MathDB

2

Part of 2004 District Olympiad

Problems(6)

AM = AN iff MN //KL (2004 Romania District VII P2)

Source:

5/24/2020
Let ABCABC be a triangle and DD a point on the side BCBC. The angle bisectors of ∠ADB,∠ADC\angle ADB ,\angle ADC intersect AB,ACAB ,AC at points M,NM ,N respectively. The angle bisectors of ∠ABD,∠ACD\angle ABD , \angle ACD intersects DM,DNDM , DN at points K,LK , L respectively. Prove that AM=ANAM = AN if and only if MNMN and KLKL are parallel.
geometryangle bisectorequal segmentsparallel
a + b + c + d = 2004, a^2 - b^2 + c^2 - d^2 =2004 2004 Romania District VIII p2

Source:

8/15/2024
The real numbers a,b,c,da, b, c, d satisfy a>b>c>da > b > c > d and a+b+c+d=2004   and   a2−b2+c2−d2=2004.a + b + c + d = 2004 \,\,\, and \,\,\, a^2 - b^2 + c^2 - d^2 = 2004. Answer, with proof, to the following questions: a) What is the smallest possible value of aa? b) What is the number of possible values of aa?
algebrainequalities
determining the vertices knowing orthocenter, circumcenter, midpoind of side

Source: Romanian District Olympiad 2004, Grade IX, Poblem 2

10/7/2018
Find the possible coordinates of the vertices of a triangle of which we know that the coordinates of its orthocenter are (−3,10), (-3,10), those of its circumcenter is (−2,−3), (-2,-3), and those of the midpoint of some side is (1,3). (1,3).
analytic geometrygeometrycircumcircle
If, a_1!=a_2!a_3!...a_n! then find n.

Source: Romanian District Olympiad 2004, Grade X, Problem 2

10/7/2018
Find all natural numbers for which there exist that many distinct natural numbers such that the factorial of one of these is equal to the product of the factorials of the rest of them.
factorialalgebraequalities
Romania District Olympiad 2004 - Grade XI

Source:

4/10/2011
a) Let x1,x2,x3,y1,y2,y3∈Rx_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R} and aij=sin⁡(xi−yj), i,j=1,3‾a_{ij}=\sin(x_i-y_j),\ i,j=\overline{1,3} and A=(aij)∈M3A=(a_{ij})\in \mathcal{M}_3 Prove that det⁡A=0\det A=0.
b) Let z1,z2,…,z2n∈C∗, n≥3z_1,z_2,\ldots,z_{2n}\in \mathbb{C}^*,\ n\ge 3 such that ∣z1∣=∣z2∣=…=∣zn+3∣|z_1|=|z_2|=\ldots=|z_{n+3}| and arg⁡z1≥arg⁡z2≥…≥arg⁡(zn+3)\arg z_1\ge \arg z_2\ge \ldots\ge \arg(z_{n+3}). If bij=∣zi−zj+n∣, i,j=1,n‾b_{ij}=|z_i-z_{j+n}|,\ i,j=\overline{1,n} and B=(bij)∈MnB=(b_{ij})\in \mathcal{M}_n, prove that det⁡B=0\det B=0.
vectorlinear algebramatrixlinear algebra unsolved
Sufficient condition for a continuous function to be constant

Source: Romanian District Olympiad 2004, Grade XII, Problem 2

10/7/2018
Let f:[0,1]⟶R f:[0,1]\longrightarrow\mathbb{R} be a continuous function such that ∫01f(x)g(x)dx=∫01f(x)dx⋅∫01g(x)dx, \int_0^1 f(x)g(x)dx =\int_0^1 f(x)dx\cdot\int_0^1 g(x)dx , for all functions g:[0,1]⟶R g:[0,1]\longrightarrow\mathbb{R} that are continuous and non-differentiable.
Prove that f f is constant.
functionintegrationreal analysisriemann integral