of divisors of n!

Source: Romanian NO, grade ix, p.2

10/3/2019
Prove that any natural number smaller or equal than the factorial of a natural number n n is the sum of at most n n distinct divisors of the factorial of n. n.
number theory
(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) >= 6 2011 Romania NMO VIII p2

Source:

8/15/2024
Let a,b,ca, b, c be distinct positive integers.
a) Prove that a2b2+a2c2+b2c2≥9a^2b^2 + a^2c^2 + b^2c^2 \ge 9.
b) if, moreover, ab+ac+bc+3=abc>0,ab + ac + bc +3 = abc > 0, show that (a−1)(b−1)+(a−1)(c−1)+(b−1)(c−1)≥6.(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.
algebrainequalities
Three roots of unity whose sum is 1

Source: Romanian NO 2011, grade x, p.2

10/3/2019
Find all numbers n n for which there exist three (not necessarily distinct) roots of unity of order n n whose sum is 1. 1.
complex numbersalgebra
Romania National Olympiad 2011 - Grade XI - problem 2

Source:

4/19/2011
Let u:[a,b]→Ru:[a,b]\to\mathbb{R} be a continuous function that has finite left-side derivative ul′(x)u_l^{\prime}(x) in any point x∈(a,b]x\in (a,b] . Prove that the function uu is monotonously increasing if and only if ul′(x)≥0u_l^{\prime}(x)\ge 0 , for any x∈(a,b]x\in (a,b] .
functioncalculusderivativeinequalitiesabstract algebrareal analysisreal analysis unsolved
Functions characterized locally by definite integrals

Source: Romanian NO 2011, grade xii, p.2

10/3/2019
Let be a continuous function f:[0,1]⟶(0,∞) f:[0,1]\longrightarrow\left( 0,\infty \right) having the property that, for any natural number n≥2, n\ge 2, there exist n−1 n-1 real numbers 0<t1<t2<⋯<tn−1<1, 0<t_1<t_2<\cdots <t_{n-1}<1, such that ∫0t1f(t)dt=∫t1t2f(t)dt=∫t2t3f(t)dt=⋯=∫tn−2tn−1f(t)dt=∫tn−11f(t)dt. \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt.
Calculate lim⁡n→∞n1f(0)+∑i=1n−11f(ti)+1f(1). \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} .
functionreal analysisintegrationcalculus