Let $\mathbb{Q}$ be the set of rational numbers, $\mathbb{Z}$ be the set of integers. On the coordinate plane, given positive integer $m$, define $$A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.$$ For segment $MN$, define $f_m(MN)$ as the number of points on segment $MN$ belonging to set $A_m$. Find the smallest real number $\lambda$, such that for any line $l$ on the coordinate plane, there exists a constant $\beta (l)$ related to $l$, satisfying: for any two points $M,N$ on $l$, $$f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)$$

Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?

Determine all triples of positive integers $(A,B,C)$ for which some function $f \colon \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ satisfies \[ f^{f(y)} (y + f(2x)) + f^{f(y)} (2y) = (Ax+By)^{C} \] for all nonnegative integers $x$ and $y$, where $f^k$ as usual denotes $f$ composed $k$ times. Proposed by [i]Benny Wang[/i]

George has $150$ cups of flour and $200$ eggs. He can make a cupcake with $3$ cups of flour and $2$ eggs, or he can make an omelet with $4$ eggs. What is the maximum number of treats (both omelets and cupcakes) he canmake?

Let $ABC$ be an acute triangle with circumcenter $O$, and let $D$, $E$, and $F$ be the feet of altitudes from $A$, $B$, and $C$ to sides $BC$, $CA$, and $AB$, respectively. Denote by $P$ the intersection of the tangents to the circumcircle of $ABC$ at $B$ and $C$. The line through $P$ perpendicular to $EF$ meets $AD$ at $Q$, and let $R$ be the foot of the perpendicular from $A$ to $EF$. Prove that $DR$ and $OQ$ are parallel.

A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$. Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$

In the figure below, the three small circles are congruent and tangent to each other. The large circle is tangent to the three small circles. [asy] import graph; unitsize(20); real r = sqrt(3) / 2; filldraw(Circle((0, 0), 1 + r), gray); filldraw(Circle(dir(90), r), white); filldraw(Circle(dir(210), r), white); filldraw(Circle(dir(330), r), white); [/asy] The area of the large circle is 1. What is the area of the shaded region?

A(x,y), B(x,y), and C(x,y) are three homogeneous real-coefficient polynomials of x and y with degree 2, 3, and 4 respectively. we know that there is a real-coefficient polinimial R(x,y) such that $B(x,y)^2-4A(x,y)C(x,y)=-R(x,y)^2$. Proof that there exist 2 polynomials F(x,y,z) and G(x,y,z) such that $F(x,y,z)^2+G(x,y,z)^2=A(x,y)z^2+B(x,y)z+C(x,y)$ if for any x, y, z real numbers $A(x,y)z^2+B(x,y)z+C(x,y)\ge 0$

Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a,b,c)$ such that $2a+3b+5c=200$ and $a+b+c=N$. [i]Proposed by Kyle Lee[/i]

Determine the set of all real numbers $\alpha$ with the following property: For each positive $c$ there exists a rational number $\frac mn~(m\in\mathbb Z,n\in\mathbb N)$ different than $\alpha$ such that $$\left|\alpha-\frac mn\right|<\frac cn.$$

If the conditions \[\begin{array}{rcl} f(2x+1)+g(3-x) &=& x \\ f((3x+5)/(x+1))+2g((2x+1)/(x+1)) &=& x/(x+1) \end{array}\] hold for all real numbers $x\neq 1$, what is $f(2013)$? $ \textbf{(A)}\ 1007 \qquad\textbf{(B)}\ \dfrac {4021}{3} \qquad\textbf{(C)}\ \dfrac {6037}7 \qquad\textbf{(D)}\ \dfrac {4029}{5} \qquad\textbf{(E)}\ \text{None of above} $

A sphere of radius $1$ is internally tangent to all four faces of a regular tetrahedron. Find the tetrahedron's volume.

Let $(p_1,p_2,..., p_n)$ be a random permutation of the set $\{1,2,...,n)$. If $n$ is odd, prove that the product $(p_1-1)\cdot (p_2-2)\cdot ...\cdot (p_n-n)$ is an even number. @below fixed.

Ten points are spaced equally around a circle. How many different chords can be formed by joining any 2 of these points? (A chord is a straight line joining two points on the circumference of a circle.) $\textbf{(A)}\ 9 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 17 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 55$

The number of distinct solutions of the equation $\big|x-|2x+1|\big| = 3$ is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }4$

Prove that if $ a \geq 0 $ and $ b \geq 0 $, then $$ \sqrt{a^2 + b^2} \geq a + b - (2 - \sqrt{2}) \sqrt{ab}.$$

Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying \[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases} \] for all nonnegative integers $ p$, $ q$, $ r$.

Let $f(x)=x^2-6x+5$. On the plane $(x, y)$ draw a set of points $M(x, y)$ whose coordinates satisfy the inequalities $$\begin{cases} f(x)+f(y)\le 0 \\ f(x)-f(y)\ge 0 \end{cases}$$

[u]Round 1 [/u] [b]p1.[/b] In the Hundred Acre Wood, all the animals are either knights or liars. Knights always tell the truth and liars always lie. One day in the Wood, Winnie-the-Pooh, a knight, decides to visit his friend Rabbit, also a noble knight. Upon arrival, Pooh finds his friend sitting at a round table with $5$ other guests. One-by-one, Pooh asks each person at the table how many of his two neighbors are knights. Surprisingly, he gets the same answer from everybody! "Oh bother!" proclaims Pooh. "I still don't have enough information to figure out how many knights are at this table." "But it's my birthday," adds one of the guests. "Yes, it's his birthday!" agrees his neighbor. Now Pooh can tell how many knights are at the table. Can you? [b]p2.[/b] Harry has an $8 \times 8$ board filled with the numbers $1$ and $-1$, and the sum of all $64$ numbers is $0$. A magical cut of this board is a way of cutting it into two pieces so that the sum of the numbers in each piece is also $0$. The pieces should not have any holes. Prove that Harry will always be able to find a magical cut of his board. (The picture shows an example of a proper cut.) [img]https://cdn.artofproblemsolving.com/attachments/4/b/98dec239cfc757e6f2996eef7876cbfd79d202.png[/img] [b]p3.[/b] Several girls participate in a tennis tournament in which each player plays each other player exactly once. At the end of the tournament, it turns out that each player has lost at least one of her games. Prove that it is possible to find three players $A$, $B$, and $C$ such that $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. [b]p4.[/b] $120$ bands are participating in this year's Northwest Grunge Rock Festival, and they have $119$ fans in total. Each fan belongs to exactly one fan club. A fan club is called crowded if it has at least $15$ members. Every morning, all the members of one of the crowded fan clubs start arguing over who loves their favorite band the most. As a result of the fighting, each of them leaves the club to join another club, but no two of them join the same one. Is it true that, no matter how the clubs are originally arranged, all these arguments will eventually stop? [b]p5.[/b] In Infinite City, the streets form a grid of squares extending infinitely in all directions. Bonnie and Clyde have just robbed the Infinite City Bank, located at the busiest intersection downtown. Bonnie sets off heading north on her bike, and, $30$ seconds later, Clyde bikes after her in the same direction. They each bike at a constant speed of $1$ block per minute. In order to throw off any authorities, each of them must turn either left or right at every intersection. If they continue biking in this manner, will they ever be able to meet? [u]Round 2 [/u] [b]p6.[/b] In a certain herd of $33$ cows, each cow weighs a whole number of pounds. Farmer Dan notices that if he removes any one of the cows from the herd, it is possible to split the remaining $32$ cows into two groups of equal total weight, $16$ cows in each group. Show that all $33$ cows must have the same weight. [b]p7.[/b] Katniss is thinking of a positive integer less than $100$: call it $x$. Peeta is allowed to pick any two positive integers $N$ and $M$, both less than $100$, and Katniss will give him the greatest common divisor of $x+M$ and $N$ . Peeta can do this up to seven times, after which he must name Katniss' number $x$, or he will die. Can Peeta ensure his survival? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $O$ be the circumcenter of an acute triangle $ABC$. Let $D$ be the intersection of the bisector of the angle $A$ and $BC$. Suppose that $\angle ODC = 2 \angle DAO$. The circumcircle of $ABD$ meets the line segment $OA$ and the line $OD$ at $E (\neq A,O)$, and $F(\neq D)$, respectively. Let $X$ be the intersection of the line $DE$ and the line segment $AC$. Let $Y$ be the intersection of the bisector of the angle $BAF$ and the segment $BE$. Prove that $\frac{\overline{AY}}{\overline{BY}}= \frac{\overline{EX}}{\overline{EO}}$.

The equation $x^3-3x^2+1=0$ has three real solutions $x_1<x_2<x_3$. Show that for any positive integer $n$, the number $\left\lceil x_3^n\right\rceil$ is a multiple of $3$.

Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.

Consider a triangle with sides of length \( a \), \( b \), and \( c \) that satisfy the following conditions: \[ a + b = c + 3 \quad c^2 + 9 = 2ab \] Find the area of the triangle.

Calculate the following $$P=(4\sin^2{0} -3)(4\sin^2\frac{\pi}{2^{2005}} -3)(4\sin^2\frac{2\pi}{2^{2005}} -3)(4\sin^2\frac{3\pi}{2^{2005}} -3)...$$ $$...\,\,\,\,(4\sin^2\frac{(2^{2004}-1)\pi}{2^{2005}} -3)(4\sin^2\frac{\pi}{2} -3)$$

Let $ABC$ be an acute triangle such that $\angle BAC \neq 60^\circ$. Let $D,E$ be points such that $BD,CE$ are tangent to the circumcircle of $ABC$ and $BD=CE=BC$ ($A$ is on one side of line $BC$ and $D,E$ are on the other side). Let $F,G$ be intersections of line $DE$ and lines $AB,AC$. Let $M$ be intersection of $CF$ and $BD$, and $N$ be intersection of $CE$ and $BG$. Prove that $AM=AN$.