MathDB

2

Part of 2019 District Olympiad

Problems(6)

AE = CF, BF = EF, <EAB= <BAC,<FAC =<CAD (2019 Romania District VII p2)

Source:

5/21/2020
Consider DD the midpoint of the base [BC][BC] of the isosceles triangle ABC in which ∠BAC<90o\angle BAC < 90^o. On the perpendicular from BB on the line BCBC consider the point EE such that ∠EAB=∠BAC\angle EAB= \angle BAC, and on the line passing though CC parallel to the line ABAB we consider the point FF such that FF and DD are on different side of the line ACAC and ∠FAC=∠CAD\angle FAC = \angle CAD. Prove that AE=CFAE = CF and BF=EFBF = EF
geometryanglesequal anglesequal segments
3(AM^2 + B&#039;P^2 + CN^2)&gt;=2D&#039;B^2, cube if (2019 Romania District VIII p2)

Source:

5/23/2020
Let ABCDA′B′C′D′ABCDA'B'C'D' be a rectangular parallelepiped and M,N,PM,N, P projections of points A,CA, C and B′B' respectively on the diagonal BD′BD'.
a) Prove that BM+BN+BP=BD′BM + BN + BP = BD'.
b) Prove that 3(AM2+B′P2+CN2)≥2D′B23 (AM^2 + B'P^2 + CN^2)\ge 2D'B^2 if and only if ABCDA′B′C′D′ABCDA'B'C'D' is a cube.
parallelepipedgeometry3D geometrycubegeometric inequality
Romanian District Olympiad 2019 - Grade 9 - Problem 2

Source: Romanian District Olympiad 2019 - Grade 9 - Problem 2

3/18/2019
Let HH be the orthocenter of the acute triangle ABC.ABC. In the plane of the triangle ABCABC we consider a point XX such that the triangle XAHXAH is right and isosceles, having the hypotenuse AH,AH, and BB and XX are on each part of the line AH.AH. Prove that XA→+XC→+XH→=XB→\overrightarrow{XA}+\overrightarrow{XC}+\overrightarrow{XH}=\overrightarrow{XB} if and only if ∠BAC=45∘. \angle BAC=45^{\circ}.
geometryVectors
Romanian District Olympiad 2019 - Grade 10 - Problem 2

Source: Romanian District Olympiad 2019 - Grade 10 - Problem 2

3/17/2019
Let n∈N,n≥3.n \in \mathbb{N}, n \ge 3. a)a) Prove that there exist z1,z2,…,zn∈Cz_1,z_2,…,z_n \in \mathbb{C} such that z1z2+z2z3+…+zn−1zn+znz1=ni.\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}. b)b) Which are the values of nn for which there exist the complex numbers z1,z2,…,zn,z_1,z_2,…,z_n, of the same modulus, such that z1z2+z2z3+…+zn−1zn+znz1=ni?\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}?
complex numbersmodulusalgebra
Romanian District Olympiad 2019 - Grade 11 - Problem 2

Source: Romanian District Olympiad 2019 - Grade 11 - Problem 2

3/16/2019
Let n∈N,n≥2,n \in \mathbb{N},n \ge 2, and A,B∈Mn(R).A,B \in \mathcal{M}_n(\mathbb{R}). Prove that there exists a complex number z,z, such that ∣z∣=1|z|=1 and ℜ(det⁡(A+zB))≥det⁡(A)+det⁡(B),\Re \left( {\det(A+zB)} \right) \ge \det(A)+\det(B), where ℜ(w)\Re(w) is the real part of the complex number w.w.
characteristic polynomialInequalityDeterminantslinear algebra
Romanian District Olympiad 2019 - Grade 12 - Problem 2

Source: Romanian District Olympiad 2019 - Grade 12 - Problem 2

3/16/2019
Let nn be a positive integer and f:[0,1]→Rf:[0,1] \to \mathbb{R} be an integrable function. Prove that there exists a point c∈[0,1−1n],c \in \left[0,1- \frac{1}{n} \right], such that ∫cc+1nf(x)dx=0 \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0 or ∫0cf(x)dx=∫c+1n1f(x)dx.\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.
integrabilityfunctioncalculus