Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

a tournament is playing between n persons. Everybody plays with everybody one time. There is no draw here. A number $k$ is called $n$ good if there is any tournament such that in that tournament they have any player in the tournament that has lost all of $k$'s. prove that 1. $n$ is greater than or equal to $2^{k+1}-1$ 2.Find all $n$ such that $2$ is a n-good

There are 20 blue points on the circle and some red inside so no three are collinear. It turned out that there exists $1123$ triangles with blue vertices having 10 red points inside. Prove that all triangles have 10 red points inside

For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$ (a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$

In the future city Baltic Way there are sixteen hospitals. Every night exactly four of them must be on duty for emergencies. Is it possible to arrange the schedule in such a way that after twenty nights every pair of hospitals have been on common duty exactly once?

[b]p1.[/b] Evaluate $1+3+5+··· +2019$. [b]p2.[/b] Evaluate $1^2 -2^2 +3^2 -4^2 +...· +99^2 -100^2$. [b]p3. [/b]Find the sum of all solutions to $|2018+|x -2018|| = 2018$. [b]p4.[/b] The angles in a triangle form a geometric series with common ratio $\frac12$ . Find the smallest angle in the triangle. [b]p5.[/b] Compute the number of ordered pairs $(a,b,c,d)$ of positive integers $1 \le a,b,c,d \le 6$ such that $ab +cd$ is a multiple of seven. [b]p6.[/b] How many ways are there to arrange three birch trees, four maple, and five oak trees in a row if trees of the same species are considered indistinguishable. [b]p7.[/b] How many ways are there for Mr. Paul to climb a flight of 9 stairs, taking steps of either two or three at a time? [b]p8.[/b] Find the largest natural number $x$ for which $x^x$ divides $17!$ [b]p9.[/b] How many positive integers less than or equal to $2018$ have an odd number of factors? [b]p10.[/b] Square $MAIL$ and equilateral triangle $LIT$ share side $IL$ and point $T$ is on the interior of the square. What is the measure of angle $LMT$? [b]p11.[/b] The product of all divisors of $2018^3$ can be written in the form $2^a \cdot 2018^b$ for positive integers $a$ and $b$. Find $a +b$. [b]p12.[/b] Find the sum all four digit palindromes. (A number is said to be palindromic if its digits read the same forwards and backwards. [b]p13.[/b] How ways are there for an ant to travel from point $(0,0)$ to $(5,5)$ in the coordinate plane if it may only move one unit in the positive x or y directions each step, and may not pass through the point $(1, 1)$ or $(4, 4)$? [b]p14.[/b] A certain square has area $6$. A triangle is constructed such that each vertex is a point on the perimeter of the square. What is the maximum possible area of the triangle? [b]p15.[/b] Find the value of ab if positive integers $a,b$ satisfy $9a^2 -12ab +2b^2 +36b = 162$. [b]p16.[/b] $\vartriangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\vartriangle ADE$. [b]p17[/b]. Let $A_1, A_2,..., A_{10}$ be $10$ points evenly spaced out on a line, in that order. Points $B_1$ and $B_2$ lie on opposite sides of the perpendicular bisector of $A_1A_{10}$ and are equidistant to $l$. Lines $B_1A_1,...,B_1A_{10}$ and $B_2A_1,...· ,B_2A_{10}$ are drawn. How many triangles of any size are present? [b]p18.[/b] Let $T_n = 1+2+3··· +n$ be the $n$th triangular number. Determine the value of the infinite sum $\sum_{k\ge 1} \frac{T_k}{2^k}$. [b]p19.[/b] An infinitely large bag of coins is such that for every $0.5 < p \le 1$, there is exactly one coin in the bag with probability $p$ of landing on heads and probability $1- p$ of landing on tails. There are no other coins besides these in the bag. A coin is pulled out of the bag at random and when flipped lands on heads. Find the probability that the coin lands on heads when flipped again. [b]p20.[/b] The sequence $\{x_n\}_{n\ge 1}$ satisfies $x1 = 1$ and $(4+ x_1 + x_2 +··· + x_n)(x_1 + x_2 +··· + x_{n+1}) = 1$ for all $n \ge 1$. Compute $\left \lfloor \frac{x_{2018}}{x_{2019}} \right \rfloor$. PS. You had better use hide for answers.

The integers from $1$ to $320000$ are placed in the cells of a $8 \times 40000$ board. Prove that it is possible to permute the rows of the table so that the numbers in each column will not be sorted from the top to the bottom in increasing order.

Is it possible to write a natural number in every cell of an infinite chessboard in such a manner that for all integers $m, n > 100$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n \ ?$

You can tile a $2 \times5$ grid of squares using any combination of three types of tiles: single unit squares, two side by side unit squares, and three unit squares in the shape of an L. The diagram below shows the grid, the available tile shapes, and one way to tile the grid. In how many ways can the grid be tiled? [asy] import graph; size(15cm); pen dps = linewidth(1) + fontsize(10); defaultpen(dps); draw((-3,3)--(-3,1)); draw((-3,3)--(2,3)); draw((2,3)--(2,1)); draw((-3,1)--(2,1)); draw((-3,2)--(2,2)); draw((-2,3)--(-2,1)); draw((-1,3)--(-1,1)); draw((0,3)--(0,1)); draw((1,3)--(1,1)); draw((4,3)--(4,2)); draw((4,3)--(5,3)); draw((5,3)--(5,2)); draw((4,2)--(5,2)); draw((5.5,3)--(5.5,1)); draw((5.5,3)--(6.5,3)); draw((6.5,3)--(6.5,1)); draw((5.5,1)--(6.5,1)); draw((7,3)--(7,1)); draw((7,1)--(9,1)); draw((7,3)--(8,3)); draw((8,3)--(8,2)); draw((8,2)--(9,2)); draw((9,2)--(9,1)); draw((11,3)--(11,1)); draw((11,3)--(16,3)); draw((16,3)--(16,1)); draw((11,1)--(16,1)); draw((12,3)--(12,2)); draw((11,2)--(12,2)); draw((12,2)--(13,2)); draw((13,2)--(13,1)); draw((14,3)--(14,1)); draw((14,2)--(15,2)); draw((15,3)--(15,1));[/asy]

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

The degrees of all vertices of a graph are not less than 100 and not more than 200. Prove that its vertices can be divided into connected pairs and triples.

There are an even number of points in a plane. No three of them lie on one straight line. Half of the points are red, the other half are blue. Prove that there exists a connecting line of a red and a blue point such that in each of the half-planes bounded by that line the number of red points is equal to the number of blue points.

In how many ways can $8$ people be divided into pairs?

Let $n$ be a positive integer. On a blackboard is a circle, and around it $n$ non-negative integers are written. Raphinha plays a game in which an operation consists of erasing a number $a$ neighboring $b$ and $c$, with $b \geq c$, and writing in its place $b + c$ if $b + c \leq 5a/4$ and $b - c$ otherwise. Your goal is to make all the numbers on the board equal $0$. Find all $n$ such that Raphinha always manages to reach her goal.

$p$ parallel lines are drawn in the plane and $q$ lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?

[b]p1.[/b] Darth Vader urgently needed a new Death Star battle station. He sent requests to four planets asking how much time they would need to build it. The Mandalorians answered that they can build it in one year, the Sorganians in one and a half year, the Nevarroins in two years, and the Klatoonians in three years. To expedite the work Darth Vader decided to hire all of them to work together. The Rebels need to know when the Death Star is operational. Can you help the Rebels and find the number of days needed if all four planets work together? We assume that one year $= 365$ days. [b]p2.[/b] Solve the inequality: $\left( \sin \frac{\pi}{12} \right)^{\sqrt{1-x}} > \left( \sin \frac{\pi}{12} \right)^x$ [b]p3.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 3x - 6$ [b]p4.[/b] Solve the system of inequalities on $[0, 2\pi]$: $$\sin (2x) \ge \sin (x)$$ $$\cos (2x) \le \cos (x)$$ [b]p5.[/b] The planet Naboo is under attack by the imperial forces. Three rebellian camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle by a defensive field. What is the maximal area that they may need to cover? [b]p6.[/b] The Lake Country on the planet Naboo has the shape of a square. There are nine roads in the country. Each of the roads is a straight line that divides the country into two trapezoidal parts such that the ratio of the areas of these parts is $2:5$. Prove that at least three of these roads intersect at one point. PS. You should use hide for answers.

Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$. Show that for all positive integers $r$, \[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]

In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members. [i]A. Skopenkov[/i]

$P$ is a convex polyhedron such that: [b](1)[/b] every vertex belongs to exactly $3$ faces. [b](1)[/b] For every natural number $n$, there are even number of faces with $n$ vertices. An ant walks along the edges of $P$ and forms a non-self-intersecting cycle, which divides the faces of this polyhedron into two sides, such that for every natural number $n$, the number of faces with $n$ vertices on each side are the same. (assume this is possible) Show that the number of times the ant turns left is the same as the number of times the ant turn right.

We will say that a rearrangement of the letters of a word has no [i]fixed letters[/i] if, when the rearrangement is placed directly below the word, no column has the same letter repeated. For instance $HBRATA$ is a rearragnement with no fixed letter of $BHARAT$. How many distinguishable rearrangements with no fixed letters does $BHARAT$ have? (The two $A$s are considered identical.)

There are 100 apples on the table with total weight of 10 kg. Each apple weighs no less than 25 grams. The apples need to be cut for 100 children so that each of the children gets 100 grams. Prove that you can do it in such a way that each piece weighs no less than 25 grams.

On each square of an $n$x$n$ board sleeps a dragon. Two dragons are called neighbors if their squares have a side in common. Each turn, Minnie wakes up a dragon which has a living neighbor and Max directs it towards one of its living neighbors. The dragon than breathes fire on that neighbor and destroys it, and then goes back to sleep. Minnie's goal is to minimize the snoring of the dragons and leave as few living dragons as possible. Max is a member of PETD (People for the Ethical Treatment of Dragons), and he wants to save as many dragons as he can. How many dragons will stay alive at the end if 1. $n=4$? 2. $n=5$?

[u]Fermi Questions[/u] [b]p1.[/b] What is $\sin (1000)$? (note: that's $1000$ radians, not degrees) [b]p2.[/b] In liters, what is the volume of $10$ million US dollars' worth of gold? [b]p3.[/b] How many trees are there on Earth? [b]p4.[/b] How many prime numbers are there between $10^8$ and $10^9$? [b]p5.[/b] What is the total amount of time spent by humans in spaceflight? [b]p6.[/b] What is the global domestic product (total monetary value of all goods and services produced in a country's borders in a year) of Bangladesh in US dollars? [b]p7.[/b] How much time does the average American spend eating during their lifetime, in hours? [b]p8.[/b] How many CHMMC-related emails did the directors receive or send in the last month? [u]Suspiciously Familiar. . .[/u] [b]p9.[/b] Suppose a farmer learns that he will die at the end of the year (day $365$, where today is day $0$) and that he has $100$ sheep. He decides to sell all his sheep on one day, and that his utility is given by $ab$ where $a$ is the money he makes by selling the sheep (which always have a fixed price) and $b$ is the number of days he has left to enjoy the profit; i.e., $365 - k$ where $k$ is the day number. If every day his sheep breed and multiply their numbers by $(421 + b)/421$ (yes, there are small, fractional sheep), on which day should he sell out? [b]p10.[/b] Suppose in your sock drawer of $14$ socks there are $5$ different colors and $3$ different lengths present. One day, you decide you want to wear two socks that have either different colors or different lengths but not both. Given only this information, what is the maximum number of choices you might have? [u]I'm So Meta Even This Acronym[/u] [b]p11.[/b] Let $\frac{s}{t}$ be the answer of problem $13$, written in lowest terms. Let $\frac{p}{q}$ be the answer of problem $12$, written in lowest terms. If player $1$ wins in problem $11$, let $n = q$. Otherwise, let $n = p$. Two players play a game on a connected graph with $n$ vertices and $t$ edges. On each player's turn, they remove one edge of the graph, and lose if this causes the graph to become disconnected. Which player (first or second) wins? [b]p12.[/b] Let $\frac{s}{t}$ be the answer of problem $13$, written in lowest terms. If player $1$ wins in problem $11$, let $n = t$. Otherwise, let $n = s$. Find the maximum value of $$\frac{x^n}{1 + \frac12 x + \frac14 x^2 + ...+ \frac{1}{2^{2n}} x^{2n}}$$ for $x > 0$. [b]p13.[/b] Let $\frac{p}{q}$ be the answer of problem $12$, written in lowest terms. Let $y$ be the largest integer such that $2^y$ divides $p$. If player $1$ wins in problem $11$, let $z = q$. Otherwise, let $z = p$. Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{z}{n + 2}+\frac{2z}{n + 1}-\frac{z}{n}$$ What is $a_y$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Odin and Evelyn are playing a game, Odin going first. There are initially $3k$ empty boxes, for some given positive integer $k$. On each player’s turn, they can write a non-negative integer in an empty box, or erase a number in a box and replace it with a strictly smaller non-negative integer. However, Odin is only ever allowed to write odd numbers, and Evelyn is only allowed to write even numbers. The game ends when either one of the players cannot move, in which case the other player wins; or there are exactly $k$ boxes with the number $0$, in which case Evelyn wins if all other boxes contain the number $1$, and Odin wins otherwise. Who has a winning strategy? $Agnijo \ Banerjee \ , United \ Kingdom$

Prove that the set of positive integers $\mathbb Z^+$ can be represented as a sum of five pairwise distinct subsets with the following property: each $5$-tuple of numbers of form $(n,2n,3n,4n,5n)$, where $n\in\mathbb Z^+$, contains exactly one number from each of these five subsets.