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.
Let $(x_n)$ be defined by $x_1=2$ and $x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$ for $n\ge1$.
(a) Prove that $(x_n)$ has a finite limit and find it.
(b) Prove that $n\le x_1+x_2+\cdots+x_n\le n+1$ for every positive integer $n$.
Let $ABC$ be an acute scalene triangle and $D$ a point on side $BC$. Choose $E$ on side $AB$ and $F$ on side $AC$ so that $\angle DEB=\angle DFC$. The lines $DF,DE$ meet $AB,AC$ at $M,N$, respectively. Let $(I_1),(I_2)$ be the circumcircles of $DEM,DFN$, respectively. Let $(J_1)$ be internally tangent to $(I_1)$ at $D$ and tangent to $AB$ at $K$, and let $(J_2)$ be internally tangent to $(I_2)$ at $D$ and tangent to $AC$ at $H$. Let $P\ne D$ be the other intersection of $(I_1),(I_2)$, and $Q\ne D$ the other intersection of $(J_1),(J_2)$.
(a) Prove that $D,P,Q$ are collinear.
(b) The circumcircle of $AEF$ meets the circumcircle of $AHK$ and the line $AQ$ again at $G,L$, respectively, with $G,L\ne A$. Prove that the tangent at $D$ to the circumcircle of $DQG$ meets $EF$ at a point on the circumcircle of $DLG$.
An investor has two rectangular plots of land, each measuring $120\,\mathrm m\times100\,\mathrm m$.
(a) On the first plot, the investor wants to build a house with a rectangular footprint measuring $25\,\mathrm m\times35\,\mathrm m$, and nine circular flowerbeds outside the house, each of diameter $5\,\mathrm m$. Prove that wherever the nine flowerbeds are built first, enough space remains for the house.
(b) On the second plot, the investor wants a pond shaped as a convex polygon, such that from any point of the remaining land one can reach the pond's bank by walking no more than $5\,\mathrm m$. Prove that the pond's perimeter is at least $(440-20\sqrt2)\,\mathrm m$.
In the Cartesian plane $Oxy$, let $(C)$ be the graph of $y=\sqrt[3]{x^2}$. A variable line $d$ meets $(C)$ at three distinct points with abscissas $x_1,x_2,x_3$.
(a) Prove that $\sqrt[3]{\dfrac{x_1x_2}{x_3^2}}+\sqrt[3]{\dfrac{x_2x_3}{x_1^2}}+\sqrt[3]{\dfrac{x_3x_1}{x_2^2}}$ is constant.
(b) Prove that $\sqrt[3]{\dfrac{x_1^2}{x_2x_3}}+\sqrt[3]{\dfrac{x_2^2}{x_3x_1}}+\sqrt[3]{\dfrac{x_3^2}{x_1x_2}}\lt -\dfrac{15}4$.
Let $n,d$ be positive integers. Let $S_n(d)$ be the set of all ordered tuples $(x_1,\ldots,x_d)$ satisfying:
(i) $x_i\in\{1,2,\ldots,n\}$ for $1\le i\le d$;
(ii) $x_i\ne x_{i+1}$ for $1\le i\le d-1$;
(iii) there are no indices $1\le i\lt j\lt k\lt l\le d$ such that $x_i=x_k$ and $x_j=x_l$.
(a) Find $|S_3(5)|$.
(b) Prove that $S_n(d)$ is nonempty if and only if $d\le2n-1$.
Let $(x_n)$ be defined by $x_0=2$, $x_1=1$ and $x_{n+2}=x_{n+1}+x_n$ for all $n\ge0$.
(a) For $n\ge1$, prove that if $x_n$ is prime, then either $n$ is prime or $n$ has no odd prime divisor.
(b) Find all pairs of nonnegative integers $(m,n)$ such that $x_n$ is divisible by $x_m$.
Let $ABC$ be an acute scalene triangle with centroid $G$, inscribed in a circle $(O)$. Let $H_a,H_b,H_c$ be the feet of the altitudes from $A,B,C$, and let $D,E,F$ be the midpoints of $BC,CA,AB$, respectively. The rays $GH_a,GH_b,GH_c$ meet $(O)$ at $X,Y,Z$, respectively.
(a) Prove that the circumcircle of $XCE$ passes through the midpoint of $BH_a$.
(b) Let $M,N,P$ be the midpoints of $AX,BY,CZ$, respectively. Prove that $DM,EN,FP$ are concurrent.