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 $C$.

Let $A_1A_2A_3A_4$ be a tetrahedron. Consider the sphere centered at $A_1$ which is tangent to the face $A_2A_3A_4$ of the tetrahedron. Show that the surface area of the part of the sphere which is inside the tetrahedron is less than the area of the triangle $A_2A_3A_4$. [i]Sorin Rădulescu[/i]

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$ Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$

Two circles that are not equal are tangent externally at point $R$. Suppose point $P$ is the intersection of the external common tangents of the two circles. Let $A$ and $B$ are two points on different circles so that $RA$ is perpendicular to $RB$. Show that the line $AB$ passes through $P$.

Let $p(n)$ denote the product of decimal digits of a positive integer $n$. Computer the sum $p(1)+p(2)+\ldots+p(2001)$.

Let $\omega (n)$ denote the number of prime divisors of the integer $n>1$. Find the least integer $k$ such that the inequality $2^{\omega (n) } \leq k \cdot n^{\frac 1 4}$ holds for all $n > 1.$ Pierre.

Two cubical dice each have removable numbers $ 1$ through $ 6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $ 7$? $ \textbf{(A)}\ \frac{1}{9} \qquad \textbf{(B)}\ \frac{1}{8} \qquad \textbf{(C)}\ \frac{1}{6} \qquad \textbf{(D)}\ \frac{2}{11} \qquad \textbf{(E)}\ \frac{1}{5}$

For any positive integer number $k$, the factorial $k!$ is defined as a product of all integers between $1$ and $k$ inclusive: $k!=k\times{(k-1)}\times\dots\times{1}$. Let $s(n)$ denote the sum of the first $n$ factorials, i.e. $$s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}$$ Find the remainder when $s(2024)$ is divided by $8$

Alice thinks of an integer $1 \le n \le 2048$. Bob asks $k$ true or false questions about Alice's integer; Alice then answers each of the questions, but she may lie on at most one question. What is the minimum value of $k$ for which Bob can guarantee he knows Alice's integer after she answers?

You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$.