Let $\pi$ be a given permutation of the set $\{1, 2, \dots, n\}$. Determine the smallest possible value of \[ \sum_{i=1}^n |\pi(i) - \sigma(i)|, \] where $\sigma$ is a permutation chosen from the set of all $n$-cycles. Express the result in terms of the number and lengths of the cycles in the disjoint cycle decomposition of $\pi$, including the fixed points.

The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$. (L.Steingarts)

Let $\triangle ABC$ be a triangle with $AB=6, BC=8, AC=10$, and let $D$ be a point such that if $I_A, I_B, I_C, I_D$ are the incenters of the triangles $BCD,$ $ ACD,$ $ ABD,$ $ ABC$, respectively, the lines $AI_A, BI_B, CI_C, DI_D$ are concurrent. If the volume of tetrahedron $ABCD$ is $\frac{15\sqrt{39}}{2}$, then the sum of the distances from $D$ to $A,B,C$ can be expressed in the form $\frac{a}{b}$ for some positive relatively prime integers $a,b$. Find $a+b$. [i]Proposed by [b]FedeX333X[/b][/i]

Let $ABC$ a triangle with $AB=BC$ and incircle $\omega$. Let $M$ the mindpoint of $BC$; $P, Q$ points in the sides $AB, AC$ such that $PQ\parallel BC$, $PQ$ is tangent to $\omega$ and $\angle CQM=\angle PQM$. Find the perimeter of triangle $ABC$ knowing that $AQ=1$.

Do there exist numbers $a,b \in R$ and surjective function $f: R \to R$ such that $f(f(x)) = bx f(x) +a$ for all real $x$? I.Voronovich

Fill in each empty white circle with a number from $1$ to $16$ so that each number is used exactly once. One number has been given to you. If a square has a given number inside and its four vertices contain the numbers $a, b, c, d$ in clockwise order, then the number inside the square must be equal to $(a + c)(b + d)$. There is a unique solution, but 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] unitsize(1cm); draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((1,0)--(1,3)); draw((2,0)--(2,3)); string[][] givens = {{"","638","650"},{"50","","338"},{"77","130",""}}; string[][] numbers = {{"","","",""},{"","","",""},{"","","",""},{"","","","5"}}; for(int i=0; i < 4; ++i) { for(int j=0; j < 4; ++j) { filldraw(circle((i,j),0.3), white); label(numbers[3-j][i], (i,j)); } } for(int i=0; i < 3; ++i){ for(int j=0; j < 3; ++j){ label(givens[2-j][i], (i + 0.5, j + 0.5)); } } [/asy]

How many ordered triples of integers $ (a, b, c)$ satisfy\[|a \plus{} b| \plus{} c \equal{} 19\quad\text{and}\quad ab \plus{} |c| \equal{} 97?\] $\textbf{(A)}\ 0\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

Compute the expected sum of elements in a subset of $\{1, 2, 3, . . . , 2020\}$ (including the empty set) chosen uniformly at random.

Let $ABCDE$ be a regular pentagon, and $F$ some point on the arc $AB$ of the circumcircle of $ABCDE$. Show that \[ \frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2}, \] and that $FD + FB + FA = FE + FC$.

Let $S$ be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant $r$, such that there exists one way to colour all the points in $S$ with three colous so that the distance between any two points with same colour is less than $r$.

Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6 \times 6$ table. Their $108$ coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column the sum of all trinomials has a real root. [i]K. Kokhas & F. Petrov[/i]

Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in\{0, 1, \ldots n\}$, then\[\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.\]

Does there exist an operation $*$ on the set of all integers such that the following conditions hold simultaneously: $(1)$ for all integers $x,y,z$, $(x*y)*z=x*(y*z)$; $(2)$ for all integers $x$ and $y$, $x*x*y=y*x*x=y$?

For a natural number $d$, $M_d$ denotes the set of natural numbers which are not representable as the sum of at least two consecutive terms of an arithmetic progression with the common difference d whose terms are integers. Prove that each $c\in M_3$ can be written in the form $c=ab$, where $a\in M_1$ and $b\in M_2\setminus\{2\}$.

Let $S$ be the set of all polynomials $Q(x,y,z)$ with coefficients in $\{0,1\}$ such that there exists a homogeneous polynomial $P(x,y,z)$ of degree $2016$ with integer coefficients and a polynomial $R(x,y,z)$ with integer coefficients so that \[P(x,y,z) Q(x,y,z) = P(yz,zx,xy)+2R(x,y,z)\] and $P(1,1,1)$ is odd. Determine the size of $S$. Note: A homogeneous polynomial of degree $d$ consists solely of terms of degree $d$. [i]Proposed by Vincent Huang[/i]

Construct a triangle given the radii of the excircles.

The midpoints of alternative sides of a convex hexagon are connected by line segments. Prove that the intersection points of the medians of the two triangles obtained coincide.

Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ? (S. Fomin , Leningrad)

Prove that there are no positive integers $a,b$ such that $(15a +b)(a +15b)$ is a power of $3.$

An integer $ n$, $ n\ge2$ is called [i]friendly[/i] if there exists a family $ A_1,A_2,\ldots,A_n$ of subsets of the set $ \{1,2,\ldots,n\}$ such that: (1) $ i\not\in A_i$ for every $ i\equal{}\overline{1,n}$; (2) $ i\in A_j$ if and only if $ j\not\in A_i$, for every distinct $ i,j\in\{1,2,\ldots,n\}$; (3) $ A_i\cap A_j$ is non-empty, for every $ i,j\in\{1,2,\ldots,n\}$. Prove that: (a) 7 is a friendly number; (b) $ n$ is friendly if and only if $ n\ge7$. [i]Valentin Vornicu[/i]

Sir Alex is coaching a soccer team of $n$ players of distinct heights. He wants to line them up so that for each player $P$, the total number of players that are either to the left of $P$ and taller than $P$ or to the right of $P$ and shorter than $P$ is even. In terms of $n$, how many possible orders are there? [i]Michael Ren[/i]

Let $P_n$ denote the $n$-th Pell number defined by $P_{n+1}=2P_n+P_{n-1}$, $P_0=0$, $P_1=1$. Furthermore, let $T_n$ denote the $n$-th triangular number, that is $T_n=\binom{n+1}2$. Show that $$\sum_{n=0}^\infty4T_n\cdot\frac{P_n}{3^{n+2}}=P_3+P_4$$ [i]Proposed by Ángel Plaza[/i]

Inside a convex quadrilateral \( ABCD \), a point \( P \) is chosen such that \[ \angle PAD = \angle PAB = \angle PBC = \angle PCB = \angle PDA = 30^\circ. \] Prove that \( \angle CDP = 30^\circ \). [i]Proposed by Vadym Solomka[/i]

The negation of the statement "No slow learners attend this school" is: $ \textbf{(A)}\ \text{All slow learners attend this school} \\ \textbf{(B)}\ \text{All slow learners do not attend this school} \\ \textbf{(C)}\ \text{Some slow learners attend this school} \\ \textbf{(D)}\ \text{Some slow learners do not attend this school} \\ \textbf{(E)}\ \text{No slow learners do not attend this school}$

Find the remainder when \[ \sum_{i=2}^{63} \frac{i^{2011}-i}{i^2-1}. \] is divided by 2016. [i]Author: Alex Zhu[/i]