Let $x_0,x_1,\dots, x_{n-1}$ be real numbers such that $0<|x_0|<|x_1|<\dots<|x_{n-1}|$. We will write the sum of the elements of each one of the $2^n$ subsets of $\{x_0,x_1,\dots,x_{n-1}\}$ in a paper. Prove that the $2^n$ written numbers are consecutive elements of a arithmetic progression if and only if the ratios $$|\frac{x_i}{x_j}|, 0\leq j<i\leq n-1$$ are equal(s) to the ratio(s) obtained with the numbers $2^0,2^1,\dots,2^{n-1}$. Note: The sum of the elements of the empty set is $0$.

Starting with a list of three numbers, the “[i]Make-My-Day[/i]” procedure creates a new list by replacing each number by the sum of the other two. For example, from $\{1, 3, 8\}$ “[i]Make-My-Day[/i]” gives $\{11, 9, 4\}$ and a new “[i]MakeMy-Day[/i]” leads to $\{13, 15, 20\}$. If we begin with $\{20, 1, 8\}$, what is the maximum difference between two numbers on the list after $2018$ consecutive “[i]Make-My-Day[/i]”s?

Prove that for all positive integers $m$ and $n$, $$\frac1m\cdot\binom{2n}0-\frac1{m+1}\cdot\binom{2n}1+\frac1{m+2}\cdot\binom{2n}2-\ldots+\frac1{m+2n}\cdot\binom{2n}{n2}>0$$

Some circles of radius 2 are drawn on the plane. Prove that the numerical value of the total area covered by these circles is at least as big as the total length of arcs bounding the area.

Real numbers $a, b, c$ are such that $$a^2+c-bc = b^2+a-ca = c^2+b-ab$$ Does it follow that $a=b=c$? [i]Proposed by Mykhailo Shtandenko[/i]

Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that $2023$ red chameleons can agree on a sequence of bites between themselves, after which they will all turn blue. What is the smallest $k$ that can guarantee that $k$ red chameleons, biting only each other, can turn blue? (For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)

Andy flips a strange coin for which the probability of flipping heads is $\frac{1}{2^k+1}$, where $k$ is the number of heads that appeared previously. If Andy flips the coin repeatedly until he gets heads 10 times, what is the expected number of total flips he performs? [i]Proposed by Harry Kim[/i]

A prime number $p$ and a positive integer $n$ are given. Prove that one can colour every one of the numbers $1,2,\ldots,p-1$ using one of the $2n$ colours so that for any $i=2,3,\ldots,n$ the sum of any $i$ numbers of the same colour is not divisible by $p$.

Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$. [i]Ciprus[/i]

Prove that there exist $0 < \alpha< \beta<1$ numbers have the following properties. (i) for any sufficiently large n, n points can be specified in $\Bbb R^3$ , so that each point is equidistant from at least $n^\alpha$ other points. (ii) the above statement is no longer true with $n^\beta$ instead of $n^\alpha$

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

4. Given is an acute-angled triangle $A B C, H$ is its orthocenter. Let $P$ be an arbitrary point inside (and not on the sides) of the triangle $A B C$ that belongs to the circumcircle of the triangle $A B H$. Let $A^{\prime}, B^{\prime}$, $C^{\prime}$ be projections of point $P$ to the lines $B C, C A, A B$. Prove that the circumcircle of the triangle $A^{\prime} B^{\prime} C^{\prime}$ passes through the midpoint of segment $C P$. Alexey Zaslavsky

Consider the numbers from $1$ to $16$. The "solitar" game consists in the arbitrary grouping of the numbers in pairs and replacing each pair with the great prime divisor of the sum of the two numbers (i.e from $(1,2); (3,4); (5,6);...;(15,16)$ the numbers which result are $3,7,11,5,19,23,3,31$). The next step follows from the same procedure and the games continues untill we obtain only one number. Which is the maximum numbers with which the game ends.

Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$ a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$ b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$

Which of the following is equivalent to $$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$$ $\textbf{(A) }3^{127}+2^{127} \qquad \textbf{(B) }3^{127}+2^{127}+2\cdot 3^{63}+3\cdot 2^{63} \qquad \textbf{(C) }3^{128}-2^{128} \qquad \textbf{(D) }3^{128}+2^{128} \qquad \textbf{(E) }5^{127}$

All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined? \[ \begin{tabular}{c c c c} {} & \textbf{Adams} & \textbf{Baker} & \textbf{Adams and Baker} \\ \textbf{Boys:} & 71 & 81 & 79 \\ \textbf{Girls:} & 76 & 90 & ? \\ \textbf{Boys and Girls:} & 74 & 84 & \\ \end{tabular} \] $ \textbf{(A)}\ 81 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 84 \quad\textbf{(E)}\ 85 $

Let $[a]$ represent the largest integer less than or equal to $a$, for any real number $a$. Let $\{a\} = a - [a]$. Are there positive integers $m,n$ and $n+1$ real numbers $x_0,x_1,\hdots,x_n$ such that $x_0=428$, $x_n=1928$, $\frac{x_{k+1}}{10} = \left[\frac{x_k}{10}\right] + m + \left\{\frac{x_k}{5}\right\}$ holds? Justify your answer.

In a scalene triangle $ABC$ with incenter $I$, the incircle is tangent to sides $CA$ and $AB$ at points $E$ and $F$. The tangents to the circumcircle of triangle $AEF$ at $E$ and $F$ meet at $S$. Lines $EF$ and $BC$ intersect at $T$. Prove that the circle with diameter $ST$ is orthogonal to the nine-point circle of triangle $BIC$. [i]Proposed by Evan Chen[/i]

How many paths are there from $A$ to $B$ that consist of $5$ horizontal segments and $5$ vertical segments of length $1$ each? (see figure) [img]https://cdn.artofproblemsolving.com/attachments/4/2/5b476ca2a232fc67fb2e2f6bb06111cab60692.png[/img]

Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that $\bullet$ $P$ passes through $A$ and $B$, $\bullet$ the focus $F$ of $P$ lies on $E$, $\bullet$ the orthocenter $H$ of $\vartriangle F AB$ lies on the directrix of $P$. If the major and minor axes of $E$ have lengths $50$ and $14$, respectively, compute $AH^2 + BH^2$.

Let $\Gamma$ be the circumcircle of a triangle $ABC$ and let $E$ and $F$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with $\Gamma$. If $EF$ is tangent to the incircle $\gamma$ of $\triangle ABC$, then find the value of $\angle BAC$.

Which of these describes the graph of $x^2(x+y+1)=y^2(x+y+1)$ ? $\textbf{(A)}\ \text{two parallel lines}$\\ $\textbf{(B)}\ \text{two intersecting lines}$\\$\textbf{(C)}\ \text{three lines that all pass through a common point}$\\ $\textbf{(D)}\ \text{three lines that do not all pass through a common point}$\\$\textbf{(E)}\ \text{a line and a parabola}$

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

If a square is drawn externally on each side of a parallelogram, prove that: (a) The quadrilateral formed by the centers of these squares is also a square. (b) The diagonals of the new square formed are concurrent with the diagonals of the original parallelogram.

Let us say that a deck of $52$ cards is arranged in a “regular” way if the ace of spades is on the very top of the deck and any two adjacent cards are either of the same value or of the same suit (top and bottom cards regarded adjacent as well). Prove that the number of ways to arrange a deck in regular way is a) divisible by $12!$ (3) b) divisible by $13!$ (5)