Let \(ABC\) be an acute scalene triangle. Let \(D,E,F\) be the midpoints of \(BC,CA,AB\), and let \(O,O'\) be its circumcenter and nine-point center. A point \(P\) lies inside triangle \(DEF\). Lines \(DP,EP,FP\) meet the nine-point circle again at \(D',E',F'\). Reflect \(A,B,C\) in \(D',E',F'\) to obtain \(A',B',C'\), respectively.
(a) If \(PO=PO'\), prove that the circle \((A'B'C')\) passes through \(O\).
(b) Reflect \(A',B',C'\) in lines \(OD,OE,OF\), obtaining \(X,Y,Z\). Let \(H\) be the orthocenter, and put \(M=HX\cap BC\), \(N=HY\cap CA\), \(K=HZ\cap AB\). Prove that \(M,N,K\) are collinear.
Editorial domain note: part (b), as printed, permits positions where its lines or intersections do not exist. The proof below concerns configurations in which the named intersections are finite and defined; a concrete exception is exhibited in the solution.
For a positive integer $m$, consider an $m\times2018$ grid with some cells blank and the others containing 0 or 1. Call the grid complete if every binary string $S$ of length 2018 can be obtained by filling the blank cells in some row with zeros and ones. A row already equal to $S$ also counts. Call the grid minimal if it is complete but ceases to be complete after any row is removed.
(a) For every integer $0\le k\le2018$, prove that a minimal $2^k\times2018$ grid exists with exactly $k$ columns containing both 0 and 1.
(b) If a minimal $m\times2018$ grid has exactly $k$ columns containing both 0 and 1, prove that $m\le2^k$.
A binary string of length 2018 is $x_1x_2\cdots x_{2018}$ with each $x_i\in\{0,1\}$.
Let $n\ge3$ be an integer and let $A_n$ be the set of positive integers less than $n$ and relatively prime to $n$. Define
$$P_n(x)=\sum_{k\in A_n}x^{k-1}.$$
(a) Prove that $P_n(x)$ is divisible by $x^r+1$ for some positive integer $r$.
(b) Find all such $n$ for which $P_n(x)$ is irreducible in $\mathbb Z[x]$.
Let $a\in[\tfrac12,\tfrac23]$. For $n=0,1,2,\ldots$, define
$$u_n=\frac3{2^{n+1}}(-1)^{\lfloor2^{n+1}a\rfloor},\qquad v_n=\frac3{2^{n+1}}(-1)^{n+\lfloor2^{n+1}a\rfloor}.$$
(a) Prove that
$$\left(\sum_{i=0}^{2018}u_i\right)^2+\left(\sum_{i=0}^{2018}v_i\right)^2\le72a^2-48a+10+\frac2{4^{2019}}.$$
(b) Find all values of $a$ for which equality holds.
The vertices of an $m\times n$ rectangular grid $ABCD$ are road junctions, with $(m+1)(n+1)$ junctions in all. We want a route starting at $A$, moving along grid edges parallel to the sides of the rectangle, visiting every junction exactly once, and then returning to $A$.
(a) Prove that such a route exists if and only if $m$ or $n$ is odd.
(b) For $m,n$ satisfying (a), what is the smallest number of junctions at which the route turns?
Let \(ABC\) be an acute triangle with circumcircle \((O)\). Its \(A\)-excircle \((J)\) touches lines \(BC,CA,AB\) at \(D,E,F\), respectively.
(a) Let \(L\) be the midpoint of \(BC\). The circle with diameter \(LJ\) meets lines \(DE,DF\) again at \(K,H\), respectively. Prove that \((BDK)\) and \((CDH)\) have a common point other than \(D\) on \((J)\).
(b) Suppose \(G=EF\cap BC\), and line \(GJ\) meets \(AB,AC\) at \(M,N\). Choose \(P\) on line \(JB\) and \(Q\) on line \(JC\) so that \(AP\perp AB\) and \(AQ\perp AC\). Put \(T=PM\cap QN\). Let \(S\) be the midpoint of the major arc \(BC\) of \((O)\), and \(I\) the incenter. Prove that \(SI\) meets \(AT\) on \((O)\).
Editorial qualification: all named lines, second intersections and circles must be defined and nondegenerate. Acuteness alone does not ensure this, even in part (a). The solution gives explicit counterexamples to the unrestricted printed domain.