Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?

Real numbers $b>a>0$ are given. Find the number $r$ in $[a,b]$ which minimizes the value of $\max\left\{\left|\frac{r-x}x\right||a\le x\le b\right\}$.

Write a positive integer in each box so that: All six numbers are different. The sum of the six numbers is $100$. If each number is multiplied by its neighbor (in a clockwise direction) and the six results of those six multiplications are added, the smallest possible value is obtained. Explain why a lower value cannot be obtained. [img]https://cdn.artofproblemsolving.com/attachments/7/1/6fdadd6618f91aa03cdd6720cc2d6ee296f82b.gif[/img]

Aumann, Bill, and Charlie each roll a fair $6$-sided die with sides labeled $1$ through $6$ and look at their individual rolls. Each flips a fair coin and, depending on the outcome, looks at the roll of either the player to his right or the player to his left, without anyone else knowing which die he observed. Then, at the same time, each of the three players states the expected value of the sum of the rolls based on the information he has. After hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Then, for the third time, after hearing what everyone said, the three players again state the expected value of the sum of the rolls based on the information they have. Prove that Aumann, Bill, and Charlie say the same number the third time.

Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$. (Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ ) Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$, their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$. Can we find two non-zero super-integers with zero product? (a zero super-integer has all its digits zero)

It is possible to place the $2014$ points in the plane so that we can construct $1006^2$ parralelograms with vertices among these points, so that the parralelograms have area 1?

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $ m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is $ \textbf{(A)}\ \frac {1}{2m \plus{} 1} \qquad \textbf{(B)}\ m \qquad \textbf{(C)}\ 1 \minus{} m \qquad \textbf{(D)}\ \frac {1}{4m} \qquad \textbf{(E)}\ \frac {1}{8m^2}$

A function $f:\{ 1,2,3,\cdots ,2016\}\rightarrow \{ 1,2,3,\cdots , 2016\}$ is called [i]good[/i] if the function $g(n)=|f(n)-n|$ is injective. Furthermore, a good function $f$ is called [i]excellent[/i] if there exists another good function $f'$ such that $f(n)-f'(n)$ is nonzero for exactly one value of $n$. Let $N$ be the number of good functions that are not excellent. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Nathan Ramesh

Let $f(n) = \sum^n_{i=1}\frac{gcd(i,n)}{n}$. Find the sum of all positive integers $ n$ for which $f(n) = 6$.

There are ten points marked on a circumference, numbered from $1$ to $10$ and join all points with segments. I color the segments, with red someones and others with blue. Without changing the colors of the segments, renumber all the points from the $1$ to the $10$. Will be possible to color the segments and to renumber the points so that those numbers that were jointed with red are jointed now with blue and the numbers that were jointed with blue they are jointed now with red?

Let $ H$ be the intersection of altitudes in triangle $ ABC$. If $ \angle B\equal{}\angle C\equal{}\alpha$ and $ O$ is the center of circle passing through $ A$, $ H$ and $ C$, then find $ \angle HOC$ in terms of $ \alpha$. $\textbf{(A)}\ 90{}^\circ \minus{}\alpha \qquad\textbf{(B)}\ 90{}^\circ \plus{}\frac{\alpha }{2} \qquad\textbf{(C)}\ 180{}^\circ \minus{}\alpha \\ \qquad\textbf{(D)}\ 180{}^\circ \minus{}\frac{\alpha }{2} \qquad\textbf{(E)}\ 180{}^\circ \minus{}2\alpha$

The set of all $10$-digit numbers may be represented as a union of two subsets: the subset $M$ consisting of all $10$-digit numbers, each of which may be represented as a product of two $5$-digit numbers, and the subset $N$ , containing the remaining $10$-digit numbers . Which of the sets $M$ and $N$ contains more elements? (S. Fomin , Leningrad)

Let $n$ be a non-negative integer. Find all non-negative integers $a,b,c,d$ such that \[a^2+b^2+c^2+d^2=7\cdot 4^n\]

Prove that there exists a set of infinitely many positive integers such that the elements of no finite subset of this set add up to a perfect square.

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

Prove that if $ a $ and $ b $ are real numbers and the polynomial $ ax^3 - ax^2 + 9bx - b $ has three positive roots, then they are equal.

An epidemic influenza broke out in the elves city. First day some of them were infected by the external source of infection and nobody later was infected by the external source. The elf is infected when visiting his ill friend. In spite of the situation every healthy elf visits all his ill friends every day. The elf is ill one day exactly, and has the immunity at least on the next day. There is no graftings in the city. Prove that a) If there were some elves immunised by the external source on the first day, the epidemic influenza can continue arbitrary long time. b) If nobody had the immunity on the first day, the epidemic influenza will stop some day.

Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point $(a, b)$ is one greater than the average of the numbers written on points $ (a+1 , b-1) $ and $ (a-1,b+1)$ . Which numbers could he write on point $(121, 212)$? Note: The elements of the first quadrant are points where both of the coordinates are non- negative.

Suppose $a_{1} < a_{2}< \cdots < a_{2024}$ is an arithmetic sequence of positive integers, and $b_{1} <b_{2} < \cdots <b_{2024}$ is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences. [i]Ray Li[/i]

We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

Each different letter in the following addition represents a different decimal digit. The sum is a six-digit integer whose digits are all equal. $$\begin{tabular}{ccccccc} & P & U & R & P & L & E\\ + & & C & O & M & E & T \\ \hline \\ \end{tabular}$$ Find the greatest possible value that the five-digit number $COMET$ could represent.

Let $ABC$ be an acute triangle with $AC > AB$. Let $\Gamma$ be the circle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smaller arc $BC$ of this circle. Let $I$ be the incenter of $ABC$ and let $E$ and $F$ be points on sides $AB$ and $AC$, respectively, such that $AE = AF$ and $I$ lies on the segment $EF$. Let $P$ be the second intersection point of the circumcircle of the triangle $AEF$ with $\Gamma$ with $P \ne A$. Let $G$ and $H$ be the intersection points of the lines $PE$ and $PF$ with $\Gamma$ different from $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines AB and $AC$, respectively. Show that the line $JK$ passes through the midpoint of $BC$.

Let $b=\log_53$. What is $\log_b\left(\log_35\right)$? $\text{(A) }-1\qquad\text{(B) }-\frac{3}{5}\qquad\text{(C) }0\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }1$

Let $ f:[-1,1]\longrightarrow\mathbb{R} $ a derivable function and a non-negative integer $ n. $ Show that there is a $ c\in [-1,1] $ so that: $$ \int_{-1}^1 x^{2n+1} f(x)dx =\frac{2}{2n+3}f'(c). $$

A rectangular prism with dimensions $20$ cm by $1$ cm by $7$ cm is made with blue $1$ cm unit cubes. The outside of the rectangular prism is coated in gold paint. If a cube is chosen at random and rolled, what is the probability that the side facing up is painted gold?