Keith decides that a sequence of digits is [i]slick[/i] if every pair of adjacent digits in the sequence is divisible by either $23$ or $17.$ What is the greatest possible number of $2$s in a $2021$-digit long slick sequence?

There are 2020 inhabitants in a town. Before Christmas, they are all happy; but if an inhabitant does not receive any Christmas card from any other inhabitant, he or she will become sad. Unfortunately, there is only one post company which offers only one kind of service: before Christmas, each inhabitant may appoint two different other inhabitants, among which the company chooses one to whom to send a Christmas card on behalf of that inhabitant. It is known that the company makes the choices in such a way that as many inhabitants as possible will become sad. Find the least possible number of inhabitants who will become sad.

For every natural number $ x$, let $ Q(x)$ be the sum and $ P(x)$ the product of the (decimal) digits of $ x$. Show that for each $ n \in \mathbb{N}$ there exist infinitely many values of $ x$ such that: $ Q(Q(x))\plus{}P(Q(x))\plus{}Q(P(x))\plus{}P(P(x))\equal{}n$.

The figure shows all 6 colorings with for different colors of a $1\times 1$ square divided in four $\tfrac{1}{2} \times \tfrac{1}{2}$ cells (two colorings are considered equal if one is the result of rotating the other). Each of the $1\times 1$ colorings will be used as a piece for a puzzle. The pieces can be rotated but not reflected. Two pieces [i]fit[/i] if when sharing a side, the touching $\tfrac{1}{2} \times \tfrac{1}{2}$ cells are the same color respectively (see examples). ¿Is it possible to assemble a $3 \times 2$ puzzle using each of the 6 pieces exactly once and such that every pair of adjacent pieces fit? [img]https://imagizer.imageshack.com/img922/6019/ZUKcED.jpg[/img]

Do there exist three different prime numbers $p$, $q$ and $r$ such that $p^2 + d$ is divisible by $qr$, $q^2 + d$ is divisible by $rp$ and $r^2 + d$ is divisible by $pq$, if (a) $d = 10$; (b) $d = 11$? (V Senderov)

How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable) $\textbf{(A)}\ 210\qquad\textbf{(B)}\ 420\qquad\textbf{(C)}\ 630\qquad\textbf{(D)}\ 840\qquad\textbf{(E)}\ 1050$

Given an $n$-gon ($n\ge 4$), consider a set $T$ of triangles formed by vertices of the polygon having the following property: Every two triangles in T have either two common vertices, or none. Prove that $T$ contains at most $n$ triangles.

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Show that there exist integers $m$ and $n$ such that \[\dfrac{m}{n}=\sqrt[3]{\sqrt{50}+7}-\sqrt[3]{\sqrt{50}-7}.\]

$f(x)$ is a function defined on $\mathbb{R}$. $f(1)=1$, and for all $x\in\mathbb{R}$, $f(x+5)\geq x+5,f(x+1)\leq f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)=$________.

Let triangle $ABC$ be perpendicular at $A.$ Let $M$ be the midpoint of segment $\overline{BC}.$ Point $D$ lies on side $\overline{AC}$ and satisfies $|AD|=|AM|.$ Let $P \neq C$ be the intersection of the circumcircle of triangles $AMC$ and $BDC.$ Prove that $CP$ bisects the angle at $C$ of triangle $ABC.$

For positive integer $n$ let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999,488$ and $a_7 = 9,999,999,488$. Find the value of $n$ so that an is divisible by the highest power of $2$.

In an acute traingle $ABC$ with $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.

In a convex quadrilateral $ABCD$, $\angle ABD=30^\circ$, $\angle BCA=75^\circ$, $\angle ACD=25^\circ$ and $CD=CB$. Extend $CB$ to meet the circumcircle of triangle $DAC$ at $E$. Prove that $CE=BD$. [i]Proposed by BJ Venkatachala[/i]

Let $A_1A_2$ and $B_1B_2$ be the diagonals of a rectangle, and let $O$ be its center. Find and construct the set of all points $P$ that satisfy simultaneously the four inequaliies: $$A_1P > OP , \\A_2P > OP, \ \ B_1P > OP , \ \ B_2P > OP.$$

Let $A$ be a set of $20$ distinct positive integers which are all no greater than $397$. Prove that for any positive integer $n$ it is possible to pick four (not necessarily distinct) elements $x_1, x_2, x_3, x_4$ of $A$ satisfying $x_1 \neq x_2$ and $$(x_1-x_2)n\equiv x_3-x_4 \pmod{397}.$$

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.

The sum of the roots of the equation $ 4x^2\plus{}5\minus{}8x\equal{}0$ is equal to: $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ -5 \qquad \textbf{(C)}\ -\dfrac{5}{4} \qquad \textbf{(D)}\ -2 \qquad \textbf{(E)}\ \text{None of these}$

Can a regular triangle be placed inside a regular hexagon in such a way that all vertices of the triangle were seen from each vertex of the hexagon? (Point $A$ is seen from $B$, if the segment $AB$ dots not contain internal points of the triangle.)

For any nonnegative integer $r$, let $S_r$ be a function whose domain is the natural numbers that satisfies $$S_r(p^{\alpha}) = \begin{cases} 0\,\, if \,\, if \,\, p \le r \\ p^{{\alpha}-1}(p -r) \,\, if \,\,p > r \end{cases}$$ for all primes $p$ and positive integers ${\alpha}$, and that $S_r(ab) = S_r(a)Sr_(b)$ whenever $a$ and $b$ are relatively prime. Now, suppose there are $n$ squirrels at a party. Each squirrel is labeled with a unique number from the set $\{1, 2,..., n\}$. Two squirrels are friends with each other if and only if the difference between their labels is relatively prime to $n$. For example, if $n = 10$, then the squirrels with labels $3$ and $10$ are friends with each other because $10 - 3 = 7$, and $7$ is relatively prime to $10$. Fix a positive integer $m$. Define a clique of size $m$ to be any set of m squirrels at the party with the property that any two squirrels in the clique are friends with each other. Determine, with proof, a formula (using $S_r$) for the number of cliques of size $m$ at the squirrel party.

Let $f_1(n)$ be the number of divisors that $n$ has, and define $f_k(n)=f_1(f_{k-1}(n))$. Compute the smallest integer $k$ such that $f_k(2013^{2013})=2$.

Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized? [i]Proposed by Lewis Chen[/i]

Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.

Let $ABC$ be an acute triangle. Let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively and let $H$ be the orthocenter of $\triangle ABC$. Let $X$ be an arbitrary point on the circumcircle of $\triangle DEF$ and let the circumcircles of $\triangle EHX$ and $\triangle FHX$ intersect the second time the lines $CF$ and $BE$ second at $Y$ and $Z$, respectively. Prove that the line $YZ$ passes through the midpoint of $BC$.