Consider a $ 2018\times 2018$. board. An "LC-tile" is a tile consisting of $9$ unit squares, having the shape as in the gure below. What is the maximum number of "LC-tiles" that can be placed on the board without superposing them? (Each of the $9$ unit squares of the tile must cover one of the unit squares of the board; a tile may be rotated, turned upside down, etc.) [img]https://cdn.artofproblemsolving.com/attachments/7/4/a2f992bc0341def1a6e5e26ba8a9eb3384698a.png [/img] Alexandru Girban

Find the sum of all integers $x$, $x \ge 3$, such that $201020112012_x$ (that is, $201020112012$ interpreted as a base $x$ number) is divisible by $x-1$

Show that there is no infinite sequence an of natural numbers such that \[a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2\] for all $n\geq 2$

Find all real solutions to the equation $(x^2-9x+19)^{x^2-3x+2} = 1$.

For any positive integer $n$, let $c(n)$ be the largest divisor of $n$ not greater than $\sqrt{n}$ and let $s(n)$ be the least integer $x$ such that $n < x$ and the product $nx$ is divisible by an integer $y$ where $n < y < x$. Prove that, for every $n$, $s(n) = (c(n) + 1) \cdot \left( \frac{n}{c(n)}+1\right)$

Two $5\times1$ rectangles have 2 vertices in common as on the picture. (a) Determine the area of overlap (b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$. [img]https://cdn.artofproblemsolving.com/attachments/9/0/4f1721c7ccdecdfe4d9cc05a17a553a0e9f670.png[/img]

Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.

Two numbers whose sum is $ 6$ and the absolute value of whose difference is $ 8$ are roots of the equation: $ \textbf{(A)}\ x^2\minus{}6x\plus{}7\equal{}0 \qquad \textbf{(B)}\ x^2\minus{}6x\minus{}7\equal{}0 \qquad \textbf{(C)}\ x^2\plus{}6x\minus{}8\equal{}0 \\ \textbf{(D)}\ x^2\minus{}6x\plus{}8\equal{}0 \qquad \textbf{(E)}\ x^2\plus{}6x\minus{}7\equal{}0$

Let $ ABC $ be a triangle and the point $ D $ is on the segment $ BC $ such that $ AD $ is the interior bisector of $ \angle A $. We stretch $ AD $ such that it meets the circumcircle of $ \Delta ABC $ at $ M $. We draw a line from $ D $ such that it meets the lines $ MB,MC $ at $ P,Q $, respectively ($ M $ is not between $ B,P $ and also is not between $ C,Q $). Prove that $ \angle PAQ\geq\angle BAC $.

A [i]string of length $n$[/i] is a sequence of $n$ characters from a specified set. For example, $BCAAB$ is a string of length 5 with characters from the set $\{A,B,C\}$. A [i]substring[/i] of a given string is a string of characters that occur consecutively and in order in the given string. For example, the string $CA$ is a substring of $BCAAB$ but $BA$ is not a substring of $BCAAB$. [list=a][*]List all strings of length 4 with characters from the set $\{A,B,C\}$ in which both the strings $AB$ and $BA$ occur as substrings. (For example, the string $ABAC$ should appear in your list.) [*]Determine the number of strings of length 7 with characters from the set $\{A,B,C\}$ in which $CC$ occures as a substring. [*]Let $f(n)$ be the number of strings of length $n$ with characters from the set $\{A,B,C\}$ such that [list][*]$CC$ occurs as a substring, and[*]if either $AB$ or $BA$ occurs as a substring then there is an occurrence of the substring $CC$ to its left.[/list] (for example, when $n\;=\;6$, the strings $CCAABC$ and $ACCBBB$ and $CCABCC$ satisfy the requirements, but the strings $BACCAB$ and $ACBBAB$ and $ACBCAC$ do not). Prove that $f(2097)$ is a multiple of $97$.[/list]

Show that a disk of radius $ 2$ can be covered by seven (possibly overlapping) disks of radius $ 1$.

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $a_1,a_2 \cdots a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let $(i)$ $\sum_{i=1}^{n}a_{2i-1}^2 =x$ $(ii)$ $\sum _{i=1}^{n}a_{2i}^2=y$ $(iii)$ $a_n+a_{n+1}=z$ Express $d$ in terms of $x,y,z,n$

Maria and Bilyana play the following game. Maria has $2024$ fair coins and Bilyana has $2023$ fair coins. They toss every coin they have. Maria wins if she has strictly more heads than Bilyana, otherwise Bilyana wins. What is the probability of Maria winning this game?

Distinct positive integers $A$ and $B$ are given$.$ Prove that there exist infinitely many positive integers that can be represented both as $x_{1}^2+Ay_{1}^2$ for some positive coprime integers $x_{1}$ and $y_{1},$ and as $x_{2}^2+By_{2}^2$ for some positive coprime integers $x_{2}$ and $y_{2}.$ [i](Golovanov A.S.)[/i]

A polynomial $P(x)$ of degree $n \ge 5$ with integer coefficients has $n$ distinct integer roots, one of which is $0$. Find all integer roots of the polynomial $P(P(x))$.