MathDB

3

Part of 2004 District Olympiad

Problems(6)

1 < \sqrt{1 + \sqrt{n}} < 2

Source: 2004 Romania District VII p3

8/15/2024
One considers the set A={n∈N∗∣1<1+n<2}A = \left\{ n \in N^* \big | 1 < \sqrt{1 + \sqrt{n}} < 2 \right\} a) Find the set AA. b) Find the set of numbers n∈An \in A such that n⋅∣1−1+n∣<1?\sqrt{n} \cdot \left| 1-\sqrt{1 + \sqrt{n}}\right| <1 ?
algebrainequalities
no of arithmetical sets in A_n is greater than 2004

Source: 2004 Romania District VIII p3

8/15/2024
It is said that a set of three different numbers is an arithmetical set if one of the three numbers is the average of the other two. Consider the set An={1,2,...,n}A_n = \{1, 2,..., n\}, where nn is a positive integer, n≥3n\ge 3. a) How many arithmetical sets are in A10A_{10}? b) Find the smallest nn, such that the number of arithmetical sets in AnA_n is greater than 20042004.
combinatorics
{x&sup2;}+{x}=0.99 is true for some x, but {x&sup2;}+{x}=1 isn&acute;t, in Q

Source: Romanian District Olympiad 2004, Grade IX, Problem 3

10/7/2018
a) Show that there are infinitely many rational numbers x>0 x>0 such that {x2}+{x}=0.99. \left\{ x^2 \right\} +\{ x \} =0.99. b) Show that there are no rational numbers x>0 x>0 such that {x2}+{x}=1. \left\{ x^2 \right\} +\{ x \} =1.
{} \{\} denotes the usual fractional part.
fractional partalgebra
Midpoints on a tetahedron

Source: Romanian District Olympiad 2004, Grade X, Problem 3

10/7/2018
On the tetrahedron ABCD ABCD make the notation M,N,P,Q, M,N,P,Q, for the midpoints of AB,CD,AC, AB,CD,AC, respectively, BD. BD. Additionally, we know that MN MN is the common perpendicular of AB,CD, AB,CD, and PQ PQ is the common perpendicular of AC,BD. AC,BD. Show that AB=CD,BC=DA,AC=BD. AB=CD, BC=DA, AC=BD.
geometry3D geometrytetrahedrongeometric transformationrotation
Romania District Olympiad 2004 - Grade XI

Source:

4/10/2011
Let f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} a function such that f(a+b2)∈{f(a),f(b)}, (∀)a,b∈Rf\left(\frac{a+b}{2}\right)\in \{f(a),f(b)\},\ (\forall)a,b\in \mathbb{R}.
a) Give an example of a non-constant function that satisfy the hypothesis.
b)If ff is continuous, prove that ff is constant.
functionreal analysisreal analysis unsolved
A condition for the commutativity of a unitary ring

Source: RMO 2004 - District Round

2/27/2007
Let (A,+,⋅)\left( A,+, \cdot \right) be a ring that verifies the following properties: (i) it has a unit, 11, and its order is pp, a prime number; (ii) there is B⊂A, ∣B∣=pB \subset A, \, |B| = p, such that: for all x,y∈Ax,y \in A, there is b∈Bb \in B such that xy=byxxy = byx. Prove that AA is commutative. Ion Savu
superior algebrasuperior algebra unsolved