MathDB

1

Part of 1999 China Team Selection Test

Problems(2)

another circle which passes through E and F

Source: China TST 1999, problem 4

5/22/2005
A circle is tangential to sides ABAB and ADAD of convex quadrilateral ABCDABCD at GG and HH respectively, and cuts diagonal ACAC at EE and FF. What are the necessary and sufficient conditions such that there exists another circle which passes through EE and FF, and is tangential to DADA and DCDC extended?
geometryperpendicular bisectorpower of a pointradical axisgeometry unsolved
Sum of differences of fourth and fifth powers

Source: China TST 1999, problem 1

5/22/2005
For non-negative real numbers x1,x2,…,xnx_1, x_2, \ldots, x_n which satisfy x1+x2+⋯+xn=1x_1 + x_2 + \cdots + x_n = 1, find the largest possible value of ∑j=1n(xj4−xj5)\sum_{j = 1}^{n} (x_j^{4} - x_j^{5}).
inequalitiessmoothingChina TST