MathDB

2

Part of 2014 Romania National Olympiad

Problems(6)

right isosceles wanted, square, rhombus (2014 Romanian NMO grade VII P2)

Source:

5/28/2020
Outside the square ABCDABCD, the rhombus BCMNBCMN is constructed with angle BCMBCM obtuse . Let PP be the intersection point of the lines BMBM and ANAN . Prove that DM⊥CPDM \perp CP and the triangle DPMDPM is right isosceles .
geometryrhombussquareperpendicularisoscelesright triangle
angles between plane, distance, cube (2014 Romanian NMO grade VIII P2)

Source:

5/28/2020
Let ABCDA′B′C′D′ABCDA'B'C'D' be a cube with side AB=aAB = a. Consider points E∈(AB)E \in (AB) and F∈(BC)F \in (BC) such that AE+CF=EFAE + CF = EF.
a) Determine the measure the angle formed by the planes (D′DE)(D'DE) and (D′DF)(D'DF).
b) Calculate the distance from D′D' to the line EFEF.
geometry3D geometrycubeangledistance
prove non-equalities

Source: Romanian National Olympiad 2014, Grade IX, Problem 2

3/2/2019
Let a a be an odd natural that is not a perfect square, and m,n∈N. m,n\in\mathbb{N} . Then
a) {m(a+a)}≠{n(a−a)} \left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\} b) [m(a+a)]≠[n(a−a)] \left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right]
Here, {},[] \{\},[] denotes the fractionary, respectively the integer part.
algebra
functional inequality

Source: Romania National Olympiad 2014, Grade X, Problem 2

3/2/2019
Let be a function f:N⟶N f:\mathbb{N}\longrightarrow\mathbb{N} satisfying (i)f(1)=1 \text{(i)} f(1)=1 (ii)f(p)=1+f(p−1), \text{(ii)} f(p)=1+f(p-1), for any prime p p (iii)f(p1p2⋯pu)=f(p1)+f(p2)+⋯f(pu), \text{(iii)} f(p_1p_2\cdots p_u)=f(p_1)+f(p_2)+\cdots f(p_u), for any natural number u u and any primes p1,p2,…,pu. p_1,p_2,\ldots ,p_u.
Show that 2f(n)≤n3≤3f(n), 2^{f(n)}\le n^3\le 3^{f(n)}, for any natural n≥2. n\ge 2.
function
All derivable functions satisfying f²=f

Source: Romanian National Olympiad 2014, Grade XI, Problem 2

3/3/2019
Find all derivable functions that have real domain and codomain, and are equal to their second functional power.
functionalgebrareal analysis
Characterization of integrands whose primitives can be found by substitution

Source: Romania National Olympiad 2014, Grade XII, Problem 2

3/3/2019
Let I,J I,J be two intervals, φ:J⟶R \varphi :J\longrightarrow\mathbb{R} be a continuous function whose image doesn't contain 0, 0, and f,g:I⟶J f,g:I\longrightarrow J be two differentiable functions such that f′=φ∘f,g′=φ∘g f'=\varphi\circ f,g'=\varphi\circ g and such that the image of f−g f-g contains 0. 0. Show that f f and g g are the same function.
functionreal analysis