Two circles $A$ and $B$, both with radius $1$, touch each other externally. Four circles $P, Q, R$ and $S$, all four with the same radius $r$, lie such that $P$ externally touches on $A, B, Q$ and $S$, $Q$ externally touches on $P, B$ and $R$, $R$ externally touches on $A, B, Q$ and $S$, $S$ externally touches on $P, A$ and $R$. Calculate the length of $r.$ [asy] unitsize(0.3 cm); pair A, B, P, Q, R, S; real r = (3 + sqrt(17))/2; A = (-1,0); B = (1,0); P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180)); R = -P; Q = (r + 2,0); S = (-r - 2,0); draw(Circle(A,1)); draw(Circle(B,1)); draw(Circle(P,r)); draw(Circle(Q,r)); draw(Circle(R,r)); draw(Circle(S,r)); label("$A$", A); label("$B$", B); label("$P$", P); label("$Q$", Q); label("$R$", R); label("$S$", S); [/asy]

$ABC$ is a triangle that is inscribed in a circle. The angle bisectors of $A, B, C$ meet the circle at $D, E, F$, respectively. Show that $AD$ is perpendicular to $EF$.

$A,B,C$ and $D$ are four points in that order on the circumference of a circle $K$. $AB$ is perpendicular to $BC$ and $BC$ is perpendicular to $CD$. $X$ is a point on the circumference of the circle between $A$ and $D$. $AX$ extended meets $CD$ extended at $E$ and $DX$ extended meets $BA$ extended at $F$. Prove that the circumcircle of triangle $AXF$ is tangent to the circumcircle of triangle $DXE$ and that the common tangent line passes through the center of the circle $K$.

Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016. $P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]

[color=darkred]The sides $AD$ and $CD$ of a tangent quadrilateral $ABCD$ touch the incircle $\varphi$ at $P$ and $Q,$ respectively. If $M$ is the midpoint of the chord $XY$ determined by $\varphi$ on the diagonal $BD,$ prove that $\angle AMP = \angle CMQ.$[/color]

Let $a_1,a_2,a_3,\cdots$ be the sequence of positive integers such that $$a_1=1 , a_{k+1}=a^3_k+1, $$ for all positive integers $k.$ Prove that for every prime number $p$ of the form $3l +2,$ where $l$ is a non-negative integer ,there exists a positive integer $n$ such that $a_n$ is divisible by $p.$

There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps. what does barycenter of n distinct points mean?

Let $\mathcal{R}$ be the region in the plane bounded by the graphs of $y=x$ and $y=x^2$. Compute the volume of the region formed by revolving $\mathcal{R}$ around the line $y=x$.

Let $m$ be a positive integer. Prove that there are integers $a, b, k$, such that both $a$ and $b$ are odd, $k\geq0$ and \[2m=a^{19}+b^{99}+k\cdot2^{1999}\]

How many positive integers less than $2019$ are divisible by either $18$ or $21$, but not both?

Let $ x $ be a positive real number such that the numbers $ x^{-1} $, $ x $, and $ x^{2018} $ have the same fractional part: $$ \{x^{-1}\} = \{x\} = \{x^{2018}\}. $$ Prove that $ x = 1 $. [b]Note:[/b] If $ x $ is a real number, its fractional part is $ \{x\} = x - \lfloor x \rfloor $, where $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $ x $.

On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth? $\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

Fill in each space of the grid with either a $0$ or a $1$ so that all $16$ strings of four consecutive numbers across and down are distinct. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable). [asy] draw((8,0)--(8,4)--(1,4)--(1,9)--(0,9) -- (0,5) -- (5,5)--(5,0)--(9,0)--(9,1)--(4,1)--(4,8)--(0,8)); draw((0,6)--(4,6)); draw((0,7)--(4,7)); draw((4,3)--(8,3)); draw((4,2)--(8,2)); draw((2,4)--(2,8)); draw((3,4)--(3,8)); draw((6,0)--(6,4)); draw((7,0)--(7,4)); label("0",(0.5, 8.5)); label("",(0.5, 7.5)); label("0",(0.5, 6.5)); label("1",(0.5, 5.5)); label("1",(1.5, 7.5)); label("",(1.5, 6.5)); label("",(1.5, 5.5)); label("0",(1.5, 4.5)); label("0",(2.5, 7.5)); label("1",(2.5, 6.5)); label("",(2.5, 5.5)); label("",(2.5, 4.5)); label("",(3.5, 7.5)); label("",(3.5, 6.5)); label("0",(3.5, 5.5)); label("1",(3.5, 4.5)); label("",(4.5, 4.5)); label("",(4.5, 3.5)); label("",(4.5, 2.5)); label("0",(4.5, 1.5)); label("0",(5.5, 3.5)); label("",(5.5, 2.5)); label("",(5.5, 1.5)); label("",(5.5, 0.5)); label("",(6.5, 3.5)); label("",(6.5, 2.5)); label("",(6.5, 1.5)); label("",(6.5, 0.5)); label("",(7.5, 3.5)); label("0",(7.5, 2.5)); label("",(7.5, 1.5)); label("1",(7.5, 0.5)); label("",(8.5, 0.5)); [/asy]

Draw an equilateral triangle with center $O$. Rotate the equilateral triangle $30^\circ, 60^\circ, 90^\circ$ with respect to $O$ so there would be four congruent equilateral triangles on each other. Look at the diagram. If the smallest triangle has area $1$, the area of the original equilateral triangle could be expressed as $p+q\sqrt r$ where $p,q,r$ are positive integers and $r$ is not divisible by a square greater than $1$. Find $p+q+r$.

For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$? $\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xf(x-y))+yf(x)=x+y+f(x^2),$$ for all real numbers $x$ and $y.$