Let the integer sequence $(a_n)_{n\ge0}$ be defined by $a_0=1$ and
$$a_n=a_{n-1}+a_{\lfloor n/3\rfloor}\qquad(n\ge1).$$
Prove that for every prime $p\le13$, there are infinitely many integers $k\ge0$ for which $p\mid a_k$. Here $\lfloor x\rfloor$ is the integer part of $x$.
Two circles intersect at two distinct points $A,B$. A common tangent touches them at $P,T$. The tangents at $P,T$ to the circumcircle of $APT$ intersect at $S$. Let $H$ be the reflection of $B$ across $PT$. Prove that $A,S,H$ are collinear.
A club has 42 members. Among any 31 members, there is a boy and a girl who know each other. Prove that one can choose 12 pairs of boys and girls who know each other.
Editorial clarification: the pairs must be disjoint, so each member occurs in at most one pair.
Consider real numbers $a,b,c$ satisfying $21ab+2bc+8ca\le12$. Find the minimum of
$$P(a,b,c)=\frac1a+\frac1b+\frac1c.$$
Editorial note: the supplied statement has domain and objective discrepancies compared with corroborating sources. Interpret reciprocals only when their denominators are nonzero; the solution separates the literal statement from explicitly labeled corrected variants.
Let $n>1$ be an integer. Let $T$ consist of the integer-coordinate points $(x,y,z)$ with $1\le x,y,z\le n$. Color points so that if $A=(x_0,y_0,z_0)$ is colored, a point $B=(x_1,y_1,z_1)$ with $x_1\le x_0$, $y_1\le y_0$, $z_1\le z_0$ cannot be colored. Find the largest possible number of colored points.
Editorial clarification: $B$ must mean a point distinct from $A$. The literal self-inclusive reading forbids every colored point. The solution records that reading and proves the intended distinct-point extremum.
Let a sequence $(a_n)_{n\ge1}$ satisfy
$$0\lt a_{n+1}-a_n\le2001\qquad(n\ge1).$$
Prove that there are infinitely many pairs of positive integers $(p,q)$ such that $p\lt q$ and $a_p\mid a_q$.
Editorial note: the supplied page does not specify that the terms are integers. The standard divisibility statement is proved below under the explicitly stated integer-term assumption, corroborated by another transcription.
Two circles with centers $O,O'$ intersect at $A,B$. The line $TT'$ is tangent to the first circle at $T$ and to the second at $T'$. The perpendiculars from $T,T'$ meet $OO'$ at $S,S'$. Ray $AS$ meets the first circle again at $R$, and ray $AS'$ meets the second again at $R'$. Prove that $R,B,R'$ are collinear.
Let $n$ be a positive integer and let $a,b$ be coprime integers greater than $1$. Suppose $M=a^{6^n}+b^{6^n}$ has at least two odd divisors greater than $1$. Determine the remainder when $M$ is divided by $6\cdot12^n$.
For real numbers $a,b$, define $x_0,x_1,x_2,\ldots$ by $x_0=a$, $x_{n+1}=x_n+b\sin x_n$. If $b=1$, prove that the sequence converges to a finite limit for every $a$. If $b\gt 2$, prove that it diverges for some $a$.
Find the largest value of
$$\frac1{\sqrt x}+\frac2{\sqrt y}+\frac3{\sqrt z}$$
for positive numbers $x,y,z$ satisfying
$$\begin{cases}\dfrac1{\sqrt2}\le z\le\min\{x\sqrt2,y\sqrt3\},\\[3pt]x+z\sqrt3\ge\sqrt6,\\[3pt]y\sqrt3+z\sqrt{10}\ge2\sqrt5.\end{cases}$$
Let $a_1,a_2,\ldots,a_{2n}$ be a permutation of $\{1,2,\ldots,2n\}$ such that
$$|a_i-a_{i+1}|\ne|a_j-a_{j+1}|$$
for all distinct indices for which these adjacent differences are defined. Prove that $a_1=a_{2n}+n$ if and only if $1\le a_{2i}\le n$ for every $i=1,2,\ldots,n$.