Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S.$ [i]Proposed by Alex Li[/i]

The numbers $ 1, 2,\ldots, 50 $ are written on a blackboard. Each minute any two numbers are erased and their positive difference is written instead. At the end one number remains. Which values can take this number?

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

[i]The following problem is open in the sense that the answer to part (b) is not currently known. A proof of part (a) will be awarded 5 points. Up to 7 additional points may be awarded for progress on part (b).[/i] Let $p(x)$ be a polynomial of degree $d$ with coefficients belonging to the set of rational numbers $\mathbb{Q}$. Suppose that, for each $1 \le k \le d-1$, $p(x)$ and its $k$th derivative $p^{(k)}(x)$ have a common root in $\mathbb{Q}$; that is, there exists $r_k \in \mathbb{Q}$ such that $p(r_k) = p^{(k)}(r_k) = 0$. (a) Prove that if $d$ is prime then there exist constants $a, b, c \in \mathbb{Q}$ such that \[ p(x) = c(ax + b)^d. \] (b) For which integers $d \ge 2$ does the conclusion of part (a) hold?

Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic. [i]Proposed by Alireza Dadgarnia and Amir Parsa Hosseini[/i]

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Find all positive integers $n$ such that there exists positive integers $a, b, m$ satisfying$$\left( a+b\sqrt{n}\right)^{2023}=\sqrt{m}+\sqrt{m+2022}.$$

(a) Let $a_1,a_2,...$ be a sequenceof reals with $a_1=1$ and $a_{n+1}>\frac32 a_n$ for all $n$. Prove that $\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}$ exists. (finite or infinite) (b) Prove that for all $\alpha>1$ there is a sequence $a_1,a_2,...$ with the same properties such that $\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}=\alpha$

A man makes a trip by automobile at an average speed of $ 50$ mph. He returns over the same route at an average speed of $ 45$ mph. His average speed for the entire trip is: $ \textbf{(A)}\ 47\frac{7}{19}\qquad \textbf{(B)}\ 47\frac{1}{4}\qquad \textbf{(C)}\ 47\frac{1}{2}\qquad \textbf{(D)}\ 47\frac{11}{19}\qquad \textbf{(E)}\ \text{none of these}$

Olja writes down $n$ positive integers $a_1, a_2, \ldots, a_n$ smaller than $p_n$ where $p_n$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $xy$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win? [i]Proposed by Matko Ljulj.[/i]