In the plane two circles $(C_1),(C_2)$ intersect at $P,Q$. Their common tangent nearer $P$ than $Q$ touches $(C_1)$ at $A$ and $(C_2)$ at $B$. The tangents to $(C_1),(C_2)$ at $P$ meet the other circle again at $E,F$, respectively. Let $H,K$ be points on the lines $AF,BE$, respectively, such that $AH=AP$ and $BK=BP$. Prove that $A,H,Q,K,B$ lie on one circle.
Editorial warning: a full line permits two choices at a prescribed positive distance. The statement as printed does not select the choices needed by the source proof. The solution discusses this limitation rather than silently changing the stem.
Let $k$ be a positive integer. Define $(x_n)_{n\ge1}$ by $x_1=1$, and, for each $n\ge1$, let $x_{n+1}$ be the least positive integer not in $\{x_1;x_2;x_3;\ldots;x_n;x_1+k;x_2+2k;x_3+3k;x_n+nk\}$. Prove that there exists a real number $a$ such that $x_n=\lfloor na\rfloor$ for every positive integer $n$.
Editorial notation note: the printed list omits the continuation dots between its last two shifted terms. The source solution uses all terms $x_i+ik$ for $1\le i\le n$; that convention is used below.
Let $n\ge2$ be a positive integer. Consider the polynomial $1+\square x+\square x^2+\cdots+\square x^{2000}$, with 2000 empty boxes. Players A and B alternately enter a real number into an unfilled box, A first, until all boxes are filled. Write the resulting polynomial as $P(x)=1+a_1x+\cdots+a_{2000}x^{2000}$. The player making the last move loses if $P$ has a real root $\alpha$ with $|\alpha|\lt 1$, and wins otherwise. Prove that one of the two players has a strategy guaranteeing a win.
Editorial note: the initial $n$ is printed in the source but is not used; the game has exactly 2000 boxes.
Let $a,b,c$ be pairwise coprime positive integers. A positive integer $n$ is called “stubborn” if it cannot be written as $n=abx+bcy+caz$ with positive integers $x,y,z$. How many stubborn numbers are there?
Let $a,s$ be real numbers with $s\gt 1$, and let $f:(0,+\infty)\to\mathbb R$. (1) Suppose $a\gt 1$. Prove that if (i) $f(x)^2\le ax^s f(x/a)$ for every $x\gt 0$, and (ii) $f(x)\lt 2^{2000}$ for every $0\lt x\lt 2^{-2000}$, then $f(x)\le x^s a^{1-s}$ for every $x\gt 0$. (2) Suppose $a\gt 2$. Find a function satisfying (i) for which $f(x)\gt x^s a^{1-s}$ for every $x\gt 0$.
Consider $2000$ circles of radius $1$ in the plane, no two of which are tangent, such that each circle intersects at least two other circles. Find the smallest possible number of distinct intersection points of these circles.
A sequence of positive real numbers $x_0,x_1,x_2,\ldots$ is defined by
$$x_0=b,\qquad x_{n+1}=\sqrt{c-\sqrt{c+x_n}}.$$
Find all values of $c$ such that, for every $b\in(0,c)$, this sequence exists and converges to a limit as $n$ tends to infinity.
Two circles $C,C'$ have centers $O,O'$, respectively, with intersecting reference lines $OX$ and $O'X'$. Points $M,M'$ move on $C,C'$ so that the counterclockwise directed angles $\angle XOM$ and $\angle X'O'M'$ are equal. Determine the locus of the midpoint of $MM'$. Suppose $OM$ and $O'M'$ intersect at $Q$. Prove that the circle through $Q,M,M'$ always passes through a fixed point.
Let $p(x)=x^3+153x^2-111x+38$. Prove that $p(n)$ is divisible by $3^{2000}$ for at least nine positive integers $n\lt 3^{2000}$. How many such integers are there?
Let $0\lt \alpha\lt \pi$. Prove that there is a unique quadratic polynomial $x^2+ax+b$ that is a factor of
$$p_n(x)=\sin\alpha\,x^n-\sin(n\alpha)x+\sin(n\alpha-\alpha)$$
for every integer $n\gt 2$. Prove that no linear polynomial $x+c$ divides $p_n(x)$ for every $n\gt 2$.
Find all integers $n\gt 3$ for which there exist $n$ points in space such that no three are collinear, no four are concyclic, and all circles through any three of the points have the same radius.