Let $ 0 \leq c \leq 1$, and let $ \eta$ denote the order type of the set of rational numbers. Assume that with every rational number $ r$ we associate a Lebesgue-measurable subset $ H_r$ of measure $ c$ of the interval $ [0,1]$. Prove the existence of a Lebesgue-measurable set $ H \subset [0,1]$ of measure $ c$ such that for every $ x \in H$ the set \[ \{r : \;x \in H_r\ \}\] contains a subset of type $ \eta$. [i]M. Laczkovich[/i]

Find all functions $f$ from the reals to the reals such that \[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\] for all real $x,y$.

\[\int_1^\infty \frac{\lfloor x^2\rfloor}{x^5}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

In the figure, $ \angle EAB$ and $ \angle ABC$ are right angles. $ AB \equal{} 4, BC \equal{} 6, AE \equal{} 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle ADE$ and $ \triangle BDC$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A=(0,0), B=(4,0), C=(4,6), Ep=(0,8); pair D=extension(A,C,Ep,B); draw(A--C--B--A--Ep--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$E$",Ep,N); label("$D$",D,2.5*N); label("$4$",midpoint(A--B),S); label("$6$",midpoint(B--C),E); label("$8$",(0,3),W);[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying \[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]

Find all ordered positive integer pairs of $(m,n)$ such that $2^n-1$ divides $2^m+1$.

[u]Round 5[/u] [b]p13.[/b] Sally is at the special glasses shop, where there are many different optical lenses that distort what she sees and cause her to see things strangely. Whenever she looks at a shape through lens $A$, she sees a shape with $2$ more sides than the original (so a square would look like a hexagon). When she looks through lens $B$, she sees the shape with $3$ fewer sides (so a hexagon would look like a triangle). How many sides are in the shape that has $200$ more diagonals when looked at from lense $A$ than from lense $B$? [b]p14.[/b] How many ways can you choose $2$ cells of a $5$ by $5$ grid such that they aren't in the same row or column? [b]p15.[/b] If $a + \frac{1}{b} = (2015)^{-1}$ and $b + \frac{1}{a} = (2016)^2$ then what are all the possible values of $b$? [u]Round 6[/u] [b]p16.[/b] In Canadian football, linebackers must wear jersey numbers from $30 -35$ while defensive linemen must wear numbers from $33 -38$ (both intervals are inclusive). If a team has $5$ linebackers and $4$ defensive linemen, how many ways can it assign jersey numbers to the $9$ players such that no two people have the same jersey number? [b]p17.[/b] What is the maximum possible area of a right triangle with hypotenuse $8$? [b]p18.[/b] $9$ people are to play touch football. One will be designated the quarterback, while the other eight will be divided into two (indistinct) teams of $4$. How many ways are there for this to be done? [u]Round 7[/u] [b]p19.[/b] Express the decimal $0.3$ in base $7$. [b]p20.[/b] $2015$ people throw their hats in a pile. One at a time, they each take one hat out of the pile so that each has a random hat. What is the expected number of people who get their own hat? [b]p21.[/b] What is the area of the largest possible trapezoid that can be inscribed in a semicircle of radius $4$? [u]Round 8[/u] [b]p22.[/b] What is the base $7$ expression of $1211_3 \cdot 1110_2 \cdot 292_{11} \cdot 20_3$ ? [b]p23.[/b] Let $f(x)$ equal the ratio of the surface area of a sphere of radius $x$ to the volume of that same sphere. Let $g(x)$ be a quadratic polynomial in the form $x^2 + bx + c$ with $g(6) = 0$ and the minimum value of $g(x)$ equal to $c$. Express $g(x)$ as a function of $f(x)$ (e.g. in terms of $f(x)$). [b]p24.[/b] In the country of Tahksess, the income tax code is very complicated. Citizens are taxed $40\%$ on their first $\$20, 000$ and $45\%$ on their next $\$40, 000$ and $50\%$ on their next $\$60, 000$ and so on, with each $5\%$ increase in tax rate aecting $\$20, 000$ more than the previous tax rate. The maximum tax rate, however, is $90\%$. What is the overall tax rate (percentage of money owed) on $1$ million dollars in income? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. .Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two circles $c_1$ and $c_2$ with centres $O_1$ and $O_2$, respectively, are touching externally at $P$. On their common tangent at $P$, point $A$ is chosen, rays drawn from which touch the circles $c_1$ and $c_2$ at points $P_1$ and $P_2$ both different from $P$. It is known that $\angle P_1AP_2 = 120^o$ and angles $P_1AP$ and $P_2AP$ are both acute. Rays $AP_1$ and $AP_2$ intersect line $O_1O_2$ at points $G_1$ and $G_2$, respectively. The second intersection between ray $AO_1$ and $c_1$ is $H_1$, the second intersection between ray $AO_2$ and $c_2$ is $H_2$. Lines $G_1H_1$ and $AP$ intersect at $K$. Prove that if $G_1K$ is a tangent to circle $c_1$, then line $G_2A$ is tangent to circle $c_2$ with tangency point $H_2$.

$40$ people, numbered $1$ through $40$ counterclockwise, sit around a circular table. They begin playing a game. Each person is initially considered "alive". Starting with person $1$, the first person eliminates the closest "alive" person to their right (so Person $1$ eliminates Person $2$). Then the next "alive" person, moving counterclockwise, eliminates the closest "alive" person to their right (so since Person $2$ is eliminated, Person $3$ eliminates Person $4$). This process continues until there is only $1$ "alive" person remaining. What is the number of the last "alive" person? [asy] usepackage("cancel", "makeroom, thicklines"); usepackage("bm"); size(15cm); picture p; draw(p, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { label(p, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(p, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(p, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(p, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); add(shift(-20, 0) * p); draw((-11, 0)--(-8,0), EndArrow); picture q; draw(q, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { label(q, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); if(i != 1) label(q, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(q, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(q, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); for(int i = 0; i < 1; i+=2) { //label(q, "\bm\xcancel{~}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(q, "\xcancel{2}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80)); } add(q); draw((9,0)--(12,0), EndArrow); picture r; draw(r, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { if(i % 2 == 1) label(r, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); if(i % 2 != 1) label(r, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(r, scale(2)*"$\cdot$", 6 *dir(180 / n * i)); } draw(r, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); for(int i = 0; i < 4; i+=2) { label(r, "\xcancel{" + string(40 - i) +"}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(r, "\xcancel{" + string(i + 1) + "}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80)); } add(shift(20, 0) * r); [/asy] [center]In the last step here, Person $39$ eliminates Person $40$. Next turn, Person $1$ eliminates the closest person to his right, Person $3$.[/center]

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality $$ \left ( \frac{1+a}{b}+2 \right ) \left ( \frac{1+b}{c}+2 \right ) \left ( \frac{1+c}{a}+2 \right )\geq 216.$$ When does equality hold? [i]Authored by Anastasija Trajanova[/i]

If $ \frac {x}{y} \equal{} \frac {3}{4}$, then the incorrect expression in the following is: $ \textbf{(A)}\ \frac {x \plus{} y}{y} \equal{} \frac {7}{4} \qquad\textbf{(B)}\ \frac {y}{y \minus{} x} \equal{} \frac {4}{1} \qquad\textbf{(C)}\ \frac {x \plus{} 2y}{x} \equal{} \frac {11}{3}$ $ \textbf{(D)}\ \frac {x}{2y} \equal{} \frac {3}{8} \qquad\textbf{(E)}\ \frac {x \minus{} y}{y} \equal{} \frac {1}{4}$

[b]p1.[/b] Chad lives on the third floor of an apartment building with ten floors. He leaves his room and goes up two floors, goes down four floors, goes back up five floors, and finally goes down one floor, where he finds Jordan's room. On which floor does Jordan live? [b]p2.[/b] A real number $x$ satisfies the equation $2014x + 1337 = 1337x + 2014$. What is $x$? [b]p3.[/b] Given two points on the plane, how many distinct regular hexagons include both of these points as vertices? [b]p4.[/b] Jordan has six different files on her computer and needs to email them to Chad. The sizes of these files are $768$, $1024$, $2304$, $2560$, $4096$, and $7680$ kilobytes. Unfortunately, the email server holds a limit of $S$ kilobytes on the total size of the attachments per email, where $S$ is a positive integer. It is additionally given that all of the files are indivisible. What is the maximum value of S for which it will take Jordan at least three emails to transmit all six files to Chad? [b]p5.[/b] If real numbers $x$ and $y$ satisfy $(x + 2y)^2 + 4(x + 2y + 2 - xy) = 0$, what is $x + 2y$? [b]p6.[/b] While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How many possible games could they have played? Two games are considered the same if and only if they include the exact same sequence of scoring. [b]p7.[/b] For a positive integer $m$, we define $m$ as a factorial number if and only if there exists a positive integer $k$ for which $m = k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$. We define a positive integer $n$ as a Thai number if and only if $n$ can be written as both the sum of two factorial numbers and the product of two factorial numbers. What is the sum of the five smallest Thai numbers? [b]p8.[/b] Chad and Jordan are in the Exeter Space Station, which is a triangular prism with equilateral bases. Its height has length one decameter and its base has side lengths of three decameters. To protect their station against micrometeorites, they install a force field that contains all points that are within one decameter of any point of the surface of the station. What is the volume of the set of points within the force field and outside the station, in cubic decameters? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges connecting these vertices are colored with different colors on at least one of the cards. What is the minimum number of cards in the deck?

Let $ ABC$ be a triangle with area $ S$ and points $ D,E,F$ on the sides $ BC,CA,AB$. Perpendiculars at points $ D,E,F$ to the $ BC,CA,AB$ cut circumcircle of the triangle $ ABC$ at points $ (D_1,D_2), (E_1,E_2), (F_1,F_2)$. Prove that: $ |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S$

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

15 positive integers, all less than 1998(and no one equal to 1), are relatively prime (no pair has a common factor > 1). Show that at least one of them must be prime.