MathDB

2

Part of 1994 IberoAmerican

Problems(2)

9th ibmo - brazil 1994/q2.

Source: 9th ibero Fortaleza-ceara, Brazil, September 17th - 25th

5/7/2006
Let ABCD ABCD a cuadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on AB AB, that is tangent to the other three sides of the cuadrilateral. (i) Show that AB \equal{} AD \plus{} BC. (ii) Calculate, in term of x \equal{} AB and y \equal{} CD, the maximal area that can be reached for such quadrilateral.
geometryrectangleinradiusincenterratiotrigonometryperimeter
9th ibmo - brazil 1994/q5.

Source: Spanish Communities

5/7/2006
Let nn and rr two positive integers. It is wanted to make rr subsets A1, A2,…,ArA_1,\ A_2,\dots,A_r from the set {0,1,⋯ ,n−1}\{0,1,\cdots,n-1\} such that all those subsets contain exactly kk elements and such that, for all integer xx with 0≤x≤n−10\leq{x}\leq{n-1} there exist x1∈A1, x2∈A2…,xr∈Arx_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r (an element of each set) with x=x1+x2+⋯+xrx=x_1+x_2+\cdots+x_r.
Find the minimum value of kk in terms of nn and rr.
functionceiling functioncombinatorics unsolvedcombinatorics