In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? $ \text{(A)}\ \frac{3}{8}\qquad\text{(B)}\ \frac{1}{2}\qquad\text{(C)}\ \frac{3}{4}\qquad\text{(D)}\ 2\frac{2}{3}\qquad\text{(E)}\ 3\frac{1}{3} $

In the game of [i]Ring Mafia[/i], there are $2019$ counters arranged in a circle. $673$ of these counters are mafia, and the remaining $1346$ counters are town. Two players, Tony and Madeline, take turns with Tony going first. Tony does not know which counters are mafia but Madeline does. On Tony’s turn, he selects any subset of the counters (possibly the empty set) and removes all counters in that set. On Madeline’s turn, she selects a town counter which is adjacent to a mafia counter and removes it. Whenever counters are removed, the remaining counters are brought closer together without changing their order so that they still form a circle. The game ends when either all mafia counters have been removed, or all town counters have been removed. Is there a strategy for Tony that guarantees, no matter where the mafia counters are placed and what Madeline does, that at least one town counter remains at the end of the game? [i]Proposed by Andrew Gu[/i]

Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$

On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely? (V.E. Kolosov)

A $20 \times 20$ rectangular grid has been given. It is known that one of the grid's unit squares contains a hidden treasure. To find the treasure, we have been given an opportunity to order several scientific studies at the same time, results of which will be known only after some time. For each study we must choose one $1 \times 4$ rectangle, and the study will tell whether the rectangle contains the treasure. The $1 \times 4$ rectangle can be either horizontal or vertical, and it can extend over a side of the $20 \times 20$ grid, coming back in at the opposite side (you can think of the $20 \times 20$ grid as a torus - the opposite sides are connected). What is the minimal amount of studies that have to ordered for us to precisely determine the unit square containing the treasure?

Point $A$ is located at $(0,0)$. Point $B$ is located at $(2,3)$ and is the midpoint of $AC$. Point $D$ is located at $(10,0)$. What are the coordinates of the midpoint of segment $CD$?

Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1.$$

Let $ABC$ be a triangle, let $M$ be the midpoint of $AB$, and let $N$ be the midpoint of $CM$. Let $X$ be a point satisfying both $\angle XMC = \angle MBC$ and $\angle XCM = \angle MCB$ such that $X$ and $B$ lie on opposite sides of $CM$. Let $\omega$ be the circumcircle of triangle $AMX$. $(a)$ Show that $CM$ is tangent to $\omega$. $(b)$ Show that the lines $NX$ and $AC$ intersect on $\omega$

In the figure below the triangles $BCD, CAE$ and $ABF$ are equilateral, and the triangle $ABC$ is right-angled with $\angle A = 90^o$. Prove that $|AD| = |EF|$. [img]https://1.bp.blogspot.com/-QMMhRdej1x8/XzP18QbsXOI/AAAAAAAAMUI/n53OsE8rwZcjB_zpKUXWXq6bg3o8GUfSwCLcBGAsYHQ/s0/2019%2Bmohr%2Bp5.png[/img]

Let $ABC$ be a right triangle with the hypotenus $BC.$ Let $BE$ be the bisector of the angle $\angle ABC.$ The circumcircle of the triangle $BCE$ cuts the segment $AB$ again at $F.$ Let $K$ be the projection of $A$ on $BC.$ The point $L$ lies on the segment $AB$ such that $BL=BK.$ Prove that $\frac{AL}{AF}=\sqrt{\frac{BK}{BC}}.$

Real numbers \( a, b, c \) satisfy the following conditions: \[ 1000 < |a| < 2000, \quad 1000 < |b| < 2000, \quad 1000 < |c| < 2000, \] and \[ \frac{ab^2}{a+b} + \frac{bc^2}{b+c} + \frac{ca^2}{c+a} = 0. \] What are the possible values of the expression \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{a}? \] [i]Proposed by Vadym Solomka[/i]

Tommy takes a 25-question true-false test. He answers each question correctly with independent probability $\frac{1}{2}$. Tommy earns bonus points for correct streaks: the first question in a streak is worth 1 point, the second question is worth 2 points, and so on. For instance, the sequence TFFTTTFT is worth 1 + 1 + 2 + 3 + 1 = 8 points. Compute the expected value of Tommy’s score.

Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.

Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that $$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$ for all real $x, y$.

Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$. (i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices. (ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.

The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? [asy]unitsize(10mm); defaultpen(fontsize(10pt)); pen finedashed=linetype("4 4"); filldraw((1,1)--(2,1)--(2,2)--(4,2)--(4,3)--(1,3)--cycle,grey,black+linewidth(.8pt)); draw((0,1)--(0,3)--(1,3)--(1,4)--(4,4)--(4,3)-- (5,3)--(5,2)--(4,2)--(4,1)--(2,1)--(2,0)--(1,0)--(1,1)--cycle,finedashed); draw((0,2)--(2,2)--(2,4),finedashed); draw((3,1)--(3,4),finedashed); label("$1$",(1.5,0.5)); draw(circle((1.5,0.5),.17)); label("$2$",(2.5,1.5)); draw(circle((2.5,1.5),.17)); label("$3$",(3.5,1.5)); draw(circle((3.5,1.5),.17)); label("$4$",(4.5,2.5)); draw(circle((4.5,2.5),.17)); label("$5$",(3.5,3.5)); draw(circle((3.5,3.5),.17)); label("$6$",(2.5,3.5)); draw(circle((2.5,3.5),.17)); label("$7$",(1.5,3.5)); draw(circle((1.5,3.5),.17)); label("$8$",(0.5,2.5)); draw(circle((0.5,2.5),.17)); label("$9$",(0.5,1.5)); draw(circle((0.5,1.5),.17));[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

A number is [i]capicua [/i] if it is read equally from left to right as it is from right to the left. For example, $23432$ and $111111$ are capicua numbers. (a) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2022$ equal digits? (b) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2021$ equal digits?

Two circles $\Gamma_1$ and $\Gamma_2$ intersect at $A,B$. Points $C,D$ lie on $\Gamma_1$, points $E,F$ lie on $\Gamma_2$ such that $A,B$ lies on segments $CE,DF$ respectively and segments $CE,DF$ do not intersect. Let $CF$ meet $\Gamma_1,\Gamma_2$ again at $K,L$ respectively, and $DE$ meet $\Gamma_1,\Gamma_2$ at $M,N$ respectively. If the circumcircles of $\triangle ALM$ and $\triangle BKN$ are tangent, prove that the radii of these two circles are equal.

Let $ABC$ be a right triangle with hypotenuse $BC$ and center $I$. Let bisectors of the angles $\angle B$ and $\angle C$ intersect the sides $AC$ and $AB$ in$ D$ and $E$, respectively. Let $P$ and $Q$ be the feet of the perpendiculars of the points $D$ and $E$ on the side $BC$. Prove that $I$ is the circumcenter of $APQ$.

Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\] (Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)

Let $n \ge 3$ points be given in the plane, no three of which lie on the same line. Determine whether it is always possible to draw an $n$-gon whose vertices are the given points and whose sides do not intersect. [i]Remark.[/i] The $n$-gon can be concave.

Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$. (a) Prove that $P$ is independent of the choice of $A$. (b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.

Let be a natural number $ k $ and let be two infinite sequences $ \left( x_n \right)_{n\ge 1} ,\left( y_n \right)_{n\ge 1} $ such that $$ \{1\}\cap\{ x_1,x_2,\ldots ,x_k\}=\{1\}\cap\{ y_1,y_2,\ldots ,y_k\} =\{ x_1,x_2,\ldots ,x_k\}\cap\{ y_1,y_2,\ldots ,y_k\} =\emptyset , $$ and defined by the following recurrence relations: $$ x_{n+k}=\frac{y_n}{x_n} ,\quad y_{n+k} =\frac{y_n-1}{x_n-1} $$ Prove that $ \left( x_n \right)_{n\ge 1} $ and $ \left( y_n \right)_{n\ge 1} $ are periodic. [i]Dumitru Acu[/i]

Imagine a square scheme consisting of $n\times n$ fields with edge length $1$, where $n$ is an arbitrary positive integer. What is the maximum possible length of a route you can follow along the edges of the fields from point $A$ in the lower left corner to point $B$ in the upper right corner if you must never return to one point where you have been before? (The figure shows for $n = 5$ an example of a permitted route and an example of a not permitted route). [img]https://cdn.artofproblemsolving.com/attachments/6/e/92931d87f11b9fb3120b8dccc2c37c35a04456.png[/img]

In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR=6$ and $PR=7$. What is the area of the square? [asy] size(170); defaultpen(linewidth(0.6)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); dot("$Q$",Q,S); dot("$P$",P,W); dot("$R$",R,1.3*S); label("$7$",(P+R)/2,NE); label("$6$",(R+B)/2,NE); [/asy] $\textbf{(A) }85\qquad\textbf{(B) }93\qquad\textbf{(C) }100\qquad\textbf{(D) }117\qquad\textbf{(E) }125$