The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments. [asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy] How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself? [list] [*] some rotation around a point of line $\ell$ [*] some translation in the direction parallel to line $\ell$ [*] the reflection across line $\ell$ [*] some reflection across a line perpendicular to line $\ell$ [/list] $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$

Marine biologists are studying a new species of shellfish whose first generation consists of $100$ shellfish, and their colony reproduces as follows: if a given generation consists of $N$ shellfish (where $5\mid N$ always holds), they divide themselves into $N/5$ groups of $5$ shellfish each. Each group collectively produces $15$ offspring, who form the next generation. Some of the shellfish contain a pearl, but a shellfish can only contain a pearl if none of its direct ancestors contained a pearl. The value of a pearl is determined by the generation of the shellfish containing it: in the $n^{\mathrm{th}}$ generation, its value is $1/3^n$. Find the maximum possible total value of the pearls in the colony. [i]Proposed by: Csongor Beke, Cambridge[/i]

On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$. $(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$. $(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.

Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?

Let $a,b,c$ natural numbers for which $a^2 + b^2 + c^2 = (a-b) ^2 + (b-c)^ 2 + (c-a) ^2$. Show that $ab, bc, ca$ and $ab + bc + ca$ are perfect squares .

Prove that if a positive integer $n$ divides the five-digit numbers $\overline{a_1a_2a_3a_4a_5}$, $\overline{b_1b_2b_3b_4b_5}$, $\overline{c_1c_2c_3c_4c_5}$, $\overline{d_1d_2d_3d_4d_5}$, $\overline{e_1e_2e_3e_4e_5}$, then it also divides the determinant $$D=\begin{vmatrix}a_1&a_2&a_3&a_4&a_5\\b_1&b_2&b_3&b_4&b_5\\c_1&c_2&c_3&c_4&c_5\\d_1&d_2&d_3&d_4&d_5\\e_1&e_2&e_3&e_4&e_5\end{vmatrix}.$$

Let $A$ be the set of all sequences from 0’s or 1’s with length 4. What’s the minimal number of sequences that can be chosen, so that an arbitrary sequence from $A$ differs at most in 1 position from one of the chosen?

Prove that for each $n \ge 4$ a parallelogram can be dissected in $n$ cyclic quadrilaterals.

A competition room of HOMC has $m \times n$ students where $m, n$ are integers larger than $2$. Their seats are arranged in $m$ rows and $n$ columns. Before starting the test, every student takes a handshake with each of his/her adjacent students (in the same row or in the same column). It is known that there are totally $27$ handshakes. Find the number of students in the room.

Let $m,n,a_1,a_2,\dots,a_n$ be positive integers and $r$ be a real number. Prove that the equation \[\lfloor a_1x\rfloor+\lfloor a_2x\rfloor+\cdots+\lfloor a_nx\rfloor=sx+r\] has exactly $ms$ solutions in $x$, where $s=a_1+a_2+\cdots+a_n+\frac1m$. [i]Linus Tang[/i]

Let $A$ consist of $16$ elements of the set $\{1, 2, 3,..., 106\}$, so that the difference of two arbitrary elements in $A$ are different from $6, 9, 12, 15, 18, 21$. Prove that there are two elements of $A$ for which their difference equals to $3$.

We define a binary operation $\star$ in the plane as follows: Given two points $A$ and $B$ in the plane, $C = A \star B$ is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points $I, M, O$ in the plane if $I \star (M \star O) = (O \star I)\star M$ holds?

Let $f(N) = N \left( \frac{9}{10} \right)^N$ , and let $\frac{m}{n}$ denote the maximum value of $f(N)$, as $N$ ranges over the positive integers. If $m$ and $n$ are relatively prime positive integers, find the remainder when $m + n$ is divided by $1000$.

[b]p1.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p2.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from $1$ to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number $3$? The total number of all obtained points is $264$. [b]p2.[/b] There are exactly $n$ students in a high school. Girls send messages to boys. The first girl sent messages to $5$ boys, the second to $7$ boys, the third to $6$ boys, the fourth to $8$ boys, the fifth to $7$ boys, the sixth to $9$ boys, the seventh to $8$, etc. The last girl sent messages to all the boys. Prove that $n$ is divisible by $3$. [b]p4.[/b] In what minimal number of triangles can one cut a $25 \times 12$ rectangle in such a way that one can tile by these triangles a $20 \times 15$ rectangle. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[i](5 pts)[/i] For some positive integer $n$, let $P(x)$ be an $n$th degree polynomial with real coefficients. [i]Note: you may cite, without proof, the Fundamental Theorem of Algebra, which states that every non-constant polynomial with complex coefficients has a complex root.[/i] (a) [i](2 pts)[/i] Show that there is an integer $k \ge \frac{n}{2}$ and a sequence of non-constant polynomials with real coefficients $Q_1(x), Q_2(x), \dots, Q_k(x)$ such that \[ P(x) = \prod_{i = 1}^k Q_i(x). \] (b) [i](1 pt)[/i] If $n$ is odd, then show that $P(x)$ has a real root. (c) [i](2 pts)[/i] Let $a$ and $b$ be real numbers, and let $m$ be a positive integer. If $\zeta = a + bi$ is a nonreal root of $P(x)$ of multiplicity $m$, then show that $\overline{\zeta} = a - bi$ is a nonreal root of $P(x)$ of multiplicity $m$.

Let $ ABCD$ be a trapezoid such that $ AB \parallel CD$ and assume that there are points $ E$ on the line outside the segment $ BC$ and $ F$ on the segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Let $ I,J,K$ respectively be the intersection of line $ EF$ and line $ CD$, the intersection of line $ EF$ and line $ AB$, and the midpoint of segment $ EF$. Prove that $ K$ is on the circumcircle of triangle $ CDJ$ if and only if $ I$ is on the circumcircle of triangle $ ABK$. [i]Utari Wijayanti, Bandung[/i]

Several frogs are sitting on the real line at distinct integer points. In each move, one of them can take a $1$-jump towards the right as long as they are still in on distinct points. We calculate the number of ways they can make $N$ moves in this way for a positive integer $N$. Prove that if the jumps were all towards the left, we will still get the same number of ways. [i](F. Petrov)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

Prove that there exist $2001$ convex polyhedra such that any three of them do not have any common points but any two of them touch each other (i.e., have at least one common boundary point but no common inner points).

Prove that $y\sqrt{3-2x}+x\sqrt{3-2y}\le x^2+y^2$ for any number $x,y\in\left[1,\frac32\right]$. When does equality occur?

If $a=b-c, b=c-d, c=d-a$ and $abcd\ne 0$, then what is the value of $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}$?

Let $ABCD$ be a parallelogram of center $O$. Prove that for any point $M\in (AB)$, there exist unique points $N\in (OC)$ and $P\in (OD)$ such that $O$ is the center of mass of $\triangle MNP$.

Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

Each of $999$ numbers placed in a circular way is either $1$ or $-1$. (Both values appear). Consider the total sum of the products of every $10$ consecutive numbers. $(a)$ Find the minimal possible value of this sum. $(b)$ Find the maximal possible value of this sum.

Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$-hedron. On earth we don't have any other information about sides of planet tarator. We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money. On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money? [b]Note 1:[/b] Tourists spend no money during travel! [b]Note 2:[/b] The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables. [b]Note 3:[/b] The maximum must be among all possible polyhedras. Time allowed for this problem was 90 minutes.

Let $P$ and $Q$ be points on the plane $ABC$ such that $m(\widehat{BAC})=90^\circ$, $|AB|=1$, $|AC|=\sqrt 2$, $|PB|=1=|QB|$, $|PC|=2=|QC|$, and $|PA|>|QA|$. What is $|PA|/|QA|$? $ \textbf{(A)}\ \sqrt 2 +\sqrt 3 \qquad\textbf{(B)}\ 5-\sqrt 6 \qquad\textbf{(C)}\ \sqrt 6 -\sqrt 2 \qquad\textbf{(D)}\ \sqrt 6 + 1 \qquad\textbf{(E)}\ \text{None} $