Let $P = (-2, 0)$. Points $P$, $Q$, $R$ lie on the graph of the function $y = x^3 - 3x + 2$ such that $Q$ is the midpoint of segment $PR$. Compute $PR^2$. [i]Proposed by David Altizio[/i]

A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds $\frac32$ m. What can be the cross-sectional area of a cube drawn by a plane?

$ n$ points (no three being collinear) are given in a plane. Some points are connected and they form $ k$ segments. If no three of these segments form triangle ( equiv. there are no three points, such that each two of them are connected) prove that $ k \leq \left \lfloor \frac {n^{2}}{4}\right\rfloor$

Triangle $ABC$ has sidelengths $AB = 13$, $AC = 14$, and $BC = 15$ and centroid $G$. What is the area of the triangle with sidelengths $AG$, $BG$, and $CG$

Let $n>2$ be an integer$.$ We want to color in red exactly $n+1$ of the numbers $1,2,\cdots,2n-1, 2n$ so that there do not exists three distinct red integers $x,y,z$ satisfying $x+y=z.$ Prove that there is one and one only way to color the red numbers according to the given condition$.$

Point $P$ is chosen on the leg $CB$ of right triangle $ABC$ ($\angle ACB = 90^\circ$). The line $AP$ intersects the circumcircle of $ABC$ at point $Q$. Let $L$ be the midpoint of $PB$. Prove that $QL$ is tangent to a fixed circle independent of the choice of point $P$.

The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$, \[ a_{n+2} = \frac 1{a_{n+1}} + a_n . \] Find $a_{2004}$.

A single burger is not enough to satisfy a guy's hunger. The five guys go to Five Guys' Restaurant, which has $20$ different meals on the menu. Each meal costs a different integer dollar amount between $\$1$ and $\$20$. The five guys have $\$20$ to split between them, and they want to use all the money to order five different meals. How many sets of five meals can the guys choose?

If $ (x^2+1)(y^2+1)+9=6(x+y)$ where $x,y$ are real numbers, what is $x^2+y^2$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3 $

Solve the equation: $|x + 1| - |x| + 3|x - 1| - 2|x - 2| = x + 2$.

A [i]permutation[/i] of a finite set is a one-to-one function from the set onto itself. A [i]cycle[/i] in a permutation $P$ is a nonempty sequence of distinct items $x_1$, $\ldots\,$, $x_n$ such that $P(x_1) = x_2$, $P(x_2) = x_3$, $\ldots\,$, $P(x_n) = x_1$. Note that we allow the 1-cycle $x_1$ where $P(x_1) = x_1$ and the 2-cycle $x_1, x_2$ where $P(x_1) = x_2$ and $P(x_2) = x_1$. Every permutation of a finite set splits the set into a finite number of disjoint cycles. If this number equals 2, then the permutation is called [i]bi-cyclic[/i]. Compute the number of bi-cyclic permutations of the 7-element set formed by the letters of "PROBLEM".

Let $p$ be a prime number. Find all integers $k$ for which $\sqrt{k^2 -pk}$ is a positive integer.

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?