Let $n$ and $k$ be positive integers with $n\geq k\geq 2$. For $i=1,\dots,n$, let $S_i$ be a nonempty set of consecutive integers such that among any $k$ of them, there are two with nonempty intersection. Prove that there is a set $X$ of $k-1$ integers such that each $S_i$, $i=1,\dots,n$ contains at least one integer in $X$.

Given a positive integer $n \ge 2$. Find all $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$, such that $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$, $a_1$ is odd, and (1) $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$ is a positive integer; (2) One can pick $n$-tuples of integers $(k_{i,1},k_{i,2},\ldots,k_{i,n})$ for $i=1,2,\ldots,M$ such that for any $1 \le i_1 <i_2 \le M$, there exists $j \in \{1,2,\ldots,n\}$ such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

[b]p1[/b]. Consider the following four statements referring to themselves: 1. At least one of these statements is true. 2. At least two of these statements are false. 3. At least three of these statements are true. 4. All four of these statements are false. Determine which statements are true and which are false. Justify your answer. [b]p2.[/b] Let $f(x) = a_{2017}x^{2017} + a_{2016}x^{2016} + ... + a_1x + a_0$ where the coefficients $a_0, a_1, ... , a_{2017}$ have not yet been determined. Alice and Bob play the following game: $\bullet$ Alice and Bob alternate choosing nonzero integer values for the coefficients, with Alice going first. (For example, Alice’s first move could be to set $a_{18}$ to $-3$.) $\bullet$ After all of the coefficients have been chosen: - If f(x) has an integer root then Alice wins. - If f(x) does not have an integer root then Bob wins. Determine which player has a winning strategy and what the strategy is. Make sure to justify your answer. [b]p3.[/b] Suppose that a circle can be inscribed in a polygon $P$ with $2017$ equal sides. Prove that $P$ is a regular polygon; that is, all angles of $P$ are also equal. [b]p4.[/b] A $3 \times 3 \times 3$ cube of cheese is sliced into twenty-seven $ 1 \times 1 \times 1$ blocks. A mouse starts anywhere on the outside and eats one of the $1\times 1\times 1$ cubes. He then moves to an adjacent cube (in any direction), eats that cube, and continues until he has eaten all $27$ cubes. (Two cubes are considered adjacent if they share a face.) Prove that no matter what strategy the mouse uses, he cannot eat the middle cube last. [Note: One should neglect gravity – intermediate configurations don’t collapse.] p5. Suppose that a constant $c > 0$ and an infinite sequence of real numbers $x_0, x_1, x_2, ...$ satisfy $x_{k+1} =\frac{x_k + 1}{1 - cx_k}$ for all $k \ge 0$. Prove that the sequence $x_0, x_1, x_2, ....$ contains infinitely many positive terms and also contains infinitely many negative terms. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integers $n \leq 3$ such that there is a set $S_n$ formed by $n$ points of the plane that satisfy the following two conditions: Any three points are not collinear. No point is found inside the circle whose diameter has ends at any two points of $S_n$. [b]NOTE: [/b] The points on the circumference are not considered to be inside the circle.

In Venetiania, the smallest currency is the ducat. The finance minister instructs his officials as follows: "I wish six kinds of banknotes, each worth a whole number of ducats. Those six values must be such that there exists a number N with the following property: Any amount of money of $n$ ducats ($n$ positive and integer) where $n \le N$ may be paid in such a way that no more than two copies of each kind are required either to pay or to return. I also wish those six values to be as large as possible for $N$. Determine those six values and provide proof that all conditions have been met." Solve the problem of those officials

A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that: [list=1][*]the first fly is caught after a resting period of one minute; [*]the resting period before catching the $2m^\text{th}$ fly is the same as the resting period before catching the $m^\text{th}$ fly and one minute shorter than the resting period before catching the $(2m+1)^\text{th}$ fly; [*]when the chameleon stops resting, he catches a fly instantly.[/list] [list=a][*]How many flies were caught by the chameleon before his first resting period of $9$ minutes in a row? [*]After how many minutes will the chameleon catch his $98^\text{th}$ fly? [*]How many flies were caught by the chameleon after 1999 minutes have passed?[/list]

There were 20 participants in a chess tournament. Each of them played with each other twice: once as white and once as black. Let us say that participant $X{}$ is no weaker than participant $Y{}$ if $X{}$ has won at least the same number of games playing white as $Y{}$ and also has won at least the same number of games playing black as $Y{}$ . Do there exist for sure two participants $A{}$ and $B{}$ such that $A{}$ is not weaker than $B{}$? [i]Boris Frenkin[/i]

Let $P$ and $P '$ be two unequal regular $n-$gons and $A$ and $A'$two points inside $P$ and$ P '$, respectively.Suppose $\{ d_1 , d_2 , \cdots d_n \}$ are the distances from $A $ to the vertices of $P$ and $\{ d'_1 , d'_2 , \cdots d'_n \}$ are defines similarly for $P',A'$. Is it possible for $\{ d'_1 , d'_2 , \cdots d'_n \}$ to be a permutation of $\{ d_1 , d_2 , \cdots d_n \}$ ?

Let $n$ be a positive integer and $a_1,a_2,\ldots a_{2n+1}$ be positive reals. For $k=1,2,\ldots ,2n+1$, denote $b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)$, where indices are taken modulo $2n+1$. Prove that the number of indices $k$ satisfying $b_k\ge 1$ does not exceed $2\sum_{i=1}^{2n+1} a_i$.

A 9-digit number $N$ is given, whose digits are non-zero and all different.The sums of all consecutive three-digit segments in the decimal representation of number $N$ are calculated and arranged in increasing order.Is it possible to obtain the following sequences as a result of this operation? $\text{a)}$ $11,15,16,18,19,21,22$ $\text{b)}$ $11,15,16,18,19,21,23$

Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?

There are $n > 1$ points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?

In each of the cells of a $13 \times 13$ board is written an integer such that the integers in adjacent cells differ by $1$. If there are two $2$s and two $24$s on this board, how many $13$s can there be?

Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible? $\textbf{(A) } 1\qquad\textbf{(B) } 3\qquad\textbf{(C) }6 \qquad\textbf{(D) }12 \qquad\textbf{(E) }24$

A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called [i]adjacent[/i] if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.

Let $S$ be a union of finitely many disjoint subintervals of $[0,1]$ such that no two points in $S$ have distance $1/10$. Show that the total length of the intervals comprising $S$ is at most $1/2$.

$16$ progamers are playing in a single elimination tournament. Each player has a different skill level and when two play against each other the one with the higher skill level will always win. Each round, each progamer plays a match against another and the loser is eliminated. This continues until only one remains. How many different progamers can reach the round that has $2$ players remaining?

There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip $1$ moves one booth, chip $2$ moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip $1$ a bulb is lit. Where is chip $1$ on the first minute in which all bulbs are lit?

We color all the points of the plane with two colors. Prove that there are (at least) two points of the plane having the same color and at distance $1$ among them.

Ten square cardboards of $3$ centimeters on a side are cut by a line, as indicated in the figure. After the cuts, there are $20$ pieces: $10$ triangles and $10$ trapezoids. Assemble a square that uses all $20$ pieces without overlaps or gaps. [img]https://cdn.artofproblemsolving.com/attachments/7/9/ec2242cca617305b02eef7a5409e6a6b482d66.gif[/img]

Let $n \geq 2$ be an integer. There are $n$ distinct circles in general position, that is, any two of them meet in two distinct points and there are no three of them meeting at one point. Those circles divide the plane in limited regions by circular edges, that meet at vertices (note that each circle have exactly $2n-2$ vertices). For each circle, temporarily color its vertices alternately black and white (note that, doing this, each vertex is colored twice, one for each circle passing through it). If the two temporarily colouring of a vertex coincide, this vertex is definitely colored with this common color; otherwise, it will be colored with gray. Show that if a circle has more than $n-2 + \sqrt{n-2}$ gray points, all vertices of some region are grey. Observation: In this problem, a region cannot contain vertices or circular edges on its interior. Also, the outer region of all circles also counts as a region.

[u]Discrete Math Round[/u] [b]p1.[/b] A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. [b]p2.[/b] What is the probability that a randomly chosen factor of $2016$ is a perfect square? [b]p3.[/b] Compute the remainder when $$5\underbrace{666...6666}_{2016 \,\, sixes}5$$ is divided by $17$. [b]p4.[/b] At an M&M factory, two types of M&Ms are produced, red and blue. The M&Ms are transported individually on a conveyor belt. Anna is watching the conveyor belt, and has determined that four out of every five red M&Ms are followed by a blue one, while one out of every six blue M&Ms is followed by a red one. What proportion of the M&Ms are red? [b]p5.[/b] Three cards are chosen from a standard deck of $52$ without replacing them. Given that the ace of spades was chosen, what is the expected number of aces chosen? [b]p6.[/b] Moor decides that he needs a new email address, and forms the address by taking some permutation of the $12$ letters $MMMOOOOOORRR$. How many permutations of the letters will contain $MOOR$ in this exact order at least once? [b]p7.[/b] Suppose that the $8$ corners of a cube can be colored either red, green, or blue. We call a coloring of the cube rotationally symmetric if the cube can be rotated along a single axis parallel to an edge of a cube either $90^o$, $180^o$, or $270^o$, and reach the original coloring. How many rotationally symmetric colorings exist using the $3$ colors? Assume that any colorings which are identical after rotation are equivalent. [b]p8.[/b] Let $x = \frac{1}{9} + \frac{1}{99} + \frac{1}{999} + ...+ \frac{1}{999999999}$ . Compute the number of digits in the first $3000$ decimal places of the base $10$ representation of $x$ which are greater than or equal to $8$. [b]p9.[/b] Two $20$-sided dice are rolled. Their outcomes are independent and take uniformly distributed integer values from $1$ to $20$, inclusive. For each roll, let $x$ be (the sum of the dice) $\times $ (the positive difference of the dice). What is the expected value of $x$? [b]p10.[/b] Compute $$\sum^{1000}_{a=1} \sum^{1000}_{b=1} \sum^{1000}_{c=1} \left\lfloor \frac{1000}{lcm (a, b, c)} \right \rfloor \phi (a) \phi (b) \phi(c)$$ where $\phi (n) = | \{k : 1 \le k \le n, gcd (k, n) = 1\} |$ counts the integers coprime to $n$ that are less than or equal to $n$. [u]Discrete Math Tiebreaker[/u] [b]Tie 1.[/b] A certain elementary school has $48$ students in the third grade that must be organized into three classes of $16$ students each. There are three troublemakers in the grade. If the students are assigned independently and randomly to classes, what is the probability that all three trou blemakers are assigned to the same $16$ student class? [b]Tie 2.[/b] A $4$-digit number $x$ has the property that the expected value of the integer obtained from switching any two digits in $x$ is $4625$. Given that the sum of the digits of $x$ is $20$, compute $x$. [b]Tie 3.[/b] Let $S$ be the set of factors of $10^5$. The number of subsets of $S$ with a least common multiple of $10^5$ can be written as $2^n * m$, where $n$ and $m$ are positive integers and $m$ is not divisible by $2$. Compute $m + n$. PS. You should use hide for answers.

Consider a convex polyhedron $P$ with vertices $V_1,\ldots ,V_p$. The distinct vertices $V_i$ and $V_j$ are called [i]neighbours[/i] if they belong to the same face of the polyhedron. To each vertex $V_k$ we assign a number $v_k(0)$, and construct inductively the sequence $v_k(n)\ (n\ge 0)$ as follows: $v_k(n+1)$ is the average of the $v_j(n)$ for all neighbours $V_j$ of $V_k$ . If all numbers $v_k(n)$ are integers, prove that there exists the positive integer $N$ such that all $v_k(n)$ are equal for $n\ge N$ .

Evin picks distinct points $A, B, C, D, E$, and $F$ on a circle. What is the probability that there are exactly two intersections among the line segments $AB$, $CD$, and $EF$? [i]Proposed by Evin Liang[/i]