MathDB

3

Part of 2018 Romania National Olympiad

Problems(6)

parallelogram, equilateral, collinear,rhombus (2018 Romanian NMO VII P3)

Source:

6/3/2020
On the sides [AB][AB] and [BC][BC] of the parallelogram ABCDABCD are constructed the equilateral triangles ABEABE and BCF,BCF, so that the points DD and EE are on the same side of the line ABAB, and FF and DD on different sides of the line BCBC. If the points E,DE,D and FF are collinear, then prove that ABCDABCD is rhombus.
geometryparallelogramcollinearEquilateralrhombus
sum a/(a^2+7) <= 3/8 if ab + bc + ca = 3 for a,b,c>=0

Source: 2018 Romanian NMO grade VIII P4

9/4/2024
Let a,b,c≥0a, b, c \ge 0 so that ab+bc+ca=3ab + bc + ca = 3. Prove that: aa2+7+bb2+7+cc2+7≤38\frac{a}{a^2+7}+\frac{b}{b^2+7}+\frac{c}{c^2+7}\le \frac38
algebrainequalities
Romanian National Olympiad 2018 - Grade 9 - problem 3

Source: Romania NMO - 2018

4/12/2018
Let f,g:R→Rf,g : \mathbb{R} \to \mathbb{R} be two quadratics such that, for any real number r,r, if f(r)f(r) is an integer, then g(r)g(r) is also an integer. Prove that there are two integers mm and nn such that g(x)=mf(x)+n, ∀x∈Rg(x)=mf(x)+n, \: \forall x \in \mathbb{R}
quadratics
Romanian National Olympiad 2018 - Grade 10 - problem 3

Source: Romania NMO - 2018

4/12/2018
Let n∈N≥2.n \in \mathbb{N}_{\geq 2}. Prove that for any complex numbers a1,a2,…,ana_1,a_2,\ldots,a_n and b1,b2,…,bn,b_1,b_2,\ldots,b_n, the following statements are equivalent: a) ∑k=1n∣z−ak∣2≤∑k=1n∣z−bk∣2, ∀z∈C.\sum_{k=1}^n|z-a_k|^2 \leq \sum_{k=1}^n|z-b_k|^2, \: \forall z \in \mathbb{C}. b) ∑k=1nak=∑k=1nbk\sum_{k=1}^na_k=\sum_{k=1}^nb_k and ∑k=1n∣ak∣2≤∑k=1n∣bk∣2.\sum_{k=1}^n|a_k|^2 \leq \sum_{k=1}^n|b_k|^2.
complex numbers
Romanian National Olympiad 2018 - Grade 11 - problem 3

Source: Romania NMO - 2018

4/7/2018
Let f:R→Rf: \mathbb{R} \to \mathbb{R} be a function with the intermediate value property. If ff is injective on R∖Q,\mathbb{R} \setminus \mathbb{Q}, prove that ff is continuous on R.\mathbb{R}.
Julieta R. Vergulescu
Romanian National Olympiad 2018 - Grade 12 - problem 3

Source: Romania NMO - 2018

4/7/2018
Let f:[a,b]→Rf:[a,b] \to \mathbb{R} be an integrable function and (an)⊂R(a_n) \subset \mathbb{R} such that an→0.a_n \to 0. <spanclass=′latex−bold′>a)</span><span class='latex-bold'>a) </span> If A={m⋅an∣m,n∈N∗},A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \}, prove that every open interval of strictly positive real numbers contains elements from A.A. <spanclass=′latex−bold′>b)</span><span class='latex-bold'>b) </span> If, for any n∈N∗n \in \mathbb{N}^* and for any x,y∈[a,b]x,y \in [a,b] with ∣x−y∣=an,|x-y|=a_n, the inequality ∣∫xyf(t)dt∣≤∣x−y∣\left| \int_x^yf(t)dt \right| \leq |x-y| is true, prove that ∣∫xyf(t)dt∣≤∣x−y∣, ∀x,y∈[a,b]\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]
Nicolae Bourbacut
functioninequalitiescalculusintegralsreal analysis