MathDB

1

Part of 2008 Indonesia TST

Problems(3)

P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}, at least 1 real root

Source: 2008 Indonesia TST stage 2 test 2 p1

12/14/2020
A polynomial P(x)=1+x2+x5+xn1+...+xns+x2008P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008} with n1,...,nsn_1, ..., n_s are positive integers and 5<n1<...<ns<20085 < n_1 < ... <n_s < 2008 are given. Prove that if P(x)P(x) has at least a real root, then the root is not greater than 1−52\frac{1-\sqrt5}{2}
algebrapolynomial
incenter wanted , &lt;BIC = &lt; IDC, cyclic ABCD

Source: 2008 Indonesia TST stage 2 test 3 p1

12/14/2020
Let ABCDABCD be a cyclic quadrilateral, and angle bisectors of ∠BAD\angle BAD and ∠BCD\angle BCD meet at point II. Show that if ∠BIC=∠IDC\angle BIC = \angle IDC, then II is the incenter of triangle ABDABD.
geometryincentercyclic quadrilateral
exist 2 disjoint subsets of A with same sum of elements

Source: 2008 Indonesia TST stage 2 test 4 p1

12/15/2020
Let AA be the subset of {1,2,...,16}\{1, 2, ..., 16\} that has 66 elements. Prove that there exist 22 subsets of AA that are disjoint, and the sum of their elements are the same.
combinatoricsSubsets