Let $ABC$ be an equilateral triangle. For an arbitrary line $\ell$. through $B$, the orthogonal projections of $A$ and $C$ on $\ell$ are denoted by $D$ and $E$ respectively. If $D\ne E$, equilateral triangles $DEP$ and $DET$ are constructed on different sides of $\ell$. Find the loci of $P$ and $T$.

Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$, $AD$ and $BC$ at $J$, $AC$ and $BD$ at $K$, and let $N$ be a point on $AB$. Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$.

Let $ABCD$ be an isosceles trapezoid with bases $AD> BC$. Let $X$ be the intersection of the bisectors of $\angle BAC$ and $BC$. Let $E$ be the intersection of$ DB$ with the parallel to the bisector of $\angle CBD$ through $X$ and let $F$ be the intersection of $DC$ with the parallel to the bisector of $\angle DCB$ through $X$. Show that quadrilateral $AEFD$ is cyclic.

Let $H$ be the orthocentre of $\triangle ABC$ and let $P$ be a point on the circumcircle of $\triangle ABC$, distinct from $A,B,C$. Let $E$ and $F$ be the feet of altitudes from $H$ onto $AC$ and $AB$ respectively. Let $PAQB$ and $PARC$ be parallelograms. Suppose $QA$ meets $RH$ at $X$ and $RA$ meets $QH$ at $Y$. Prove that $XE$ is parallel to $YF$.

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$. Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.

Given a circle, point $A$ on it and point $M$ inside it. We consider the chords $BC$ passing through $M$. Prove that the circles passing through the midpoints of the sides of all the triangles $ABC$ are tangent to a fixed circle.

For a regular hexahedron and a regular octahedron, all their faces are regular triangles, whose lengths of each side are $a$. Their inradius are $r_1,r_2$. $\frac{r_1}{r_2}=\frac{m}{n}, \gcd(m,n)=1$. Then $mn=$________.

[u]Round 1 [/u] [b]p1. [/b]Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the 12th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p2.[/b] Show that in the sequence $10017$, $100117$, $1001117$, $...$ all numbers are divisible by $53$. [b]p3.[/b] Harry and Draco have three wands: a bamboo wand, a willow wand, and a cherry wand, all of the same length. They must perform a spell wherein they take turns picking a wand and breaking it into three parts – first Harry, then Draco, then Harry again. But in order for the spell to work, Harry has to make sure it is possible to form three triangles out of the pieces of the wands, where each triangle has a piece from each wand. How should he break the wands to ensure the success of the spell? [b]p4.[/b] A $2\times 2\times 2$ cube has $4$ equal squares on each face. The squares that share a side are called neighbors (thus, each square has $4$ neighbors – see picture). Is it possible to write an integer in each square in such a way that the sum of each number with its $4$ neighbors is equal to $13$? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/8/4/0f7457f40be40398dee806d125ba26780f9d3a.png[/img] [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2 [/u] [b]p6.[/b] A red square is placed on a table. $2010$ white squares, each the same size as the red square, are then placed on the table in such a way that the red square is fully covered and the sides of every white square are parallel to the sides of the red square. Is it always possible to remove one of the white squares so the red square remains completely covered? [b]p7.[/b] A computer starts with a given positive integer to which it randomly adds either $54$ or $77$ every second and prints the resulting sum after each addition. For example, if the computer is given the number $1$, then a possible output could be: $1$, $55$, $109$, $186$, $…$ Show that after finitely many seconds the computer will print a number whose last two digits are the same. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Ephram Chun is a senior and math captain at Lexington High School. He is well-loved by the freshmen, who seem to only listen to him. Other than being the father figure that the freshmen never had, Ephramis also part of the Science Bowl and Science Olympiad teams along with being part of the highest orchestra LHS has to offer. His many hobbies include playing soccer, volleyball, and the many forms of chess. We hope that he likes the questions that we’ve dedicated to him! [b]p1.[/b] Ephram is scared of freshmen boys. How many ways can Ephram and $4$ distinguishable freshmen boys sit together in a row of $5$ chairs if Ephram does not want to sit between $2$ freshmen boys? [b]p2.[/b] Ephram, who is a chess enthusiast, is trading chess pieces on the black market. Pawns are worth $\$100$, knights are worth $\$515$, and bishops are worth $\$396$. Thirty-four minutes ago, Ephrammade a fair trade: $5$ knights, $3$ bishops, and $9$ rooks for $8$ pawns, $2$ rooks, and $11$ bishops. Find the value of a rook, in dollars. [b]p3.[/b] Ephramis kicking a volleyball. The height of Ephram’s kick, in feet, is determined by $$h(t) = - \frac{p}{12}t^2 +\frac{p}{3}t ,$$ where $p$ is his kicking power and $t$ is the time in seconds. In order to reach the height of $8$ feet between $1$ and $2$ seconds, Ephram’s kicking power must be between reals $a$ and $b$. Find is $100a +b$. [b]p4.[/b] Disclaimer: No freshmen were harmed in the writing of this problem. Ephram has superhuman hearing: He can hear sounds up to $8$ miles away. Ephramstands in the middle of a $8$ mile by $24$ mile rectangular grass field. A freshman falls from the sky above a point chosen uniformly and randomly on the grass field. The probability Ephram hears the freshman bounce off the ground is $P\%$. Find $P$ rounded to the nearest integer. [img]https://cdn.artofproblemsolving.com/attachments/4/4/29f7a5a709523cd563f48176483536a2ae6562.png[/img] [b]p5.[/b] Ephram and Brandon are playing a version of chess, sitting on opposite sides of a $6\times 6$ board. Ephram has $6$ white pawns on the row closest to himself, and Brandon has $6$ black pawns on the row closest to himself. During each player’s turn, their only legal move is to move one pawn one square forward towards the opposing player. Pawns cannot move onto a space occupied by another pawn. Players alternate turns, and Ephram goes first (of course). Players take turns until there are no more legal moves for the active player, at which point the game ends. Find the number of possible positions the game can end in. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Six straight lines are drawn on the plane. It is known that for any three of them there is a fourth of the same set of lines, such that all four will touch some circle. Do all six lines necessarily touch the same circle? (I. Bogdanov)

A triangle $\Delta$ with sidelengths $a\leq b\leq c$ is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of $\Delta$. Prove that $b^2>ac$.

Let $ABC$ be an equilateral triangle of side length $6$ inscribed in a circle $\omega$. Let $A_1,A_2$ be the points (distinct from $A$) where the lines through $A$ passing through the two trisection points of $BC$ meet $\omega$. Define $B_1,B_2,C_1,C_2$ similarly. Given that $A_1,A_2,B_1,B_2,C_1,C_2$ appear on $\omega$ in that order, find the area of hexagon $A_1A_2B_1B_2C_1C_2$.

[b]p1.[/b] Fill in each box with an integer from $1$ to $9$. Each number in the right column is the product of the numbers in its row, and each number in the bottom row is the product of the numbers in its column. Some numbers may be used more than once, and not every number from $1$ to $9$ is required to be used. [img]https://cdn.artofproblemsolving.com/attachments/c/0/0212181d87f89aac374f75f1f0bde6d0600037.png[/img] [b]p2.[/b] A set $X$ is called [b]prime-difference free [/b] (henceforth pdf) if for all $x, y \in X$, $|x - y|$ is not prime. Find the number n such that the following both hold. $\bullet$ There is a pdf set of size $n$ that is a subset of $\{1,..., 2016\}$, and $\bullet$ There is no pdf set of size $n + 1$ that is a subset of $\{1,..., 2016\}$. [b]p3.[/b] Let $X_1,...,X_{15}$ be a sequence of points in the $xy$-plane such that $X_1 = (10, 0)$ and $X_{15} = (0, 10)$. Prove that for some $i \in \{1, 2,..., 14\}$, the midpoint of $X_iX_{i+1}$ is of distance greater than $1/2$ from the origin. [b]p4.[/b] Suppose that $s_1, s_2,..., s_{84}$ is a sequence of letters from the set $\{A,B,C\}$ such that every four-letter sequence from $\{A,B,C\}$ occurs exactly once as a consecutive subsequence $s_k$, $s_{k+1}$, $s_{k+2}$, $s_{k+3}$. Suppose that $$(s_1, s_2, s_3, s_4, s_5) = (A,B,B,C,A).$$ What is $s_{84}$? Prove your answer. [b]p5.[/b] Determine (with proof) whether or not there exists a sequence of positive real numbers $a_1, a_2, a_3,...$ with both of the following properties: $\bullet$ $\sum^n_{i=1} a_i \le n^2$, for all $n \ge 1$, and $\bullet$ $\sum^n_{i=1} \frac{1}{a_i} \le 2016$, for all $n \ge 1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The fence consists of $25$ vertical bars. The heights of the bars are pairwise distinct positive integers from $1$ to $25$. The width of every bar is $1$. Find the maximum $S$ for which regardless of the order of the bars one can find a rectangle of area $S$ formed by the fence.

[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \] [i] Stoyan Atanasov[/i]

Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$. [i]Author: Alex Zhu[/i]

Let $AD$ be a median of $\triangle ABC$ such that $m(\widehat{ADB})=45^{\circ}$ and $m(\widehat{ACB})=30^{\circ}$. What is the measure of $\widehat{ABC}$ in degrees? $ \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 90 \qquad\textbf{(C)}\ 105 \qquad\textbf{(D)}\ 120 \qquad\textbf{(E)}\ 135 $

Let $AH$ be altitude in acute trinagle $ABC$, inscribed in circle $s$. Points $D$ and $E$ are chosen on segment $BH$. Points $X$ and $Y$ are chosen on rays $AD$ and $AE$, respectively, such that midpoints of segments $DX$ and $EY$ lies on $s$. Suppose that points $B$, $X$, $Y$ and $C$ are concyclic. Prove that $BD+BE=2CH$.

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a function having the following property: For any two points $A$ and $B$ in $\mathbb{R}^2$, the distance between $A$ and $B$ is the same as the distance between the points $f(A)$ and $f(B)$. Denote the unique straight line passing through $A$ and $B$ by $l(A,B)$ (a) Suppose that $C,D$ are two fixed points in $\mathbb{R}^2$. If $X$ is a point on the line $l(C,D)$, then show that $f(X)$ is a point on the line $l(f(C),f(D))$. (b) Consider two more point $E$ and $F$ in $\mathbb{R}^2$ and suppose that $l(E,F)$ intersects $l(C,D)$ at an angle $\alpha$. Show that $l(f(C),f(D))$ intersects $l(f(E),f(F))$ at an angle $\alpha$. What happens if the two lines $l(C,D)$ and $l(E,F)$ do not intersect? Justify your answer.

Let $c$ be the inscribed circle of the triangle $ABC$, $d$ a line tangent to $c$ which does not pass through the vertices of triangle $ABC$. Prove the existence of points $A_1,B_1, C_1$, respectively, on the lines $BC,CA,AB$ satisfying the following two properties: $(i)$ Lines $AA_1,BB_1$, and $CC_1$ are parallel. $(ii)$ Lines $AA_1,BB_1$, and $CC_1$ meet $d$ respectively at points $A' ,B'$, and $C'$ such that \[\frac{A'A_1}{A' A}=\frac{B'B_1}{B 'B}=\frac{C'C_1}{C'C}\]

In a regular $ 6n\plus{}1$-gon, $ k$ vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices.

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).

Let $ABC$ be an acute triangle. Let $P$ be a point on the circle $(ABC)$, and $Q$ be a point on the segment $AC$ such that $AP\perp BC$ and $BQ\perp AC$. Lot $O$ be the circumcenter of triangle $APQ$. Find the angle $OBC$.

The starship is in a half-space at a distance $a$ from its boundary. The crew knows about it, but has no idea in which direction to move in order to reach the boundary plane. The starship can fly in space along any trajectory, measuring the length of the path traveled, and has a sensor that sends a signal when the border has been reached. Can a starship be guaranteed to reach the border with a path no longer than $14a$?

$AB$ is the diameter of circle $O$. A random point $P$ is selected on $O$ so that $AP = 4$ and $BP = 3$. Points $C$ and $D$ are drawn on circle $O$ so that $OC$ bisects $AP$ and $OD$ bisects $BP$. What is the degree measure of $\angle COD$?