Let $ABC$ be a triangle with $\angle A\lt 60^\circ$. Let $X,Y$ be points on $AB,AC$, respectively, such that $CA+AX=CB+BX$ and $BA+AY=BC+CY$. Let $P$ be the point in the plane such that $PX\perp AB$ and $PY\perp AC$. Prove that $\angle BPC\lt 120^\circ$.
Students in a class form groups each of which contains exactly three members such that any two distinct groups have at most one member in common. Prove that, when the class size is 46, there is a set of 10 students in which no group is properly contained.
Let $\Gamma$ be the circumcircle of triangle $ABC$. A circle through $A,C$ meets sides $BC,BA$ at $D,E$, respectively. Lines $AD,CE$ meet $\Gamma$ again at $G,H$, respectively. The tangents to $\Gamma$ at $A,C$ meet line $DE$ at $L,M$, respectively. Prove that lines $LH,MG$ meet at a point on $\Gamma$.
Consider $f:\mathbb N_0\to\mathbb N_0$, where $\mathbb N_0$ is the set of nonnegative integers, defined by
$$f(0)=0,\qquad f(2n)=2f(n),\qquad f(2n+1)=n+2f(n)\quad(n\ge0).$$
(a) Determine $L=\{n:f(n)\lt f(n+1)\}$, $E=\{n:f(n)=f(n+1)\}$ and $G=\{n:f(n)\gt f(n+1)\}$.
(b) For every $k\ge0$, find a formula for $a_k=\max\{f(n):0\le n\le2^k\}$.
Let $a,b,c$ be integers satisfying $0\lt a\lt c-1$ and $1\lt b\lt c$. For $0\le k\le a$, let $r_k$, $0\le r_k\lt c$, be the remainder of $kb$ divided by $c$. Prove that the sets $\{r_0,r_1,\ldots,r_a\}$ and $\{0,1,\ldots,a\}$ are different.
