Determine all $ f:R\rightarrow R $ such that $$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$

Three cards, only one of which is an ace, are placed face down on a table. You select one, but do not look at it. The dealer turns over one of the other cards, which is not the ace (if neither are, he picks one of them randomly to turn over). You get a chance to change your choice and pick either of the remaining two face-down cards. If you selected the cards so as to maximize the chance of finding the ace on the second try, what is the probability that you selected it on the (a) first try? (b) second try?

If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right. \] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.

Let $(X,d)$ be a metric space with $d:X\times X \to \mathbb{R}_{\geq 0}$. Suppose that $X$ is connected and compact. Prove that there exists an $\alpha \in \mathbb{R}_{\geq 0}$ with the following property: for any integer $n > 0$ and any $x_1,\dots,x_n \in X$, there exists $x\in X$ such that the average of the distances from $x_1,\dots,x_n$ to $x$ is $\alpha$ i.e. $$\frac{d(x,x_1)+d(x,x_2)+\cdots+d(x,x_n)}{n} = \alpha.$$

Let $\triangle ABC$ have incenter $I$. The line $CI$ intersects the circumcircle of $\triangle ABC$ for the second time at $L$, and $CI=2IL$. Points $M$ and $N$ lie on the segment $AB$, such that $\angle AIM =\angle BIN = 90^{\circ}$. Prove that $AB=2MN$.

Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$. Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.

A $2.00 \text{L}$ balloon at $20.0^{\circ} \text{C}$ and $745 \text{mmHg}$ floats to an altitude where the temperature is $10.0^{\circ} \text{C}$ and the air pressure is $700 \text{mmHg}$. What is the new volume of the balloon? $ \textbf{(A)}\hspace{.05in}0.94 \text{L}\qquad\textbf{(B)}\hspace{.05in}1.06 \text{L}\qquad\textbf{(C)}\hspace{.05in}2.06 \text{L}\qquad\textbf{(D)}\hspace{.05in}2.20 \text{L}\qquad $

Let $ABC$ be a triangle with $AB=2$ and $AC=3$. Let $H$ be the orthocenter, and let $M$ be the midpoint of $BC$. Let the line through $H$ perpendicular to line $AM$ intersect line $AB$ at $X$ and line $AC$ at $Y$. Suppose that lines $BY$ and $CX$ are parallel. Then $[ABC]^2=\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c$ and $d$, where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $1000a+100b+10c+d$. [i]Proposed by Luke Robitaille

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Prove that \[ a^2b^2(a^2+b^2-2) \geq (a+b)(ab-1) \] for all positive real numbers $a$ and $b.$

A pile of $2000$ coins is given on a table. In each step, we choose a pile with at least three coins, remove one coin from it, and divide the rest of this pile into two piles (not necessarily of the same size). Is it possible that after several steps each pile on the table has exactly three coins?

Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

[b]p1.[/b] Aidan walks into a skyscraper’s first floor lobby and takes the elevator up $50$ floors. After exiting the elevator, he takes the stairs up another $10$ floors, then takes the elevator down $30$ floors. Find the floor number Aidan is currently on. [b]p2.[/b] Jeff flips a fair coin twice and Kaylee rolls a standard $6$-sided die. The probability that Jeff flips $2$ heads and Kaylee rolls a $4$ is $P$. Find $\frac{1}{P}$ . [b]p3.[/b] Given that $a\odot b = a + \frac{a}{b}$ , find $(4\odot 2)\odot 3$. [b]p4.[/b] The following star is created by gluing together twelve equilateral triangles each of side length $3$. Find the outer perimeter of the star. [img]https://cdn.artofproblemsolving.com/attachments/e/6/ad63edbf93c5b7d4c7e5d68da2b4632099d180.png[/img] [b]p5.[/b] In Lexington High School’sMath Team, there are $40$ students: $20$ of whom do science bowl and $22$ of whom who do LexMACS. What is the least possible number of students who do both science bowl and LexMACS? [b]p6.[/b] What is the least positive integer multiple of $3$ whose digits consist of only $0$s and $1$s? The number does not need to have both digits. [b]p7.[/b] Two fair $6$-sided dice are rolled. The probability that the product of the numbers rolled is at least $30$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] At the LHSMath Team Store, $5$ hoodies and $1$ jacket cost $\$13$, and $5$ jackets and $1$ hoodie cost $\$17$. Find how much $15$ jackets and $15$ hoodies cost, in dollars. [b]p9.[/b] Eric wants to eat ice cream. He can choose between $3$ options of spherical ice cream scoops. The first option consists of $4$ scoops each with a radius of $3$ inches, which costs a total of $\$3$. The second option consists of a scoop with radius $4$ inches, which costs a total of $\$2$. The third option consists of $5$ scoops each with diameter $2$ inches, which costs a total of $\$1$. The greatest possible ratio of volume to cost of ice cream Eric can buy is nπ cubic inches per dollar. Find $3n$. [b]p10.[/b] Jen claims that she has lived during at least part of each of five decades. What is the least possible age that Jen could be? (Assume that age is always rounded down to the nearest integer.) [b]p11.[/b] A positive integer $n$ is called Maisylike if and only if $n$ has fewer factors than $n -1$. Find the sum of the values of $n$ that are Maisylike, between $2$ and $10$, inclusive. [b]p12.[/b] When Ginny goes to the nearby boba shop, there is a $30\%$ chance that the employee gets her drink order wrong. If the drink she receives is not the one she ordered, there is a $60\%$ chance that she will drink it anyways. Given that Ginny drank a milk tea today, the probability she ordered it can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the value of $a +b$. [b]p13.[/b] Alex selects an integer $m$ between $1$ and $100$, inclusive. He notices there are the same number of multiples of $5$ as multiples of $7$ betweenm and $m+9$, inclusive. Find how many numbers Alex could have picked. [b]p14.[/b] In LMTown there are only rainy and sunny days. If it rains one day there’s a $30\%$ chance that it will rain the next day. If it’s sunny one day there’s a $90\%$ chance it will be sunny the next day. Over n days, as n approaches infinity, the percentage of rainy days approaches $k\%$. Find $10k$. [b]p15.[/b] A bag of letters contains $3$ L’s, $3$ M’s, and $3$ T’s. Aidan picks three letters at random from the bag with replacement, and Andrew picks three letters at random fromthe bag without replacement. Given that the probability that both Aidan and Andrew pick one each of L, M, and T can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p16.[/b] Circle $\omega$ is inscribed in a square with side length $2$. In each corner tangent to $2$ of the square’s sides and externally tangent to $\omega$ is another circle. The radius of each of the smaller $4$ circles can be written as $(a -\sqrt{b})$ where $a$ and $b$ are positive integers. Find $a +b$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/c76a780ac857f745067a8d6c4433efdace2dbb.png[/img] [b]p17.[/b] In the nonexistent land of Lexingtopia, there are $10$ days in the year, and the Lexingtopian Math Society has $5$ members. The probability that no two of the LexingtopianMath Society’s members share the same birthday can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Let $D(n)$ be the number of diagonals in a regular $n$-gon. Find $$\sum^{26}_{n=3} D(n).$$ [b]p19.[/b] Given a square $A_0B_0C_0D_0$ as shown below with side length $1$, for all nonnegative integers $n$, construct points $A_{n+1}$, $B_{n+1}$, $C_{n+1}$, and $D_{n+1}$ on $A_nB_n$, $B_nC_n$, $C_nD_n$, and $D_nA_n$, respectively, such that $$\frac{A_n A_{n+1}}{A_{n+1}B_n}=\frac{B_nB_{n+1}}{B_{n+1}C_n} =\frac{C_nC_{n+1}}{C_{n+1}D_n}=\frac{D_nD_{n+1}}{D_{n+1}A_n} =\frac34.$$ [img]https://cdn.artofproblemsolving.com/attachments/6/a/56a435787db0efba7ab38e8401cf7b06cd059a.png[/img] The sum of the series $$\sum^{\infty}_{i=0} [A_iB_iC_iD_i ] = [A_0B_0C_0D_0]+[A_1B_1C_1D_1]+[A_2B_2C_2D_2]...$$ where $[P]$ denotes the area of polygon $P$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] Let $m$ and $n$ be two real numbers such that $$\frac{2}{n}+m = 9$$ $$\frac{2}{m}+n = 1$$ Find the sum of all possible values of $m$ plus the sumof all possible values of $n$. [b]p21.[/b] Let $\sigma (x)$ denote the sum of the positive divisors of $x$. Find the smallest prime $p$ such that $$\sigma (p!) \ge 20 \cdot \sigma ([p -1]!).$$ [b]p22.[/b] Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. Let $M$ be the midpoint of side $\overline{AB}$. Suppose there exists a point X on the circle passing through points $A$, $M$, and $C$ such that $BMCX$ is a parallelogram and $M$ and $X$ are on opposite sides of line $BC$. Let segments $\overline{AX}$ and $\overline{BC}$ intersect at a point $Y$ . Given that $BY = 8$, find $AY^2$. [b]p23.[/b] Kevin chooses $2$ integers between $1$ and $100$, inclusive. Every minute, Corey can choose a set of numbers and Kevin will tell him how many of the $2$ chosen integers are in the set. How many minutes does Corey need until he is certain of Kevin’s $2$ chosen numbers? [b]p24.[/b] Evaluate $$1^{-1} \cdot 2^{-1} +2^{-1} \cdot 3^{-1} +3^{-1} \cdot 4^{-1} +...+(2015)^{-1} \cdot (2016)^{-1} \,\,\, (mod \,\,\,2017).$$ [b]p25.[/b] In scalene $\vartriangle ABC$, construct point $D$ on the opposite side of $AC$ as $B$ such that $\angle ABD = \angle DBC = \angle BC A$ and $AD =DC$. Let $I$ be the incenter of $\vartriangle ABC$. Given that $BC = 64$ and $AD = 225$, find$ BI$ . [img]https://cdn.artofproblemsolving.com/attachments/b/1/5852dd3eaace79c9da0fd518cfdcd5dc13aecf.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.