The set $ S$ is a finite subset of $ [0,1]$ with the following property: for all $ s\in S$, there exist $ a,b\in S\cup\{0,1\}$ with $ a, b\neq s$ such that $ s \equal{}\frac{a\plus{}b}{2}$. Prove that all the numbers in $ S$ are rational.

$(CZS 5)$ A convex quadrilateral $ABCD$ with sides $AB = a, BC = b, CD = c, DA = d$ and angles $\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,$ and $\delta = \angle CDA$ is given. Let $s = \frac{a + b + c +d}{2}$ and $P$ be the area of the quadrilateral. Prove that $P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}$

A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to $\gamma_1$ and $\gamma_2$.

Let $[ABC]$ be a triangle and $X, Y$ and $Z$ points on the sides $[AB], [BC]$ and $[AC]$, respectively. Prove that circumcircles of triangles $AXZ, BXY$ and $CYZ$ intersect at a point.

There is a board of $2008\times 2008$ and $2008$ pieces, one in each row and each column of the board. It is allowed to do one of the following movements: a) Take two steps to the right and $10$ up. b) Take two steps to the right and $6$ steps down. c) Take two steps to the left and $6$ steps up. d) Take two steps to the left and $10$ steps down. If the path down cannot be completed, it is skipped to the upper part along the same column and the route continues normally, similarly in the other directions. In each play you will move a checker using any of the allowed operations. Would it be possible that at some point, after a finite number of played, the pieces are located forming a square of side $44$ in the upper left corner of the board and the remaining $72$ are in the last row in the first $72$ boxes?

The altitudes of the acute-angled non-isosceles triangle $ ABC $ intersect at the point $ H $. On the segment $ C_1H $, where $ CC_1 $ is the altitude of the triangle, the point $ K $ is marked. Points $ L $ and $ M $ are the feet of perpendiculars from point $ K $ on straight lines $ AC $ and $ BC $, respectively. The lines $ AM $ and $ BL $ intersect at $ N $. Prove that $ \angle ANK = \angle HNL $.

Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$

Points $D$ and $E$ divides the base $BC$ of an isosceles triangle $ABC$ into three equal parts and $D$ is between $B$ and $E.$ Show that $\angle BAD<\angle DAE.$

We say that a polynomial $p$ is respectful if $\forall x, y \in Z$, $y - x$ divides $p(y) - p(x)$, and $\forall x \in Z$, $p(x) \in Z$. We say that a respectful polynomial is disguising if it is nonzero, and all of its non-zero coefficients lie between $0$ and $ 1$, exclusive. Determine $\sum deg(f)\cdot f(2)$, where the sum includes all disguising polynomials $f$ of degree at most $5$.

Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$. Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.

Prove that for every positive integer $n$, the decimal expression of $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}$ is periodic .

How many distinct real roots does the equation $ \sqrt{x\plus{}4\sqrt{x\minus{}4} } \minus{}\sqrt{x\plus{}2\sqrt{x\minus{}1} } \equal{}1$ have? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Does such a convex (all angles less than $180^o$) pentagon $ABCDE$, such that all angles $ABD$, $BCE$, $CDA$, $DEB$ and $EAC$ are obtuse?

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]

There exists a unique strictly increasing arithmetic sequence $\{a_i\}_{i=1}^{100}$ of positive integers such that \[a_1+a_4+a_9+\cdots+a_{100}=\text{1000},\] where the summation runs over all terms of the form $a_{i^2}$ for $1\leq i\leq 10$. Find $a_{50}$. [i]Proposed by David Altizio and Tony Kim[/i]

Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

Let $ABCDEF$ be a regular hexagon with sidelength s. The points $P$ and $Q$ are on the diagonals $BD$ and $DF$, respectively, such that $BP = DQ = s$. Prove that the three points $C$, $P$ and $Q$ are on a line. [i](Walther Janous)[/i]

Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose \[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\] Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$

Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$?

Let $ G$ be anon-commutative group. Consider all the one-to-one mappings $ a\rightarrow a'$ of $ G$ onto itself such that $ (ab)'\equal{}b'a'$ (i.e. the anti-automorphisms of $ G$). Prove that this mappings together with the automorphisms of $ G$ constitute a group which contains the group of the automorphisms of $ G$ as direct factor.

Let $S$ be a set of $9$ distinct integers all of whose prime factors are at most $3.$ Prove that $S$ contains $3$ distinct integers such that their product is a perfect cube.

Find the smallest positive value taken by $a^3 + b^3 + c^3 - 3abc$ for positive integers $a$, $b$, $c$ . Find all $a$, $b$, $c$ which give the smallest value

Let $ABCD$ be a convex quadrilateral in which $$\angle BAC = 48^{\circ}, \angle CAD = 66^{\circ}, \angle CBD = \angle DBA.$$Prove that $\angle BDC = 24^{\circ}$.

On a horizontally placed number line, a pile of \( t_i > 0 \) tokens is placed on each number \( i \in \{1, 2, \ldots, s\} \). As long as at least one pile contains at least two tokens, we repeat the following procedure: we choose such a pile (say, it consists of \( k \geq 2 \) tokens), and move the top token from the selected pile \( k - 1 \) unit positions to the right along the number line. What is the largest natural number \( N \) on which a token can be placed? (Express \( N \) as a function of \( (t_i;\ i = 1, \ldots, s) \).)

Let $X$ be a set containing $n$ elements. Find the number of ordered triples $(A,B, C)$ of subsets of $X$ such that $A$ is a subset of $B$ and $B$ is a proper subset of $C$.