VN TST 2015

Let $\alpha$ be the positive root of $x^2+x=5$. Suppose that $n$ is a positive integer and that nonnegative integers $c_0,c_1,\ldots,c_n$ satisfy
$$c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015.$$
(a) Prove that $c_0+c_1+\cdots+c_n\equiv2\pmod3$.
(b) Find the least possible value of $c_0+c_1+\cdots+c_n$.

A circle $(O)$ has a fixed chord $BC$ which is not a diameter. A point $A$ moves on the major arc $BC$ so that $ABC$ is acute and $AB\lt AC$. Let $I$ be the midpoint of $BC$ and $H$ the orthocenter of $ABC$. Ray $IH$ meets $(O)$ again at $K$. Line $AH$ meets line $BC$ at $D$, and line $KD$ meets $(O)$ again at $M$. The line through $M$ perpendicular to $BC$ meets $AI$ at $N$.
(a) Prove that $N$ belongs to a fixed circle as $A$ varies.
(b) The circle tangent to $AK$ at $A$ and passing through $N$ meets $AB,AC$ again at $P,Q$, respectively. Let $J$ be the midpoint of $PQ$. Prove that $AJ$ passes through a fixed point.

A positive integer $k$ has property $T(m)$ if, for every positive integer $a$, there is a positive integer $n$ such that
$$1^k+2^k+\cdots+n^k\equiv a\pmod m.$$
(a) Find all positive integers $k$ with property $T(20)$.
(b) Find the smallest positive integer $k$ with property $T(20^{15})$.

One hundred students take part in an oral examination. The panel consists of 25 examiners. Each student is questioned by one examiner. Every student likes at least 10 members of the panel.
(a) Prove that seven examiners can be selected so that every student likes at least one of them.
(b) Prove that the examination can be scheduled so that each student is questioned by exactly one examiner whom the student likes, and each examiner questions at most 10 students.

Let \(ABC\) be an acute scalene triangle and let \(P\) be an interior point such that \(\angle APB=\angle APC=\alpha>180^\circ-\angle BAC\). The circumcircle of \(APB\) meets \(AC\) again at \(E\), and the circumcircle of \(APC\) meets \(AB\) again at \(F\). Let \(Q\) lie inside \(AEF\) with \(\angle AQE=\angle AQF=\alpha\). Reflect \(Q\) in \(EF\) to obtain \(D\). The internal bisector of \(\angle EDF\) meets \(AP\) at \(T\).
(a) Prove \(\angle DET=\angle ABC\) and \(\angle DFT=\angle ACB\).
(b) The line \(PA\) meets the lines \(DE,DF\) at \(M,N\), respectively. Let \(I,J\) be the incenters of \(PEM,PFN\), and let \(K\) be the circumcenter of \(DIJ\). The line \(DT\) meets this circumcircle again at \(H\). Prove that \(HK\) passes through the incenter of \(DMN\).
Editorial warning: the source statement is retained, but part (b) is false without an additional configuration restriction. The solutions distinguish the valid part (a), the source's restricted-configuration argument, and an exact counterexample to the printed part (b). Here \(AC,AB\) must mean their supporting lines: the given angle inequality forces \(E,F\) onto the opposite extensions beyond \(A\).

Find the smallest positive integer $n$ for which there exist real numbers $a_1,a_2,\ldots,a_n$ satisfying
$$a_1+\cdots+a_n>0,\qquad a_1^3+\cdots+a_n^3\lt 0,\qquad a_1^5+\cdots+a_n^5>0.$$