Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $D,E,F$ be the reflections of $H$ in $BC,CA,AB$, respectively, and let $A^{\prime},B^{\prime},C^{\prime}$ be the reflections of $A,B,C$ in those same respective lines. Let $S$ be the circumcenter of $A^{\prime} B^{\prime} C^{\prime}$, and $H^{\prime}$ the orthocenter of $DEF$. Let $J$ be the center of the circle through the three perpendicular projections of $H$ onto $B^{\prime} C^{\prime},C^{\prime} A^{\prime},A^{\prime} B^{\prime}$. Prove that $HJ\parallel H^{\prime} S$.
At an applied-mathematics summer camp there are $8m+1$ boys, where $m>5$, and some girls. Each girl knows exactly three boys, and every two boys have exactly one girl whom they both know. Let $n$ be the largest number of girls that can be selected so that each boy knows at most one selected girl. Prove that $n\ge2m+1$.
Find all positive integers $k$ for which there are infinitely many positive integers $n$ such that $\binom{(2025+k)n}{2025n}$ is not divisible by $kn+1$.
An $n\times n$ board has rows and columns numbered $1$ through $n$; the cell in row $i$, column $j$ is denoted $(i,j)$. A subset $A$ of its cells is called good if, for any two cells $(x_1,y),(x_2,y)$ in $A$, none of the cells $(u,v)$ satisfying
$$x_1\lt u\le x_2,\quad v\lt y,$$
or
$$x_1\le u\lt x_2,\quad v>y,$$
belongs to $A$. Find the smallest number of pairwise disjoint good subsets such that every board cell belongs to exactly one subset.
Let $P(x)=x^4-x^3+x$.
a) Prove that for every positive real $a$, the polynomial $P(x)-a$ has exactly one positive root.
b) Define $a_1=1/3$ and, for $n\ge1$, let $a_{n+1}$ be the positive root of $P(x)-a_n$. Prove that $(a_n)$ has a finite limit and find it.
For each integer $n\ge0$, put
$$u_n=(2+\sqrt5)^n+(2-\sqrt5)^n.$$
a) Prove that $u_n$ is a positive integer for every $n\ge0$. What is the largest possible remainder when $u_n$ is divided by 24?
b) Find all ordered pairs of positive integers $(a,b)$ with $a,b\lt 500$ such that for every odd $n$,
$$u_n\equiv a^n-b^n\pmod{1111}.$$
Let $ABC$ be an acute scalene triangle inscribed in $(O)$, with orthocenter $H$. Line $AH$ meets $(O)$ again at $D\ne A$. Let $E,F$ be the midpoints of $AB,AC$, respectively. The line through $H$ perpendicular to $HF$ meets $BC$ at $K$.
a) Line $DK$ meets $(O)$ again at $Y\ne D$. Prove that the intersection of $BY$ with the perpendicular bisector of $BK$ lies on the circumcircle of $OFY$.
b) The line through $H$ perpendicular to $HE$ meets $BC$ at $L$. Line $DL$ meets $(O)$ again at $Z\ne D$. Let $M,N,P$ be the intersections of the pairs $(BZ,OE)$, $(CY,OF)$, $(BY,CZ)$, respectively. Put $T=YZ\cap MN$, and let $d$ be the line through $T$ perpendicular to $OA$. Prove that $d$ passes through the midpoint of $AP$.
Let $ABC$ be an acute scalene triangle with altitudes $AD,BE,CF$, where $D\in BC,E\in CA,F\in AB$. Let $H,O,I$ be its orthocenter, circumcenter and incenter. Let $M,N,P$ be the midpoints of $BC,CA,AB$, and put $X=AI\cap NP$, $Y=BI\cap PM$, $Z=CI\cap MN$.
a) Prove that the circumcircles of $AXD,BYE,CZF$ have two common points lying on $OH$.
b) Lines $XP,YM,ZN$ meet these three circles again at $X',Y',Z'$, respectively, distinct from $X,Y,Z$. Let $J$ be the reflection of $I$ in $O$. Prove that $X',Y',Z'$ are collinear on a line perpendicular to $HJ$.
Let $k$ be a positive integer. A $3k\times3k$ board has cells $(i,j)$, where i counts columns from left to right and j counts rows from bottom to top. Place $4k$ marbles, at most one per cell, subject to:
- Every row and every column contains at least one marble.
- Every marble shares a row or a column with at least one other marble.
a) For $k=1$, how many placements of four marbles satisfy these conditions? Two placements are different if some cell is occupied in one but not in the other.
b) For general $k\ge1$, determine the largest natural number N such that, however N distinct cells are marked, there is a valid placement of $4k$ marbles avoiding every marked cell.