MathDB

4

Part of 2006 Romania National Olympiad

Problems(5)

Inequality with variables between 1/2 and 1

Source: Romanian NMO 2006, Grade 8, Problem 4

4/18/2006
Let a,b,c∈[12,1]a,b,c \in \left[ \frac 12, 1 \right]. Prove that 2≤a+b1+c+b+c1+a+c+a1+b≤3. 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . selected by Mircea Lascu
inequalitiescalculusderivativefunctiongeometrytrigonometryvector
A set with at least two elements

Source: Romanian NMO 2006, Grade 7, Problem 4

4/18/2006
Let AA be a set of positive integers with at least 2 elements. It is given that for any numbers a>ba>b, a,b∈Aa,b \in A we have [a,b]a−b∈A\frac{ [a,b] }{ a- b } \in A, where by [a,b][a,b] we have denoted the least common multiple of aa and bb. Prove that the set AA has exactly two elements. Marius Gherghu, Slatina
calculusintegrationnumber theorygreatest common divisorleast common multiplerelatively prime
Some sets of a plane

Source: RMO 2006, 10th grade

4/17/2006
Let n∈N\displaystyle n \in \mathbb N, n≥2\displaystyle n \geq 2. Determine n\displaystyle n sets Ai\displaystyle A_i, 1≤i≤n\displaystyle 1 \leq i \leq n, from the plane, pairwise disjoint, such that: (a) for every circle C\displaystyle \mathcal C from the plane and for every i∈{1,2,…,n}\displaystyle i \in \left\{ 1,2,\ldots,n \right\} we have Ai∩Int(C)≠ϕ\displaystyle A_i \cap \textrm{Int} \left( \mathcal C \right) \neq \phi; (b) for all lines d\displaystyle d from the plane and every i∈{1,2,…,n}\displaystyle i \in \left\{ 1,2,\ldots,n \right\}, the projection of Ai\displaystyle A_i on d\displaystyle d is not d\displaystyle d.
analytic geometrycombinatorics proposedcombinatorics
Almost strictly increasing functions

Source: RMO 2006

4/18/2006
Let f:[0,∞)→Rf: [0,\infty)\to\mathbb R be a function such that for any x>0x>0 the sequence {f(nx)}n≥0\{f(nx)\}_{n\geq 0} is increasing. a) If the function is also continuous on [0,1][0,1] is it true that ff is increasing? b) The same question if the function is continuous on Q∩[0,∞)\mathbb Q \cap [0, \infty).
functionreal analysisreal analysis unsolved
A mean value problem \int_0^1 f(x)dx=0 and \int_{0}^{c}xf(x)

Source: Romanian National Olympiad, 2006

4/17/2006
Let f:[0,1]→Rf: [0,1]\to\mathbb{R} be a continuous function such that ∫01f(x)dx=0. \int_{0}^{1}f(x)dx=0. Prove that there is c∈(0,1)c\in (0,1) such that ∫0cxf(x)dx=0. \int_{0}^{c}xf(x)dx=0. Cezar Lupu, Tudorel Lupu
integrationfunctioncalculusderivativetrigonometryreal analysisreal analysis unsolved