MathDB

2

Part of 2018 Romania National Olympiad

Problems(6)

area chasing, square, rhombus, symmetric (2018 Romanian NMO VII P2)

Source:

6/3/2020
In the square ABCDABCD the point EE is located on the side [AB][AB], and FF is the foot of the perpendicular from BB on the line DEDE. The point LL belongs to the line DEDE, such that FF is between EE and LL, and FL=BFFL = BF. NN and PP are symmetric of the points A,FA , F with respect to the lines DE,BLDE, BL, respectively. Prove that:
a) The quadrilateral BFLPBFLP is square and the quadrilateral ALNDALND is rhombus. b) The area of the rhombus ALNDALND is equal to the difference between the areas of the squares ABCDABCD and BFLPBFLP.
geometryrhombusareasquaresymmetry
min (a-b)^2 + 2(a-c)^2+ 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2

Source: 2018 Romanian NMO grade VIII P2

9/4/2024
Let a,b,c,da, b, c, d be natural numbers such that a+b+c+d=2018a + b + c + d = 2018. Find the minimum value of the expression: E=(a−b)2+2(a−c)2+3(a−d)2+4(b−c)2+5(b−d)2+6(c−d)2.E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.
algebrainequalitiesnumber theory
Romanian National Olympiad 2018 - Grade 9 - problem 2

Source: Romania NMO - 2018

4/12/2018
Let a,b,c≥0a,b,c \geq 0 and a+b+c=3.a+b+c=3. Prove that a1+b+b1+c+c1+a≥11+b+11+c+11+a\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \geq \frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+a}
inequalities
Romanian National Olympiad 2018 - Grade 10 - problem 2

Source: Romania NMO - 2018

4/12/2018
Let ABCABC be a triangle, OO its circumcenter and R=1R=1 its circumradius. Let G1,G2,G3G_1,G_2,G_3 be the centroids of the triangles OBC,OACOBC, OAC and OAB.OAB. Prove that the triangle ABCABC is equilateral if and only if AG1+BG2+CG3=4AG_1+BG_2+CG_3=4
geometrycircumcirclecomplex numbers
Romanian National Olympiad 2018 - Grade 11 - problem 2

Source: Romania NMO - 2018

4/7/2018
Let x>0.x>0. Prove that 2−x+2−1/x≤1.2^{-x}+2^{-1/x} \leq 1.
calculusinequalities
Romanian National Olympiad 2018 - Grade 12 - problem 2

Source: Romania NMO - 2018

4/7/2018
Let F\mathcal{F} be the set of continuous functions f:R→Rf: \mathbb{R} \to \mathbb{R} such that ef(x)+f(x)≥x+1, ∀x∈Re^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R} For f∈F,f \in \mathcal{F}, let I(f)=∫0ef(x)dxI(f)=\int_0^ef(x) dx Determine min⁡f∈FI(f).\min_{f \in \mathcal{F}}I(f).
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functioncalculusintegrationreal analysiscollege contests