In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of edges $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ [asy] size(6cm); pair A,B,C,D,EE,F,G,H,I,J; C = (0,0); B = (-1,1); D = (2,0.5); A = B+D; G = (0,2); F = B+G; H = G+D; EE = G+B+D; I = (D+H)/2; J = (B+F)/2; filldraw(C--I--EE--J--cycle,lightgray,black); draw(C--D--H--EE--F--B--cycle); draw(G--F--G--C--G--H); draw(A--B,dashed); draw(A--EE,dashed); draw(A--D,dashed); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(I); dot(J); label("$A$",A,E); label("$B$",B,W); label("$C$",C,S); label("$D$",D,E); label("$E$",EE,N); label("$F$",F,W); label("$G$",G,N); label("$H$",H,E); label("$I$",I,E); label("$J$",J,W); [/asy] $\textbf{(A) } \frac{5}{4} \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{25}{16} \qquad \textbf{(E) } \frac{9}{4}$

In a game of rock-paper-scissors with $n$ people, the following rules are used to determine a champion: (a) In a round, each person who has not been eliminated randomly chooses one of rock, paper, or scissors to play. (b) If at least one person plays rock, at least one person plays paper, and at least one person plays scissors, then the round is declared a tie and no one is eliminated. If everyone makes the same move, then the round is also declared a tie. (c) If exactly two moves are represented, then everyone who made the losing move is eliminated from playing in all further rounds (for example, in a game with $8$ people, if $5$ people play rock and $3$ people play scissors, then the $3$ who played scissors are eliminated). (d) The rounds continue until only one person has not been eliminated. That person is declared the champion and the game ends. If a game begins with $4$ people, what is the expected value of the number of rounds required for a champion to be determined? [i]In the game of rock-paper-scissors, two players each choose one of rock, paper, or scissors to play. Rock beats scissors, scissors beats paper, and paper beats rock. If the players play the same thing, the match is considered a draw.[/i]

In $\triangle ABC$, let point $D$ be on $\overline{BC}$ such that the perimeters of $\triangle ADB$ and $\triangle ADC$ are equal. Let point $E$ be on $\overline{AC}$ such that the perimeters of $\triangle BEA$ and $\triangle BEC$ are equal. Let point $F$ be the intersection of $\overline{AB}$ with the line that passes through $C$ and the intersection of $\overline{AD}$ and $\overline{BE}$. Given that $BD = 10$, $CD = 2$, and $BF/FA = 3$, what is the perimeter of $\triangle ABC$?

Find all $c\in\mathbb R$ for which there exists an infinitely differentiable function $f:\mathbb R\to\mathbb R$ such that for all $n\in\mathbb N$ and $x\in\mathbb R$ we have $$f^{(n+1)}(x)>f^{(n)}(x)+c.$$

Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.

Albert and Betty are playing the following game. There are $100$ blue balls in a red bowl and $100$ red balls in a blue bowl. In each turn a player must make one of the following moves: a) Take two red balls from the blue bowl and put them in the red bowl. b) Take two blue balls from the red bowl and put them in the blue bowl. c) Take two balls of different colors from one bowl and throw the balls away. They take alternate turns and Albert starts. The player who first takes the last red ball from the blue bowl or the last blue ball from the red bowl wins. Determine who has a winning strategy.

If a polynomial $P$ with integer coefficients has three distinct integer zeroes , then show that $P(n)\neq 1$ for any integer $n$.

Let $f:\mathbb{R}^{\geq 0}\to\mathbb{R}$ be a function such that $f(x)-3x$ and $f(x)-x^3$ are ascendant functions. Prove that $f(x)-x^2-x$ is an ascendant function, too. (We call the function $g(x)$ ascendant, when for every $x\leq{y}$ we have $g(x)\leq{g(y)}$.)

Evaluate $$2023 \cdot \frac{2023^6 +27}{(2023^2 +3)(2024^3 -1)}-2023^2.$$ [i]Proposed by Evin Liang[/i]

The points $A, B, C, D$ lie on the line $\ell$ in this order. The points $P$ and $Q$ are chosen on one side of the line $\ell$, and the point $R$ is chosen on the other side so that: $$\angle APB = \angle CPD = \angle QBC = \angle QCB = \angle RAD = \angle RDA$$ Prove that the points $P, Q, R$ lie on the same line. [i]Proposed by Mykhailo Shtandenko, Fedir Yudin[/i]

A cyclic quadrilateral $ABCD$ is given. Point $K_1, K_2$ lie on the segment $AB$, points $L_1, L_2$ on the segment $BC$, points $M_1, M_2$ on the segment $CD$ and points $N_1, N_2$ on the segment $DA$. Moreover, points $K_1, K_2, L_1, L_2, M_1, M_2, N_1, N_2$ lie on a circle $\omega$ in that order. Denote by $a, b, c, d$ the lengths of the arcs $N_2K_1, K_2L_1, L_2M_1, M _2N_1$ of the circle $\omega$ not containing points $K_2, L_2, M_2, N_2$, respectively. Prove that \begin{align*} a+c=b+d. \end{align*}

Let $a,b,c\not= 0$ and $x,y,z\in\mathbb{R}^+$ such that $x+y+z=3$. Prove that \[\frac{3}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}\] [color=#FF0000]Mod: before the edit, it was [/color] \[\frac{3}{2}\left (\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right )\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}\]

[b]p13.[/b] Emma has the five letters: $A, B, C, D, E$. How many ways can she rearrange the letters into words? Note that the order of words matter, ie $ABC DE$ and $DE ABC$ are different. [b]p14.[/b] Seven students are doing a holiday gift exchange. Each student writes their name on a slip of paper and places it into a hat. Then, each student draws a name from the hat to determine who they will buy a gift for. What is the probability that no student draws himself/herself? [b]p15.[/b] We model a fidget spinner as shown below (include diagram) with a series of arcs on circles of radii $1$. What is the area swept out by the fidget spinner as it’s turned $60^o$ ? [img]https://cdn.artofproblemsolving.com/attachments/9/8/db27ffce2af68d27eee5903c9f09a36c2a6edf.png[/img] [b]p16.[/b] Let $a,b,c$ be the sides of a triangle such that $gcd(a, b) = 3528$, $gcd(b, c) = 1008$, $gcd(a, c) = 504$. Find the value of $a * b * c$. Write your answer as a prime factorization. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $\triangle ABC$ with $BC = 1$, the internal angle bisector of $\angle A$ intersects $BC$ at $D$. $M$ is taken to be the midpoint of $BC$. Point $E$ is chosen on the boundary of $\triangle ABC$ such that $ME$ bisects its perimeter. The circumcircle $\omega$ of $\triangle DEC$ is taken, and the second intersection of $AD$ and $\omega$ is $K$, as well as the second intersection of $ME$ and $\omega$ being $L$. If $B$ lies on line $KL$ and $ED$ is parallel to $AB$, then the perimeter of $\triangle ABC$ can be written as a real number $S$. Compute $\lfloor 1000S\rfloor$. Proposed by Albert Wang (awang11)

Given a positive integer $n$, find the greatest integer $p$ with the property that for any function $f : \mathbb P(X) \to C$, where $X$ and $C$ are sets of cardinality $n$ and $p$, respectively, there exist two distinct sets $A,B \in \mathbb P(X)$ such that $f(A) = f(B) = f(A \cup B)$. ($\mathbb P(X)$ is the family of all subsets of $X$.)

In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence HHTTHHHHTHHTTTT of 15 coin tosses we observe that there are two HH, three HT, four TH, and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH, three HT, four TH, and five TT subsequences?

Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle DAC$, and let $M$ and $N$ be points on segments $BD$ and $CD$, respectively, such that $\angle MAD = \angle DAN$. Let $S, P$ and $Q$ (all different from $A$) be the intersections of the rays $AD$, $AM$ and $AN$ with $\Gamma$, respectively. Show that the intersection of $SM$ and $QD$ lies on $\Gamma$.

Allowing $x$ to be a real number, what is the largest value that can be obtained by the function $25\sin(4x)-60\cos(4x)?$

Let $CH$ be the altitude of the triangle $ABC$ drawn on the board, in which $\angle C = 90^o$, $CA \ne CB$. The mathematics teacher drew the perpendicular bisectors of segments$ CA$ and $CB$, which cut the line CH at points $K$ and $M$, respectively, and then erased the drawing, leaving only the points $C, K$ and $M$ on the board. Restore triangle $ABC$, using only a compass and a ruler.

For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

Compute the remainder when $$\sum_{n=1}^{2018} n^4$$ is divided by $53$.

Let $ABCD$ be a rectangle with $AB=9$ and $BC=3$. Suppose that $D$ is reflected across $AC$ to a point $E$. Compute the area of trapezoid $AEBC$.

There are two points $A$ and $B$ in the plane. a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.

Find all positive integer solutions $(a, b)$ to the equation $$\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)}$$ for (i) $n = 2007$; (ii) $n = 2010$.