Call a positive integer [i]monotonous[/i] if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there? $\textbf{(A)} \text{ 1024} \qquad \textbf{(B)} \text{ 1524} \qquad \textbf{(C)} \text{ 1533} \qquad \textbf{(D)} \text{ 1536} \qquad \textbf{(E)} \text{ 2048}$

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied? [i]Proposed by Gerhard Wöginger, Austria[/i]

Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.) [asy]import three; import math; size(180); defaultpen(linewidth(.8pt)); currentprojection=orthographic(2,0.2,1); triple A=(0,0,1); triple B=(sqrt(2)/2,sqrt(2)/2,0); triple C=(sqrt(2)/2,-sqrt(2)/2,0); triple D=(-sqrt(2)/2,-sqrt(2)/2,0); triple E=(-sqrt(2)/2,sqrt(2)/2,0); triple F=(0,0,-1); draw(A--B--E--cycle); draw(A--C--D--cycle); draw(F--C--B--cycle); draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]$ \textbf{(A)}\ 210 \qquad \textbf{(B)}\ 560 \qquad \textbf{(C)}\ 840 \qquad \textbf{(D)}\ 1260 \qquad \textbf{(E)}\ 1680$

BdMO National 2018 Higher Secondary P3 Nazia rolls four fair six-sided dice. She doesn’t see the results. Her friend Faria tells her that the product of the numbers is $144$. Faria also says the sum of the dice, $S$ satisfies $14\leq S\leq 18$ . Nazia tells Faria that $S$ cannot be one of the numbers in the set {$14,15,16,17,18$} if the product is $144$. Which number in the range {$14,15,16,17,18$} is an impossible value for $S$ ?

Six cards numbered 1 through 6 are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.

Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round? $ \textbf{(A)}\ \frac{5}{72}\qquad\textbf{(B)}\ \frac{5}{36}\qquad\textbf{(C)}\ \frac{1}{6}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ 1 $

We want to paint some identically-sized cubes so that each face of each cube is painted a solid color and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce?

Given are $n$ different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points $A,B$.We consider $k$ distinct lines passing through $A$ and $m$ distinct lines passing through $B$.There is no line passing through both $A$ and $B$ and all the lines passing through $k$ intersect with all the lines passing through $B$.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.

An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.

How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive? $\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$

Consider the string of length $6$ composed of three characters $a, b, c$. For each string, if two $a$s are next to each other, or two $b$s are next to each other, then replace $aa$ by $b$, and replace $bb$ by $a$. Also, if $a$ and $b$ are next to each other, or two $c$s are next to each other, remove all two of them (i.e. delete $ab, ba, cc$). Determine the number of strings that can be reduced to $c$, the string of length $1$, by the reducing processes mentioned above.

The integer $ 9$ can be written as a sum of two consecutive integers: $ 9 \equal{} 4\plus{}5.$ Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: $ 9 \equal{} 4\plus{}5 \equal{} 2\plus{}3\plus{}4.$ Is there an integer that can be written as a sum of $ 1990$ consecutive integers and that can be written as a sum of (more than one) consecutive positive integers in exactly $ 1990$ ways?

$n$ men and one woman are in the meeting room with $n+1$ chairs, each of them having their own seat. Show that the following two number of cases are equal. (1) Number of cases to choose one man to get out of the room, and make the left $n-1$ men to sit to each other's chair. (2) Number of cases to make $n+1$ people to sit to each other's chair.

Let $ n$ and $ k$ be positive integers with $ k \geq n$ and $ k \minus{} n$ an even number. Let $ 2n$ lamps labelled $ 1$, $ 2$, ..., $ 2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $ N$ be the number of such sequences consisting of $ k$ steps and resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off. Let $ M$ be number of such sequences consisting of $ k$ steps, resulting in the state where lamps $ 1$ through $ n$ are all on, and lamps $ n \plus{} 1$ through $ 2n$ are all off, but where none of the lamps $ n \plus{} 1$ through $ 2n$ is ever switched on. Determine $ \frac {N}{M}$. [i]Author: Bruno Le Floch and Ilia Smilga, France[/i]

Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?

Distinct pebbles are placed on a $1001 \times 1001$ board consisting of $1001^2$ unit tiles, such that every unit tile consists of at most one pebble. The [i]pebble set[/i] of a unit tile is the set of all pebbles situated in the same row or column with said unit tile. Determine the minimum amount of pebbles that must be placed on the board so that no two distinct tiles have the same [i]pebble set[/i]. [hide=Where's the Algebra Problem?]It's already posted [url=https://artofproblemsolving.com/community/c6h2742895_simple_inequality]here[/url].[/hide]

A [i]domino[/i] is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^2$ dominoes can be placed without overlapping on a $2n \times 2n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.

A $10$ digit number is called interesting if its digits are distinct and is divisible by $11111$. Then find the number of interesting numbers.

Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

Let $A_1, A_2, ..., A_k$ be $5$-element subsets of set $\{1, 2, ..., 23\}$ such that, for all $1 \le i < j \le k$ set $A_i \cap A_j$ has at most three elements. Show that $k \le 2018$.

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

$N$ teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order $\mathcal{A} = (a_1 \ge a_2 \ge \cdots \ge a_N )$ is known, as well as which team obtained which score. Find the number of sequences $\mathcal{A}$ such that the outcome of all matches is uniquely determined by this information. [I]Proposed by Dominic Yeo, United Kingdom.[/i]

In how many distinguishable rearrangements of the letters ABCCDEEF does the A precede both C's, the F appears between the 2 C's, and the D appears after the F?