Two transformations are said to [i]commute[/i] if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane: - A translation $2$ units to the right - A $90^\circ$- rotation counterclockwise about the origin. - A reflection across the $x$-axis, and - A dilation centered at the origin with scale factor $2$. Of the $6$ pairs of distinct transformations from this list, how many commute? $ \textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5 \qquad $

Consider a circle $k$ and a point $K$ in the plane. For any two distinct points $P$ and $Q$ on $k$, denote by $k'$ the circle through $P,Q$ and $K$. The tangent to $k'$ at $K$ meets the line $PQ$ at point $M$. Describe the locus of the points $M$ when $P$ and $Q$ assume all possible positions.

Let $a, b$ be positive real numbers, and let $x, y$ be complex numbers such that $|x| = a$ and $|y| = b$. Find the minimal and maximal value of \[\left|\frac{x + y}{1 + x\overline{y}}\right|\]

In a convex pentagon $ABCDE$, the points $M$, $N$, $P$, $Q$, $R$ are the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EA$, respectively. If the segments $AP$, $BQ$, $CR$ and $DM$ pass through a single point, prove that $EN$ contains that point as well. [i]Yugoslavia[/i]

Let $ ABCD$ be a rhombus with $ AC\equal{}16$ and $ BD\equal{}30$. Let $ N$ be a point on $ \overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \overline{AC}$ and $ \overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ PQ$? [asy]unitsize(2.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C; pair Np=waypoint(B--A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(Np--Q); draw(Np--P); label("$D$",D,SW); label("$C$",C,SE); label("$B$",B,NE); label("$A$",A,NW); label("$N$",Np,N); label("$P$",P,SW); label("$Q$",Q,SSE); draw(rightanglemark(Np,P,C,2)); draw(rightanglemark(Np,Q,D,2));[/asy]$ \textbf{(A)}\ 6.5 \qquad \textbf{(B)}\ 6.75 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 7.25 \qquad \textbf{(E)}\ 7.5$

The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.

Suppose that 13 cards numbered $1, 2, 3, \dots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes? [asy] size(11cm); draw((0,0)--(2,0)--(2,3)--(0,3)--cycle); label("7", (1,1.5)); draw((3,0)--(5,0)--(5,3)--(3,3)--cycle); label("11", (4,1.5)); draw((6,0)--(8,0)--(8,3)--(6,3)--cycle); label("8", (7,1.5)); draw((9,0)--(11,0)--(11,3)--(9,3)--cycle); label("6", (10,1.5)); draw((12,0)--(14,0)--(14,3)--(12,3)--cycle); label("4", (13,1.5)); draw((15,0)--(17,0)--(17,3)--(15,3)--cycle); label("5", (16,1.5)); draw((18,0)--(20,0)--(20,3)--(18,3)--cycle); label("9", (19,1.5)); draw((21,0)--(23,0)--(23,3)--(21,3)--cycle); label("12", (22,1.5)); draw((24,0)--(26,0)--(26,3)--(24,3)--cycle); label("1", (25,1.5)); draw((27,0)--(29,0)--(29,3)--(27,3)--cycle); label("13", (28,1.5)); draw((30,0)--(32,0)--(32,3)--(30,3)--cycle); label("10", (31,1.5)); draw((33,0)--(35,0)--(35,3)--(33,3)--cycle); label("2", (34,1.5)); draw((36,0)--(38,0)--(38,3)--(36,3)--cycle); label("3", (37,1.5)); [/asy] $\textbf{(A) }4082\qquad\textbf{(B) }4095\qquad\textbf{(C) }4096\qquad\textbf{(D) }8178\qquad\textbf{(E) }8191$

Kamil wrote on a board an expression consisting of alternating addition and subtraction of natural numbers from $1$ to $100$: \[1-2+3-4+5-6+\ldots-98+99-100.\] Then, Kamil erased one of the plus or minus signs and replaced it with an equals sign, obtaining a true equality. Which number preceded the erased sign? Find all possibilities and justify your answer.

Diameter $AB$ of a circle has length a 2-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.

Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21$. Including $\overline{AB}$ and $\overline{BC}$, how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}$? $ \textbf{(A) }5 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }15 \qquad $

Let $n$ and $k$ be positive integers and $G$ be a complete graph on $n$ vertices. Each edge of $G$ is colored one of $k$ colors such that every triangle consists of either three edges of the same color or three edges of three different colors. Furthermore, there exist two different-colored edges. Prove that $n\le(k-1)^2$. [i]Linus Tang[/i]

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$. [i]Proposed by Evan Chen[/i]

Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions: a) not a triple of successive members is chosen, b) at least one of every triple of successive members is chosen, c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.

[u]Set 3[/u] [b]3.1[/b] Annie has $24$ letter tiles in a bag; $8$ C’s, $8$ M’s, and $8$ W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw? [b]3.2[/b] Let $T$ be the answer from the previous problem. Charlotte is initially standing at $(0, 0)$ in the coordinate plane. She takes $T$ steps, each of which moves her by $1$ unit in either the $+x$, $-x$, $+y$, or $-y$ direction (e.g. her first step takes her to $(1, 0)$, $(1, 0)$, $(0, 1)$ or $(0, -1)$). After the T steps, how many possibilities are there for Charlotte’s location? [b]3.3[/b] Let $T$ be the answer from the previous problem, and let $S$ be the sum of the digits of $T$. Francesca has an unfair coin with an unknown probability $p$ of landing heads on a given flip. If she flips the coin $S$ times, the probability she gets exactly one head is equal to the probability she gets exactly two heads. Compute the probability $p$. PS. You should use hide for answers.

Let $a_1, a_2, ..., a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \le i < j \le n$, there exist $k,\ell$, both different from $i$ and $j$ such that $a_i + a_j = a_k + a_{\ell}$. What is the smallest possible value of $n$?

Let $ABCD$ to be a quadrilateral inscribed in a circle $\Gamma$. Let $r$ and $s$ to be the tangents to $\Gamma$ through $B$ and $C$, respectively, $M$ the intersection between the lines $r$ and $AD$ and $N$ the intersection between the lines $s$ and $AD$. After all, let $E$ to be the intersection between the lines $BN$ and $CM$, $F$ the intersection between the lines $AE$ and $BC$ and $L$ the midpoint of $BC$. Prove that the circuncircle of the triangle $DLF$ is tangent to $\Gamma$.

Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.

Konsistent Karl is taking this contest. He can solve the first five problems in one minute each, the next five in two minutes each, and the last five in three minutes each. What is the maximum possible score Karl can earn? (Recall that this contest is $15$ minutes long, there are $15$ problems, and the $n$th problem is worth $n$ points. Assume that entering answers and moving between or skipping problems takes no time.) [i]Proposed by Michael Tang[/i]

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

In a triangle $\triangle ABC$ with orthocenter $H$, let $BH$ and $CH$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. If the tangent line to the circumcircle of $\triangle ABC$ passing through $A$ intersects $BC$ at $P$, $M$ is the midpoint of $AH$, and $EF$ intersects $BC$ at $G$, then prove that $PM$ is parallel to $GH$. [i]Proposed by Sreejato Bhattacharya[/i]

All sides of triangle $ABC$ are different. On rays $B A$ and $C A$ the segments $B K$ and $CM$ are laid out, equal to side $BC$. Let us denote by $x$ the length of the segment $KM$. In the same way, by plotting the side $AC$ on the rays $AB$ and $CB$ from $A$ and $C$, we obtain a segment of length $y$, and by plotting the side AB on the rays $AC$ and $BC$, we obtain a segment of length $z$. a) Prove that a triangle can be formed from the segments $x$, $y$ and $z$, and this triangle is similar to triangle $ABC$. b) Find the radius of the circumcircle of a triangle with sides $x$, $y$ and $z$, if the radii of the circumscribed and inscribed circles of triangle $ABC$ are equal to $R$ and $r$ respectively.

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that for all $x,y\in\mathbb{N}$: $$0\le y+f(x)-f^{f(y)}(x)\le1$$ that here $$f^n(x)=\underbrace{f(f(\ldots(f}_{n}(x))\ldots)$$

There is a family of curves: $P_0,P_1,P_2,\cdots$. $P_0$ is a regular triangle, whose area is $1$. For all $k\in\mathbb{Z}_+$, $P_k$ is defined in this way: trisect all sides of $P_{k-1}$, and draw outward a regular triangle with side of the segment in the middle, then cut off the segment in the middle. $S_n$ is the area of $P_n$. [b](a)[/b] Find $S_n$. [b](b)[/b] Find $\lim_{n\to\infty}S_n$.

Let $ABC$ be a triangle $(A=90^{\circ})$ and $D\in (AC)$ such that $BD$ is the bisector of $B$. Prove that $BC-BD=2AB$ if and only if \[\frac{1}{BD}-\frac{1}{BC}=\frac{1}{2AB} \]

On a \( 10 \times 10 \) chessboard, there are 91 white pawns placed in different squares. Nico picks a white pawn, paints it black, and places it in an empty square, repeating the process until all pawns have been painted. Prove that at some point, there will be two pawns of different colors placed on squares that share a common edge.