Let $S$ be the sum of the first nine terms of the sequence \[x+a, x^2+2a, x^3+3a, \cdots.\] Then $S$ equals: $ \textbf{(A)}\ \frac{50a+x+x^8}{x+1} \qquad\textbf{(B)}\ 50a-\frac{x+x^{10}}{x-1}\qquad\textbf{(C)}\ \frac{x^9-1}{x+1}+45a\qquad$$\textbf{(D)}\ \frac{x^{10}-x}{x-1}+45a\qquad\textbf{(E)}\ \frac{x^{11}-x}{x-1}+45a$

[u]Part 6 [/u] [b]p16.[/b] Le[b][/b]t $p(x)$ and $q(x)$ be polynomials with integer coefficients satisfying $p(1) = q(1)$. Find the greatest integer $n$ such that $\frac{p(2023)-q(2023)}{n}$ is an integer no matter what $p(x)$ and $q(x)$ are. [b]p17.[/b] Find all ordered pairs of integers $(m,n)$ that satisfy $n^3 +m^3 +231 = n^2m^2 +nm.$ [b]p18.[/b] Ben rolls the frustum-shaped piece of candy (shown below) in such a way that the lateral area is always in contact with the table. He rolls the candy until it returns to its original position and orientation. Given that $AB = 4$ and $BD =CD = 3$, find the length of the path traced by $A$. [u]Part 7 [/u] [b]p19.[/b] In their science class, Adam, Chris, Eddie and Sam are independently and randomly assigned an integer grade between $70$ and $79$ inclusive. Given that they each have a distinct grade, what is the expected value of the maximum grade among their four grades? [b]p20.[/b] Let $ABCD$ be a regular tetrahedron with side length $2$. Let point $E$ be the foot of the perpendicular from $D$ to the plane containing $\vartriangle ABC$. There exist two distinct spheres $\omega_1$ and $\omega_2$, centered at points $O_1$ and $O_2$ respectively, such that both $O_1$ and $O_2$ lie on $\overrightarrow{DE}$ and both spheres are tangent to all four of the planes $ABC$, $BCD$, $CDA$, and $DAB$. Find the sum of the volumes of $\omega_1$ and $\omega_2$. [b]p21.[/b] Evaluate $$\sum^{\infty}_{i=0}\sum^{\infty}_{j=0}\sum^{\infty}_{k=0} \frac{1}{(i + j +k +1)2^{i+j+k+1}}.$$ [u]Part 8 [/u] [b]p22.[/b] In $\vartriangle ABC$, let $I_A$, $I_B$ , and $I_C$ denote the $A$, $B$, and $C$-excenters, respectively. Given that $AB = 15$, $BC = 14$ and $C A = 13$, find $\frac{[I_A I_B I_C ]}{[ABC]}$ . [b]p23.[/b] The polynomial $x +2x^2 +3x^3 +4x^4 +5x^5 +6x^6 +5x^7 +4x^8 +3x^9 +2x^{10} +x^{11}$ has distinct complex roots $z_1, z_2, ..., z_n$. Find $$\sum^n_{k=1} |R(z^2n))|+|I(z^2n)|,$$ where $R(z)$ and $I(z)$ indicate the real and imaginary parts of $z$, respectively. Express your answer in simplest radical form. [b]p24.[/b] Given that $\sin 33^o +2\sin 161^o \cdot \sin 38^o = \sin n^o$ , compute the least positive integer value of $n$. [u]Part 9[/u] [b]p25.[/b] Submit a prime between $2$ and $2023$, inclusive. If you don’t, or if you submit the same number as another team’s submission, you will receive $0$ points. Otherwise, your score will be $\min \left(30, \lfloor 4 \cdot ln(x) \rfloor \right)$, where $x$ is the positive difference between your submission and the closest valid submission made by another team. [b]p26.[/b] Sam, Derek, Jacob, andMuztaba are eating a very large pizza with $2023$ slices. Due to dietary preferences, Sam will only eat an even number of slices, Derek will only eat a multiple of $3$ slices, Jacob will only eat a multiple of $5$ slices, andMuztaba will only eat a multiple of $7$ slices. How many ways are there for Sam, Derek, Jacob, andMuztaba to eat the pizza, given that all slices are identical and order of slices eaten is irrelevant? If your answer is $A$ and the correct answer is $C$, the number of points you receive will be: irrelevant? If your answer is $A$ and the correct answer is $C$, the number of points you receive will be: $$\max \left( 0, \left\lfloor 30 \left( 1-2\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)$$ [b]p27.[/b] Let $ \Omega_(k)$ denote the number of perfect square divisors of $k$. Compute $$\sum^{10000}_{k=1} \Omega_(k).$$ If your answer is $A$ and the correct answer is $C$, the number of points you recieve will be $$\max \left( 0, \left\lfloor 30 \left( 1-4\sqrt{\frac{|A-C|}{C}}\right)\right\rfloor \right)$$ PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3267911p30056982]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A [i]word[/i] is a finite sequence of letters from some alphabet. A word is [i]repetitive[/i] if it is a concatenation of at least two identical subwords (for example, $ababab$ and $abcabc$ are repetitive, but $ababa$ and $aabb$ are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.) [i]Romania (Dan Schwarz)[/i]

Circles $k_1$ and $k_2$ intersect at $P$ and $Q$, and $A$ and $B$ are the tangency points of their common tangent that is closer to $P$ (where $A$ is on $k_1$ and $B$ on $k_2$). The tangent to $k_1$ at $P$ intersects $k_2$ again at $C$. The lines $AP$ and $BC$ meet at $R$. Show that the lines $BP$ and $BC$ are tangent to the circumcircle of triangle $PQR$.

Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs?

Determine the smallest positive integer, $n$, which satisfies the equation $n^3+2n^2 = b$, where $b$ is the square of an odd integer.

Let $A(XYZ)$ be the area of the triangle $XYZ$. A non-regular convex polygon $P_1 P_2 \ldots P_n$ is called [i]guayaco[/i] if exists a point $O$ in its interior such that \[A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).\] Show that, for every integer $n \ge 3$, a guayaco polygon of $n$ sides exists.

Determine the number of ways to select a sequence of $ 8$ sets $ A_1,A_2,\ldots,A_8$, such that each is a subset (possibly empty) of $ \{1,2\}$ and $ A_m$ contains $ A_n$ if $ m$ divides $ n$.

Find all positive integers $n$ such that $n!+2$ divides $(2n)!$.

$ S\subset\mathbb N$ is called a square set, iff for each $ x,y\in S$, $ xy\plus{}1$ is square of an integer. a) Is $ S$ finite? b) Find maximum number of elements of $ S$.

How many positive solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$? How many positive integer solutions?

The numbers $16$ and $25$ are a pair of consecutive perfect squares whose difference is $9$. How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$? $\textbf{(A) } 674 \qquad \textbf{(B) } 1011 \qquad \textbf{(C) } 1010 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2017$

Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?

We call [i]Pythagorean triple[/i] a triple $(x,y,z)$ of positive integers such that $x<y<z$ and $x^2+y^2=z^2$. Prove that for all $n \in \mathbb{N}$ number $2^{n+1}$ is in exactly $n$ [i]Pythagorean triples[/i]

David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\textsc{Central Michigan University}$. What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to read $\textsc{Carnegie Mellon University}$?

Let $X$ be a set with $|X| = n$ , and let $X_1 , X_2 ,... X_n$ be the $n$subsets eith $|X_j| \geq 2$, for $1 \leq j \leq n$. Suppose for each $2$ element subset $Y$ of $X$, there is a unique $j$ in the set $1,2,3....,n$ such that $Y \subset X_j$ . Prove that $X_j \cap X_k \not= \Phi$ for all $1 \leq j < k \leq n$

What is the average length in letters of a word in this question? $\text{(A) }3.5\qquad\text{(B) }4\qquad\text{(C) }4.5\qquad\text{(D) }5\qquad\text{(E) }5.5$

The real numbers $a,b,c$ with $bc\neq0$ satisfy $\frac{1-c^2}{bc}\geq0.$ Prove that $10(a^2+b^2+c^2-bc^3)\geq2ab+5ac.$

A [i] magic square[/i] is a $3\times 3$ grid of numbers in which the sums of the numbers in each row, column, and long diagonal are all equal. How many magic squares exist where each of the integers from $11$ to $19$ inclusive is used exactly once and two of the numbers are already placed as shown below? $\begin{tabular}{|l|l|l|l|} \hline & & 18 \\ \hline & 15 & \\ \hline & & \\ \hline \end{tabular}$

Sophie has $20$ indistinguishable pairs of socks in a laundry bag. She pulls them out one at a time. After pulling out $30$ socks, the expected number of unmatched socks among the socks that she has pulled out can be expressed in simplest form as $\tfrac{m}{n}$. Find $m+n$.

A man travels $ m$ feet due north at $ 2$ minutes per mile. He returns due south to his starting point at $ 2$ miles per minute. The average rate in miles per hour for the entire trip is: $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 24\qquad\\ \textbf{(E)}\ \text{impossible to determine without knowing the value of }{m}$

In my calculator, one of the keys from $1$ to $9$ does not work properly: when you press it, a digit between $1$ and $9$ appears on the screen that is not the correct one. When I tried to write the number $987654321$, a number divisible by $11$ appeared on the screen and leaves a remainder of $3$ when divided by $9$. What is the broken key? What is the number that appeared on the screen?

For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers. Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one. R. Salimov

What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$

[b]3/1/32.[/b] The bisectors of the internal angles of parallelogram $ABCD$ determine a quadrilateral with the same area as $ABCD$. Given that $AB > BC$, compute, with proof, the ratio $\frac{AB}{BC}$.