Let $n$ be a positive integer. In a village, $n$ boys and $n$ girls are living. For the yearly ball, $n$ dancing couples need to be formed, each of which consists of one boy and one girl. Every girl submits a list, which consists of the name of the boy with whom she wants to dance the most, together with zero or more names of other boys with whom she wants to dance. It turns out that $n$ dancing couples can be formed in such a way that every girl is paired with a boy who is on her list. Show that it is possible to form $n$ dancing couples in such a way that every girl is paired with a boy who is on her list, and at least one girl is paired with the boy with whom she wants to dance the most.

Let $a,b,c\ge-1$ be real numbers with $a^3+b^3+c^3=1$. Prove that $a+b+c+a^2+b^2+c^2\le4$, and determine the cases of equality. (Proposed by Karl Czakler)

[asy]size(250); void bargraph(real X, real Y, real ymin, real ymax, real ystep, real tickwidth, string yformat, Label LX, Label LY, Label[] LLX, real[] height,pen p=nullpen) { draw((0,0)--(0,Y),EndArrow); draw((0,0)--(X,0),EndArrow); label(LX,(X,0),plain.SE,fontsize(9)); label(LY,(0,Y),plain.NW,fontsize(9)); real yscale=Y/(ymax+ystep); for(real y=ymin; y<ymax; y+=ystep) { draw((-tickwidth,yscale*y)--(0,yscale*y)); label(format(yformat,y),(-tickwidth,yscale*y),plain.W,fontsize(9)); } int n=LLX.length; real xscale=X/(2*n+2); for(int i=0;i<n;++i) { real x=xscale*(2*i+1); path P=(x,0)--(x,height[i]*yscale)--(x+xscale,height[i]*yscale)--(x+xscale,0)--cycle; fill(P,p); draw(P); label(LLX[i],(x+xscale/2),plain.S,fontsize(10)); } for(int i=0;i<n;++i) draw((0,height[i]*yscale)--(X,height[i]*yscale),dashed); } string yf="%#.1f"; Label[] LX={"Spring","Summer","Fall","Winter"}; for(int i=0;i<LX.length;++i) LX[i]=rotate(90)*LX[i]; real[] H={4.5,5,4,4}; bargraph(60,50,1,5.1,0.5,2,yf,"season","hamburgers (millions)",LX,H,yellow); fill(ellipse((45,30),7,10),brown);[/asy] A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $ 25 \%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter? \[ \textbf{(A)}\ 2.5 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 3.5 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 4.5 \]

Let $a,b,c$ be the positive real numbers satisfying $a^2+b^2+c^2=3$. Prove that: $$\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.$$

Suppose you are given that for some positive integer $n$, $1! + 2! + \ldots + n!$ is a perfect square. Find the sum of all possible values of $n$.

Given a $9\times 9$ grid, we assign either $+1$ or $-1$ to each square on the grid. We define an [i]adjustment[/i] as follow: for each square on the grid, we make a product of all numbers of those squares which share a common side with the square (excluding itself).Then we have $81$ products. Next we replace all the squares’ values with their corresponding products. Determine if we can make all values in the grid equal to $1$ through finite [i]adjustments[/i].

The first four terms of an infinite sequence $S$ of decimal digits are $1$, $9$, $8$, $2$, and succeeding terms are given by the final digit in the sum of the four immediately preceding terms. Thus $S$ begins $1$, $9$, $8$, $2$, $0$, $9$, $9$, $0$, $8$, $6$, $3$, $7$, $4$, $\cdots$. Do the digits $3$, $0$, $4$, $4$ ever come up consecutively in $S$?

A function $g \colon \mathbb{R} \to \mathbb{R}$ is given such that its range is a finite set. Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfies $$2f(x+g(y))=f(2g(x)+y)+f(x+3g(y))$$ for all $x, y \in \mathbb{R}$.

Show that the set $S$ of natural numbers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two reciprocals of natural numbers ($S =\left\{n |\frac{3}{n} \neq \frac{1}{p} + \frac{1}{q} \text{ for any } p, q \in \mathbb N \right\}$) is not the union of finitely many arithmetic progressions.

Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$. Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$.

How many perfect squares have the digits $1$ through $9$ each exactly once when written in base $10$? You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, your score will be $\lfloor12.5\cdot\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor.$

[u]Round 5[/u] [b]p13.[/b] Jason flips a coin repeatedly. The probability that he flips $15$ heads before flipping $4$ tails can be expressed as $\frac{a}{2^b}$ where $a$ and $b$ are positive integers and $a$ is odd. Find $a +b$. [b]p14.[/b] Triangle $ABC$ has side lengths $AB = 3$, $BC = 3$, and $AC = 4$. Let D be the intersection of the angle bisector of $\angle B AC$ and segment $BC$. Let the circumcircle of $\vartriangle B AD$ intersect segment $AC$ at a point $E$ distinct from $A$. The length of $AE$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p15.[/b] The sum of the squares of all values of $x$ such that $\{(x -2)(x -3)\} = \{(x -1)(x -6)\}$ and $\lfloor x^2 -6x +6 \rfloor= 9$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. Note: $\{a\}$ is the fractional part function, and returns $a -\lfloor a \rfloor$ . [u]Round 6[/u] [b]p16.[/b] Maisy the Polar Bear is at the origin of the Polar Plane ($r = 0, \theta = 0$). Maisy’s location can be expressed as $(r,\theta)$, meaning it is a distance of $r$ away from the origin and at a angle of $\theta$ degrees counterclockwise from the $x$-axis. When Maisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Maisy cannot jump to any point it has been to before. Let $L(r,\theta)$ be the number of paths Maisy can take to reach point $(r,\theta)$. The sum of $L(r,\theta)$ over all points where $r$ is an integer between $1$ and $2020$ and $\theta$ is an integer between $0$ and $359$ can be written as $\frac{n^k-1}{m}$ for some minimum value of $n$, such that $n$, $k$, and $m$ are all positive integers. Find $n +k +m$. [b]p17.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past $B$ to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE = \sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. Find $DF^2$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/0ee8716cebd6701fcae6544d9e39e68fff35f5.png[/img] [b]p18.[/b] Find the number of trailing zeroes at the end of $$\prod^{2021}_{i=1} (2021i -1) = (2020)(4041)...(2021^2 -1).$$ [u]Round 7[/u] [b]p19.[/b] A function $f (n)$ is defined as follows: $$f (n) = \begin{cases} \frac{n}{3} \,\,\, if \,\,\, n \equiv 0 (mod \, 3) \\ n^2 +4n -5 \,\,\,if \,\,\,n \equiv 1 (mod \, 3) \\ n^2 +n -2 \,\,\, if \,\,\,n \equiv 2 (mod \, 3) \end{cases}$$ Find the number of integer values of $n$ between $2$ and $1000$ inclusive such that $f ( f (... f (n))) = 1$ for some number of applications of $f (n)$. [b]p20.[/b] In the diagram below, the larger circle with diameter $AW$ has radius $16$. $ABCD$ and $WXY Z$ are rhombi where $\angle B AD = \angle XWZ = 60^o$ and $AC = CY = YW$. $M$ is the midpoint of minor arc $AW$, as shown. Let $I$ be the center of the circle with diameter $OM$. Circles with center $P$ and $G$ are tangent to lines $AD$ and $WZ$, respectively, and also tangent to the circle with center $I$ . Given that $IP \perp AD$ and $IG \perp WZ$, the area of $\vartriangle PIG$ can be written as $a +b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of a prime. Find $a +b +c$. [b]p21.[/b] In a list of increasing consecutive positive integers, the first item is divisible by $1$, the second item is divisible by $4$, the third item is divisible by $7$, and this pattern increases up to the seventh item being divisible by $19$. Find the remainder when the least possible value of the first item in the list is divided by $100$. [u]Round 8[/u] [b]p22.[/b] Let the answer to Problem $24$ be $C$. Jacob never drinks more than $C$ cups of coffee in a day. He always drinks a positive integer number of cups. The probability that he drinks $C +1-X$ cups is $X$ times the probability he drinks $C$ cups of coffee for any positive number $X$ from $1$ to $C$ inclusive. Find the expected number of cups of coffee he drinks. [b]p23.[/b] Let the answer to Problem $22$ be $A$. Three lines are drawn intersecting the interior of a triangle with side lengths $26$, $28$, and $30$ such that each line is parallel and a distance A away from a respective side. The perimeter of the triangle formed by the three new lines can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$. [b]p24.[/b] Let the answer to Problem $23$ be $B$. Given that $ab-c = bc-a = ca-b$ and $a^2+b^2+c^2 = B +2$, find the sum of all possible values of $|a +b +c|$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166500p28814367]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alex writes the sixteen digits $2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9$ side by side in any order and then places a colon somewhere between two digits, so that a division task arises. Can the result of this calculation be $2$?

Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_1$ be a point on $\ell.$ Let $P_2$ be the point on $,m$ closest to $P_1,$ $P_3$ be the point on $\ell$ closest to $P_3,$ $P_4$ be the point on $m$ closest to $P_3,$ and $P_5$ be the point on $\ell$ closest to $P_4.$ Given that $P_1P_2=5, P_2P_3=3,$ and $P_3P_4=2,$ compute $P_4P_5.$

Are there positive integers $a, b, c$, such that the numbers $a^2bc+2, b^2ca+2, c^2ab+2$ be perfect squares?

$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.

Suppose that $ P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3$. Find all possible values for which $ \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3)$ can be valued.