A man has a seven-candle chandellier. The first evening he lighted one candle for one hour, the second evening he lighted two candles, also for one hour, and so on. After one hour the seventh evening, all seven candles simultaneously finished. How did the man choose the candles to light every evening?

In a country, there are some cities linked together by roads. The roads just meet each other inside the cities. In each city, there is a board which showing the shortest length of the road originating in that city and going through all other cities (the way can go through some cities more than one times and is not necessary to turn back to the originated city). Prove that 2 random numbers in the boards can't be greater or lesser than 1.5 times than each other.

Each side of a regular octagon is colored blue or yellow. In each step, the sides are simultaneously recolored as follows: if the two neighbors of a side have different colors, the side will be recolored blue, otherwise it will be recolored yellow. Show that after a finite number of moves all sides will be colored yellow. What is the least value of the number $N$ of moves that always lead to all sides being yellow?

In how many ways can we place the numbers from $1$ to $100$ in a $2\times 50$ rectangle (divided into $100$ unit squares) so that any two consecutive numbers are always placed in squares with a common side?

Let positive integer $k$ satisfies $1<k<100$. For the permutation of $1,2,...,100$ be $a_1,a_2,...,a_{100}$, take the minimum $m>k$ such that $a_m$ is at least less than $(k-1)$ numbers of $a_1,a_2,...,a_k$. We know that the number of sequences satisfies $a_m=1$ is $\frac{100!}{4}$. Find the all possible values of $k$.

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

As the figure is shown, place a $2\times 5$ grid table in horizontal or vertical direction, and then remove arbitrary one $1\times 1$ square on its four corners. The eight different shapes consisting of the remaining nine small squares are called [i]banners[/i]. [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--C--H--J--N^^B--I^^D--N^^E--M^^F--L^^G--K); draw(Aa--Ca--Ha--Ja--Aa^^Ba--Ia^^Da--Na^^Ea--Ma^^Fa--La^^Ga--Ka); [/asy] [asy] defaultpen(linewidth(0.4)+fontsize(10));size(50); pair A=(-1,1),B=(-1,3),C=(-1,5),D=(-3,5),E=(-5,5),F=(-7,5),G=(-9,5),H=(-11,5),I=(-11,3),J=(-11,1),K=(-9,1),L=(-7,1),M=(-5,1),N=(-3,1),O=(-5,3),P=(-7,3),Aa=(-1,7),Ba=(-1,9),Ca=(-1,11),Da=(-3,11),Ea=(-5,11),Fa=(-7,11),Ga=(-9,11),Ha=(-11,11),Ia=(-11,9),Ja=(-11,7),Ka=(-9,7),La=(-7,7),Ma=(-5,7),Na=(-3,7),Oa=(-5,9),Pa=(-7,9); draw(B--Ca--Ea--M--N^^B--O^^C--E^^Aa--Ma^^Ba--Oa^^Da--N); draw(L--Fa--Ha--J--L^^Ga--K^^P--I^^F--H^^Ja--La^^Pa--Ia); [/asy] Here is a fixed $9\times 18$ grid table. Find the number of ways to cover the grid table completely with 18 [i]banners[/i].

There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$ What is the least number of colours which suffices?

A Pokemon Go player starts at $(0,0)$ and carries a pedometer that records the number of steps taken. He then takes steps with length $1$ unit in the north, south, east, or west direction, such that each move after the first is perpendicular to the move before it. Somehow, the player eventually returns to $(0, 0)$, but he had visited no point (except $(0, 0)$) twice. Let $n$ be the number on the pedometer when the player returns to $(0, 0)$. Of the numbers from $1$ to $2019$ inclusive, how many can be the value of $n$?

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

At this year's Olympiad, some of the students are friends (friendship is symmetric), however there are also students which are not friends. No matter how the students are partitioned in two contest halls, there are always two friends in different halls. Let $A$ be a fixed student. Show that there exist students $B$ and $C$ such that there are exactly two friendships in the group $\{ A,B,C \}$. [i]Authored by Mirko Petrushevski[/i]

Find all integers $ n \ge 2 $ for which it is possible to divide any triangle $ T $ in triangles $ T_1, T_2, \cdots, T_n $ and choose medians $ m_1, m_2, \cdots, m_n $, one in each of these triangles, so that these $ n $ medians have equal length.

For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class. [i]Proposed by Romania[/i]

There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always possible. (D. Karpov)

The numbers $1,2,\dots, 2013$ are written on $2013$ stones weighing $1,2,\dots, 2013$ grams such that each number is used exactly once. We have a two-pan balance that shows the difference between the weights at the left and the right pans. No matter how the numbers are written, if it is possible to determine in $k$ weighings whether the weight of each stone is equal to the number that is written on the stone, what is the least possible value of $k$? $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of above} $

Let $x_1,x_2,\ldots,x_n$ be real numbers satisfying $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that for every integer $k\ge2$ there are integers $a_1,a_2,\ldots,a_n$, not all zero, such that $|a_i|\le k-1$ for all $i$, and $|a_1x_1+a_2x_2+\ldots+a_nx_n|\le{(k-1)\sqrt n\over k^n-1}$.

Three sergeants and several solders serve in a platoon. The sergeants take turns on duty. The commander has given the following orders: (a) Each day, at least one task must be issued to a soldier. (b) No soldier may have more than two task or receive more than one tasks in a single day. (c) The lists of soldiers receiving tasks for two different days must not be the same. (d) The first sergeant violating any of these orders will be jailed. Can at least one of the sergeants, without conspiring with the others, give tasks according to these rules and avoid being jailed? [i]M. Kulikov[/i]

There are $1996$ points marked on a straight line at regular intervals. Petya colors half of them red and the rest blue. Then Vasya divides them into pairs ''red'' - ''blue'' so that the sum distances between points in pairs was maximum. Prove that this maximum does not depend on what coloring Petya made.

Let $n\geq 3$ be an odd integer. On a line, $n$ points are marked in such a way that the distance between any two of them is an integer. It turns out that each marked point has an even sum of distances to the remaining $n-1$ marked points. Prove that the distance between any two marked points is even.

In a certain Zoom meeting, there are $4$ students. How many ways are there to split them into any number of distinguishable breakout rooms, each with at least $ 1$ student?

The numbers from $1$ to $2010$ inclusive are placed along a circle so that if we move along the circle in clockwise order, they increase and decrease alternately. Prove that the difference between some two adjacent integers is even.

[u]Round 1[/u] [b]p1.[/b] How many powers of $2$ are greater than $3$ but less than $2013$? [b]p2.[/b] What number is equal to six greater than three times the answer to this question? [b]p3.[/b] Surya Cup-a-tea-raju goes to Starbucks Coffee to sip coffee out of a styrofoam cup. The cup is a cylinder, open on one end, with base radius $3$ centimeters and height $10$ centimeters. What is the exterior surface area of the styrofoam cup? [u]Round 2[/u] [b]p4.[/b] Andrew has two $6$-foot-length sticks that he wishes to make into two of the sides of the entrance to his fort, with the ground being the third side. If he wants to make his entrance in the shape of a triangle, what is the largest area that he can make the entrance? [b]p5.[/b] Ethan and Devin met a fairy who told them “if you have less than $15$ dollars, I will give you cake”. If both had integral amounts of dollars, and Devin had 5 more dollars than Ethan, but only Ethan got cake, how many different amounts of money could Ethan have had? [b]p6.[/b] If $2012^x = 2013$, for what value of $a$, in terms of $x$, is it true that $2012^a = 2013^2$? [u]Round 3[/u] [b]p7.[/b] Find the ordered triple $(L, M, T)$ of positive integers that makes the following equation true: $$1 + \dfrac{1}{L + \dfrac{1}{M+\dfrac{1}{T}}}=\frac{79}{43}.$$ [b]p8.[/b] Jonathan would like to start a banana plantation so he is saving up to buy an acre of land, which costs $\$600,000$. He deposits $\$300,000$ in the bank, which gives $20\%$ interest compounded at the end of each year. At this rate, how many years will Jonathan have to wait until he can buy the acre of land? [b]p9.[/b] Arul and Ethan went swimming at their town pool and started to swim laps to see who was in better shape. After one hour of swimming at their own paces, Ethan completed $32$ more laps than Arul. However, after that, Ethan got tired and swam at half his original speed while Arul’s speed didn’t change. After one more hour, Arul swam a total of $320$ laps. How many laps did Ethan swim after two hours? [u]Round 4[/u] [b]p10.[/b] A right triangle with a side length of $6$ and a hypotenuse of 10 has circles of radius $1$ centered at each vertex. What is the area of the space inside the triangle but outside all three circles? [b]p11.[/b] In isosceles trapezoid $ABCD$, $\overline{AB} \parallel\overline{CD}$ and the lengths of $\overline{AB}$ and $\overline{CD}$ are $2$ and $6$, respectively. Let the diagonals of the trapezoid intersect at point $E$. If the distance from $E$ to $\overline{CD}$ is $9$, what is the area of triangle $ABE$? [b]p12.[/b] If $144$ unit cubes are glued together to form a rectangular prism and the perimeter of the base is $54$ units, what is the height? PS. You should use hide for answers. Rounds 6-8 are [url=https://artofproblemsolving.com/community/c3h3136014p28427163]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

You are given $N$ such that $ n \ge 3$. We call a set of $N$ points on a plane acceptable if their abscissae are unique, and each of the points is coloured either red or blue. Let's say that a polynomial $P(x)$ divides a set of acceptable points either if there are no red dots above the graph of $P(x)$, and below, there are no blue dots, or if there are no blue dots above the graph of $P(x)$ and there are no red dots below. Keep in mind, dots of both colors can be present on the graph of $P(x)$ itself. For what least value of k is an arbitrary set of $N$ points divisible by a polynomial of degree $k$?

2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.

Let $A = \{a_1, a_2, a_3, a_4, a_5\}$ be a set of $5$ positive integers. Show that for any rearrangement of $A$, $a_{i1}$, $a_{i2}$, $a_{i3}$, $a_{i4}$, $a_{i5}$, the product $$(a_{i1} -a_1) (a_{i2} -a_2) (a_{i3} -a_3) (a_{i4} -a_4) (a_{i5} -a_5)$$ is always even.