The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is [i]bad[/i] if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? $ \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $

Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.

find all functions $f(x)$ satisfying: $(\forall x\in R) f(x)+xf(1-x)=x^2+1$

There are $n$ lamps and $k$ switches in a room. Initially, each lamp is either turned on or turned off. Each lamp is connected by a wire with $2020$ switches. Switching a switch changes the state of a lamp, that is connected to it, to the opposite state. It is known that one can switch the switches so that all lamps will be turned on. Prove, that it is possible to achieve the same result by switching the switches no more than $ \left \lfloor \dfrac{k}{2} \right \rfloor$ times. [i]Proposed by T. Zimanov[/i]

Prove that there are infinitely many positive integers $n$ such that $\frac{5^n-1}{n+2}$ is an integer.

Pick a $3$-digit number $abc$, which contains no $0$'s. The probability that this is a winning number is $\frac1a\cdot\frac1b\cdot\frac1c$. However, the BMT problem writer tries to balance out the chances for the numbers in the following ways: [list] [*] For the lowest digit $n$ in the number, he rolls an $n$-sided die for each time that the digit appears, and gives the number $0$ probability of winning if an $n$ is rolled. [*] For the largest digit $m$ in the number, he rolls an $m$-sided die once and scales the probability of winning by that die roll. [/list] If you choose optimally, what is the probability that your number is a winning number?

[b]p1.[/b] A Slavic dragon has three heads. A knight fights the dragon. If the knight cuts off one dragon’s head three new heads immediately grow. Is it possible that the dragon has $2018$ heads at some moment of the fight? [b]p2.[/b] Peter has two squares $3\times 3$ and $4\times 4$. He must cut one of them or both of them in no more than four parts in total. Is Peter able to assemble a square using all these parts? [b]p3.[/b] Usually, dad picks up Constantine after his music lessons and they drive home. However, today the lessons have ended earlier and Constantine started walking home. He met his dad $14$ minutes later and they drove home together. They arrived home $6$ minutes earlier than usually. Home many minutes earlier than usual have the lessons ended? Please, explain your answer. [b]p4.[/b] All positive integers from $1$ to $2018$ are written on a blackboard. First, Peter erased all numbers divisible by $7$. Then, Natalie erased all remaining numbers divisible by $11$. How many numbers did Natalie remove? Please, explain your answer. [b]p5.[/b] $30$ students took part in a mathematical competition consisting of four problems. $25$ students solved the first problem, $24$ students solved the second problem, $22$ students solved the third, and, finally, $21$ students solved the fourth. Show that there are at least two students who solved all four problems. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The numbers 1, 2 and 3 are written on a board. Two friends are playing the following game. A player writes a number that doesn't exceed 2022 and isn't already on the board and is a sum or a product of two numbers that are written on the board. They take turns writing numbers and the winner is the player who writes 2022 on the board. Which player has a winning strategy and why? [i]Proposed by Ilija Jovcheski[/i]

Let $z=z(x,y)$ be implicit function with two variables from $2sin(x+2y-3z)=x+2y-3z$. Find $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$.

A committee with $20$ members votes for the candidates $A,B,C$ by a different election system. Each member writes his ordered prefer list to the ballot (e.g. if he writes $BAC$, he prefers $B$ to $A$ and $C$, and prefers $A$ to $C$). After the ballots are counted, it is recognized that each of the six different permutations of three candidates appears in at least one ballot, and $11$ members prefer $A$ to $B$, $12$ members prefer $C$ to $A$, $14$ members prefer $B$ to $C$. How many members are there such that $B$ is the first choice of them? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{More information is needed} $

Let \[p(x)=x^{2008}+x^{2007}+x^{2006}+\cdots+x+1,\] and let $r(x)$ be the polynomial remainder when $p(x)$ is divided by $x^4+x^3+2x^2+x+1$. Find the remainder when $|r(2008)|$ is divided by $1000$.

In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$. a) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$. b) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.

Let $n$ be an integer greater than $1$. In how many ways can we fill all the numbers $1, 2,..., 2n$ in the boxes of a grid of $2\times n$, one in each box, so that any two consecutive numbers are they in squares that share one side of the grid?

Given the number $720$, Juan must choose $4$ numbers that are divisors of $720$. He wins if none of the four chosen numbers is a divisor of the product of the other three. Decide whether Juan can win.

Let $ABCDEF$ be an inscribed hexagon with $$AB.CD.EF=BC.DE.FA$$ Let $B_1$ be the reflection point of $B$ with respect to $AC$ and $D_1$ be the reflection point of $D$ with respect to $CE,$ and finally let $F_1$ be the reflection point of $F$ with respect to $AE.$ Prove that $\triangle B_1D_1F_1\sim BDF.$

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

The altitudes of an acute triangle $ABC$ intersect at $H$. The tangent line at $H$ to the circumcircle of triangle $BHC$ intersects the lines $AB$ and $AC$ at points $Q$ and $P$ respectively. The circumcircles of triangles $ABC$ and $APQ$ intersect at point $K$ ($K\neq A$). The tangent lines at the points $A$ and $K$ to the circumcircle of triangle $APQ$ intersect at $T$. Prove that $TH$ passes through the midpoint of segment $BC$.

Do there exist such a triangle $T$, that for any coloring of a plane in two colors one may found a triangle $T'$, equal to $T$, such that all vertices of $T'$ have the same color. [b] proposed by S. Berlov[/b]

A rabbit is placed on a $2n\times 2n$ chessboard. Every time the rabbit moves to one of the adjacent squares. (Adjacent means sharing an edge). It is known that the rabbit went through every square and came back to the place where the rabbit started, and the path of the rabbit form a polygon $\mathcal{P}$. Find the maximum possible number of the vertices of $\mathcal{P}$. For example the answer for the case $n=2$ would be $12$. [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(2cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -11.3, xmax = 27.16, ymin = -11.99, ymax = 10.79; /* image dimensions */ /* draw figures */ draw((5.14,3.19)--(8.43,3.22), linewidth(1)); draw((8.43,3.22)--(11.72,3.25), linewidth(1)); draw((11.72,3.25)--(11.75,-0.04), linewidth(1)); draw((11.75,-0.04)--(11.78,-3.33), linewidth(1)); draw((11.78,-3.33)--(8.49,-3.36), linewidth(1)); draw((8.49,-3.36)--(5.2,-3.39), linewidth(1)); draw((5.2,-3.39)--(5.17,-0.1), linewidth(1)); draw((5.17,-0.1)--(5.14,3.19), linewidth(1)); draw((6.785,3.205)--(6.845,-3.375), linewidth(1)); draw((8.43,3.22)--(8.49,-3.36), linewidth(1)); draw((10.075,3.235)--(10.135,-3.345), linewidth(1)); draw((5.155,1.545)--(11.735,1.605), linewidth(1)); draw((5.17,-0.1)--(11.75,-0.04), linewidth(1)); draw((11.765,-1.685)--(5.185,-1.745), linewidth(1)); draw((5.97,2.375)--(10.905,2.42), linewidth(1)); draw((10.905,2.42)--(10.92,0.775), linewidth(1)); draw((10.92,0.775)--(9.275,0.76), linewidth(1)); draw((9.275,0.76)--(9.29,-0.885), linewidth(1)); draw((9.29,-0.885)--(10.935,-0.87), linewidth(1)); draw((10.935,-0.87)--(10.95,-2.515), linewidth(1)); draw((10.95,-2.515)--(6.015,-2.56), linewidth(1)); draw((6.015,-2.56)--(6,-0.915), linewidth(1)); draw((6,-0.915)--(7.645,-0.9), linewidth(1)); draw((7.645,-0.9)--(7.63,0.745), linewidth(1)); draw((7.63,0.745)--(5.985,0.73), linewidth(1)); draw((5.985,0.73)--(5.97,2.375), linewidth(1)); /* dots and labels */ dot((5.97,2.375),linewidth(4pt) + dotstyle); dot((5.985,0.73),linewidth(4pt) + dotstyle); dot((6,-0.915),linewidth(4pt) + dotstyle); dot((6.015,-2.56),linewidth(4pt) + dotstyle); dot((7.645,-0.9),linewidth(4pt) + dotstyle); dot((7.63,0.745),linewidth(4pt) + dotstyle); dot((9.275,0.76),linewidth(4pt) + dotstyle); dot((9.29,-0.885),linewidth(4pt) + dotstyle); dot((10.95,-2.515),linewidth(4pt) + dotstyle); dot((10.935,-0.87),linewidth(4pt) + dotstyle); dot((10.92,0.775),linewidth(4pt) + dotstyle); dot((10.905,2.42),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts: [list] (a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities $v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$; (b) $v_s,v_p$ and $v_d$, expressed in meter per second, are integers. (c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always $50$ seconds. [/list] Find all possible values of $v_s, v_p$ and $v_d$.

Let $S$ be the set of ordered integer pairs $(x, y)$ such that $0 < x < y < 42$ and there exists some integer $n$ such that $x^6-y^6 \mid n^2+2015^2$. What is the sum $\sum_{(x_i, y_i) \in S}x_iy_i$?

Two squids are forced to participate in a game. Before it begins, they will be informed of all the rules, and can discuss their strategies freely. Then, they will be locked in separate rooms, and be given distinct positive integers no larger than $2023$ as their IDs respectively. The two squids then take turns alternatively; on one's turn, the squid chooses one of the following: 1. announce a positive integer, which will be heard by the other squid; 2. declare which squid has the larger ID. If correct, they win and are released together; otherwise, they lose and are fried together. Find the smallest positive integer $N$ so that, no matter what IDs the squids have been given, they can always win in a finite number of turns, and the sum of the numbers announced during the game is no larger than $N$.

Let $n\ge 3$ be a given integer. Nonzero real numbers $a_1,..., a_n$ satisfy: $\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}}$ What values can be taken by the product $\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}$ ?

A subset of $\left\{1,2,3,\ldots,2017,2018\right\}$ has the property that none of its members are $5$ times another. What is the maximum number of elements that such a subset could have? [i]2018 CCA Math Bonanza Lightning Round #4.2[/i]

For which integers $n \ge 5$ is it possible to color the vertices of a regular$ n$-gon using at most $6$ colors in such a way that any $5$ consecutive vertices have different colors?