Řešitelský seminář, 11.4.2017
Problem 1. Let f(x) be a polynomial of degree at most n such that for k = 0,1,..., n. Find f(n + 1).
Problem 2. Let a > 0 be a real number. Find the value of the following integral
ra cos t ,
-dt
a at + 1
Problem 3. For a non-negative integer n, let f(n) be the number obtained by writing n in binary and replacing every 0 with 1 and vice versa. For example, n = 23 is 10111 in binary, so f(n) is 1000 in binary, thus /(23) = 8.
1. Compute
n
Em
where fn{k) means function f applied n-times on k.
2. Show
When does the equality holdr: Problem 4.
En
k=l
l<z<j<n
Prove íÍ2aí there exits pairwaise distinct elements x\,...xn such that Xi is a member of Si for each index i.
Domácí úloha
Problem 5. Let n > 2 and A\, Ai,... An+\ be n + 1 points in the n-dimensional Euclidean space, not lying on the same hyperplane, and let B be a point strictly inside the convex hull of A\, Ai,... An+\. Prove that \^A1BA1\ > 90°.