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]

$8$ ants are placed on the edges of the unit cube. Prove that there exists a pair of ants at a distance not exceeding $1$.

$ABC$ is a triangle with sides $1$, $2$ and $\sqrt3$. Determine the smallest possible area of an equilateral triangle with a vertex on each side of triangle $ABC$.

Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)

Let $X$ be the interior point of triangle $ABC$. prove that the product of the distances of point $ X $ from the vertices $ A, B, C $ is at least eight times greater than the product of the distances of this point from the lines $ AB, BC, CA $.

Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\] for every $k=1,2,\dots,N$.

$2018\times 2018$ field is covered with $1 \times 2$ dominos, such that every $2 \times 2$ or $1 \times 4,4 \times 1$ figure is not covered by only two dominos. Can be covered more than $99\%$ of field ?

Sandwiches at Joe's Fast Food cost $ \$3$ each and sodas cost $ \$2$ each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? $ \textbf{(A) } 31\qquad \textbf{(B) } 32\qquad \textbf{(C) } 33\qquad \textbf{(D) } 34\qquad \textbf{(E) } 35$

Consider a $ n \times n$ checkerboard with $ n > 1, n \in \mathbb{N}.$ How many possibilities are there to put $ 2n \minus{} 2$ identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. Two pebbles are on the same checkerboard diagonal if the connection segment of the midpoints of the respective fields are parallel to one of the diagonals of the $ n \times n$ square.

Consider $ 8$ points that are a knight’s move away from the origin (i.e., the eight points $\{(2, 1)$ , $(2, -1)$ , $(1, 2)$ , $(1, -2)$ , $(-1, 2)$ , $(-1, -2)$ , $(-2, 1)$, $(-2, -1)\}$). Each point has probability $\frac12$ of being visible. What is the expected value of the area of the polygon formed by points that are visible? (If exactly $0, 1, 2$ points appear, this area will be zero.)

Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)

The operation $\#$ is defined by $x\#y=\frac{x-y}{xy}$. For how many real values $a$ is $a\#\left(a\#2\right)=1$? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }4\qquad\text{(E) infinitely many}$

There are 2000 cities in a country and no roads. Prove that some cities can be connected by a road such that there would be 2 cities with 1 road passing through them, there would be 2 cities with 2 roads passim through them,...,there would be 2 cities with 1000 roads passing through them. [I]Proposed by F. Bakharev[/i]

Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)

A natural number is called [i]triangular[/i] if it may be presented in the form $\frac{n(n+1)}2$. Find all values of $a$ $(1\le a\le9)$ for which there exist a triangular number all digit of which are equal to $a$.

The numbers $25$ and $76$ have the property that when squared in base 10, their squares also end in the same two digits. A positive integer that has at most $3$ digits when expressed in base 21 and also has the property that its base $21$ square ends in the same $3$ digits is called amazing. Find the sum of all amazing numbers. Express your answer in base $21$.

In $\triangle{ABC}$, $AX=XY=YB=BC$, and $m\angle{ABC}=120^{\circ}$. What is $m\angle{BAC}$? [asy] pair A, B, C, X, Y; A = origin; X = dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(330); draw(A--B--C--cycle); draw(X--Y--B); label("$A$",A,W); label("$B$",B,N); label("$C$",C,E); label("$X$",X,NW); label("$Y$",Y,SE); [/asy] $\text{(A) }15\qquad\text{(B) }20\qquad\text{(C) }25\qquad\text{(D) }30\qquad\text{(E) }35$

A graph $G$ has $n$ vertices ($n>1$). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$. Prove that the inequality (the summation is over all edges of $G$): \[\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.\]

Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .

$AL$ is internal bisector of scalene $\triangle ABC$ ($L \in BC$). $K$ is chosen on segment $AL$. Point $P$ lies on the same side with respect to line $BC$ as point $A$ such that $\angle BPL = \angle CKL$ and $\angle CPL = \angle BKL$. $M$ is midpoint of segment $KP$, and $D$ is foot of perpendicular from $K$ on $BC$. Prove that $\angle AMD = 180^\circ - |\angle ABC - \angle ACB|$. [i]Proposed by Mykhailo Shtandenko and Fedir Yudin[/i]

Prove that $\frac 12 \cdot \sqrt{4\sin^2 36^{\circ} - 1}=\cos 72^\circ$.

$4n$ first grade students at Songkhla Primary School, including $2n$ boys and $2n$ girls, participate in a taekwondo tournament where every pair of students compete against each other exactly once. The tournament is scored as follows: $\bullet$ In a match between two boys or between two girls, a win is worth $3$ points, a draw $1$ point, and a loss $0$ points. $\bullet$ In a math between a boy and a girl, if the boy wins, he receives $2$ points, else he receives $0$ points. If the girl wins, she receives $3$ points, if she draws, she receives $2$ points, and if she loses, she receives $0$ points. After the tournament, the total score of each student is calculated. Let $P$ be the number of matches ending in a draw, and let $Q$ be the total number of matches. Suppose that the maximum total score is $4n - 1$. Find $P/Q$.