MathDB

2

Part of 1995 IberoAmerican

Problems(2)

Interesting problem of a triangle (equal chords)

Source: 10th Iberoamerican 1995 pr. B2

3/30/2004
The incircle of a triangle ABCABC touches the sides BCBC, CACA, ABAB at the points DD, EE, FF respectively. Let the line ADAD intersect this incircle of triangle ABCABC at a point XX (apart from DD). Assume that this point XX is the midpoint of the segment ADAD, this means, AX=XDAX = XD. Let the line BXBX meet the incircle of triangle ABCABC at a point YY (apart from XX), and let the line CXCX meet the incircle of triangle ABCABC at a point ZZ (apart from XX). Show that EY=FZEY = FZ.
geometrytrapezoidtrigonometrysimilar triangles
10th ibmo - chile 1995/q2.

Source: Spanish Communities

5/7/2006
Let nn be a positive integer greater than 1. Determine all the collections of real numbers x_1,\ x_2,\dots,\ x_n\geq1\mbox{ and }x_{n+1}\leq0 such that the next two conditions hold:
(i) x112+x232+⋯+xnn−12=nxn+112x_1^{\frac12}+x_2^{\frac32}+\cdots+x_n^{n-\frac12}= nx_{n+1}^\frac12
(ii) x1+x2+⋯+xnn=xn+1\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1}
inequalitiesquadraticsalgebra unsolvedalgebra