For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

Let $p=3\cdot 10^{10}+1$ be a prime and let $p_n$ denote the probability that $p\mid (k^k-1)$ for a random $k$ chosen uniformly from $\{1,2,\cdots,n\}$. Given that $p_n\cdot p$ converges to a value $L$ as $n$ goes to infinity, what is $L$? [i]Proposed by Vijay Srinivasan[/i]

Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

Let $ABC$ be an acute angled triangle. Let the altitude from $C$ to $AB$ meet $AB$ at $C'$ and have midpoint $M$, and let the altitude from $B$ to $AC$ meet $AC$ at $B'$ and have midpoint $N$. Let $P$ be the point of intersection of $AM$ and $BB'$ and $Q$ the point of intersection of $AN$ and $CC'$. Prove that the point $M, N, P$ and $Q$ lie on a circle.

For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]

Does there exist a natural number $N$ which is a power of$2$, such that one can permute its decimal digits to obtain a different power of $2$?

How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Ivan writes the matrix $\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times: [b]1.[/b] he chooses a row or column of the matrix, and [b]2.[/b] he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively. Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}$ after finitely many steps?

Let $n$ be a nonzero natural number and let $S = \{1, 2, . . . , n\}$. A $3 \times n$ board is called [i]beautiful [/i] if it can be completed with numbers from the set $S$ like this as long as the following conditions are met: $\bullet$ on each line, each number from the set S appears exactly once, $\bullet$ on each column the sum of the products of two numbers on that column is divisible by $n$ (that is, if the numbers $a, b, c$ are written on a column, it must be $ab + bc + ca$ be divisible by $n$). For which values ​​of the natural number $n$ are there beautiful tables ¸and for which values ​​do not exist? Justify your answer.

The equation $ x^{x^{x}}...\equal{}2$ is satisfied when $x$ is equal to: $\textbf{(A)}\ \infty \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt[4]{2} \qquad \textbf{(D)}\ \sqrt{2} \qquad \textbf{(E)}\ \text{None of these}$

In set $S$, there is an operation $'' \circ ''$ such that $\forall a,b \in S$, a unique $a \circ b \in S$ exists. And (i) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ (b \circ c)$. (ii) $a \circ b \neq b \circ a$ when $a \neq b$. Prove that: a.) $\forall a,b,c \in S$, $(a \circ b) \circ c = a \circ c$. b.) If $S = \{1,2, \ldots, 1990\}$, try to define an operation $'' \circ ''$ in $S$ with the above properties.

a) Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that\[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$. (It is [url=https://artofproblemsolving.com/community/c6h1268817p6621849]2015 IMO Shortlist A2 [/url]) b) The same question for if \[f(x-f(y))=f(f(x))-f(y)-2\] for all integers $x,y$

For a non-constant arithmetic progression $(a_n)$ there exists a natural $n$ such that $a_{n}+a_{n+1} = a_{1}+…+a_{3n-1}$ . Prove that there are no zero terms in this progression.

Let $F(0)=0$, $F(1)=\frac32$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$ for $n\ge2$. Determine whether or not $\displaystyle{\sum_{n=0}^{\infty}\, \frac{1}{F(2^n)}}$ is a rational number. (Proposed by Gerhard Woeginger, Eindhoven University of Technology)

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? $\textbf{(A)}\ 37\qquad\textbf{(B)}\ 63\qquad\textbf{(C)}\ 117\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 163$

In a dance, a group $S$ of $1994$ students stand in a big circle. Each student claps the hands of each of his two neighbours a number of times. For each student $x,$ let $f(x)$ be the total number of times $x$ claps the hands of his neighbours. As an example, suppose there are $3$ students $A, B$ and $C$. A claps hand with $B$ two times, $B$ claps hand with $C$ three times and $C$ claps hand with $A$ five times. Then $f(A) = 7, f(B) = 5$ and $f(C) = 8.$ (i) Prove that $\{f(x) | x \in S\}\ne\{n | n$ is an integer, $2 \le n \le 1995\}$. (ii) Find an example in which $\{f(x) | x \in S\} = \{n | n$ is an integer, $n \ne 3, 2 \le n \le 1996\}$

Prove that for all real numbers $x$ of the interval $0 < x <\pi$ the inequality $$\sin x +\frac12 \sin 2x +\frac13 \sin 3x > 0$$ holds.

$1001$ marbles are drawn at random and without replacement from a jar of $2020$ red marbles and $n$ blue marbles. Find the smallest positive integer $n$ such that the probability that there are more blue marbles chosen than red marbles is strictly greater than $\frac{1}{2}$. [i]Proposed by Taiki Aiba[/i]

Let $ C_1,C_2$ and $ C_3$ be three pairwise disjoint circles. For each pair of disjoint circles, we define their internal tangent lines as the two common tangents which intersect in a point between the two centres. For each $ i,j$, we define $ (r_{ij},s_{ij})$ as the two internal tangent lines of $ (C_i,C_j)$. Let $ r_{12},r_{23},r_{13},s_{12},s_{13},s_{23}$ be the sides of $ ABCA'B'C'$. Prove that $ AA',BB'$ and $ CC'$ are concurrent. [img]https://cdn.artofproblemsolving.com/attachments/1/2/5ef098966fc9f48dd06239bc7ee803ce4701e2.png[/img]

A Saudi company has two offices. One office is located in Riyadh and the other in Jeddah. To insure the connection between the two offices, the company has designated from each office a number of correspondents so that : (a) each pair of correspondents from the same office share exactly one common correspondent from the other office. (b) there are at least $10$ correspondents from Riyadh. (c) Zayd, one of the correspondents from Jeddah, is in contact with exactly $8$ correspondents from Riyadh. What is the minimum number of correspondents from Jeddah who are in contact with the correspondent Amr from Riyadh?

There is a central train station in point $O$, which is connected to other train stations $A_1, A_2, \ldots, A_8$ with tracks. There is also a track between stations $A_i$ and $A_{i+1}$ for each $i$ from $1$ to $8$ (here $A_9 = A_1$). The length of each track $A_iA_{i+1}$ is equal to $1$, and the length of each track $OA_i$ is equal to $2$, for each $i$ from $1$ to $8$. There are also $8$ trains $B_1, B_2, \ldots, B_8$, with speeds $1, 2, \ldots, 8$ correspondently. Trains can move only by the tracks above, in both directions. No time is wasted on changing directions. If two or more trains meet at some point, they will move together from now on, with the speed equal to that of the fastest of them. Is it possible to arrange trains into stations $A_1, A_2, \ldots, A_8$ (each station has to contain one train initially), and to organize their movement in such a way, that all trains arrive at $O$ in time $t < \frac{1}{2}$? [i](Proposed by Bogdan Rublov)[/i]

Mia is “helping” her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time? $\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5$

We will say that a list of positive integers is [i]admissible [/i] if all its numbers are less than or equal to $100$ and their sum is greater than $1810$. Find the smallest positive integer $d$ such that each admissible list can be crossed out some numbers such that the sum of the numbers left uncrossed out is greater than or equal to $1810-d$ and less than or equal to $1810+d$ .

Rose and Brunno play the game on a board shaped like a regular 1001-gon. Initially, all vertices of the board are white, and there is a chip at one of them. On each turn, Rose chooses an arbitrary positive integer \( k \), then Brunno chooses a direction: clockwise or counterclockwise, and moves the chip in the chosen direction by \( k \) vertices. If at the end of the turn the chip stands at a white vertex, this vertex is painted red. Find the greatest number of vertices that Rose can make red regardless of Brunno's actions, if the number of turns is not limited.

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.