Among the coordinates $(x,y)$ $(1\leq x,y\leq 101)$, choose some points such that there does not exist $4$ points which form a isoceles trapezium with its base parallel to either the $x$ or $y$ axis(including rectangles). Find the maximum number of coordinate points that can be chosen.

There are $12$ candies on the table, four of which are rare candies. Chad has a friend who can tell rare candies apart from regular candies, but Chad can’t. Chad’s friend is allowed to take four candies from the table, but may not take any rare candies. Can his friend always take four candies in such a way that Chad will then be able to identify the four rare candies? If so, describe a strategy. If not, prove that it cannot be done. Note that Chad does not know anything about how the candies were selected (e.g. the order in which they were selected). However, Chad and his friend may communicate beforehand.

We wish to place natural numbers around a circle such that the following property is satisfied: the absolute values of the differences of each pair of neighboring numbers are all different. a) Is it possible to place the numbers from 1 to 2009 satisfying this property b) Is it possible to suppress one of the numbers from 1 to 2009 in such a way that the remaining 2008 numbers can be placed satisfying the property

Let $n$ be an integer, $n \geq 3$. Let $f(n)$ be the largest number of isosceles triangles whose vertices belong to some set of $n$ points in the plane without three colinear points. Prove that there exists positive real constants $a$ and $b$ such that $an^{2}< f(n) < bn^{2}$ for every integer $n$, $n \geq 3$.

Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have \[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]

Let $A_1,A_2,\cdots,A_n$ be $n$ distinct points on the plane ($n>1$). We consider all the segments $A_iA_j$ where $i<j\leq{n}$ and color the midpoints of them. What's the minimum number of colored points? (In fact, if $k$ colored points coincide, we count them $1$.)

The convex $1976$-gon is divided into $1975$ triangles. Prove that there is a straight line separating one of these triangles from the rest.

There are $n$ distinct straight lines on a plane such that every line intersects exactly $12$ others. Determine all the possible values of $n$.

Let $A_n$ be the set of all sequences with length $n$ and members of the set $\{1,2…q\}$. We denote with $B_n$ a subset of $A_n$ with a minimal number of elements with the following property: For each sequence $a_1,a_2,...,a_n$ from $A_n$ there exist a sequence $b_1,b_2,...,b_n$ from $B_n$ such that $a_i\neq b_i$ for each $i=1,2,....,n$. Prove that, if $q>n$, then $|B_n |=n+1$.

$2013$ smurfs are sitting at a large round table. Each of them has two tickets. on each card represents a number from $\{1, 2, . . ., 2013\}$ such that each of the numbers from this set occurs exactly twice. Every smurf takes the card every minute with the smaller of the two numbers, it smurfs on to its left neighbor and receives a card from his right neighbor. Show that there will come a time when a smurf has two cards with the same number.

You want to color a flag like the one shown in the following image, for which four different colors are available. Two regions of the flag that share a side (or a segment of a side) must have different colors. The flag cannot be flipped, rotated, or reflected. How many different flags can be colored with these conditions? [img]https://cdn.artofproblemsolving.com/attachments/4/9/879d1e144acdbc63ee2ffe34cf13a920d5d836.png[/img]

Call a convex polyhedron a [i]footballoid [/i] if it has the following properties. (1) Any face is either a regular pentagon or a regular hexagon. (2) All neighbours of a pentagonal face are hexagonal (a [i]neighbour [/i] of a face is a face that has a common edge with it). Find all possibilities for the number of pentagonal and hexagonal faces of a footballoid.

Numbers 1 and −1 are written in the cells of a board 2000×2000. It is known that the sum of all the numbers in the board is positive. Show that one can select 1000 rows and 1000 columns such that the sum of numbers written in their intersection cells is at least 1000. [i]D. Karpov[/i]

A rectangle $\mathcal R$ is divided into a set $\mathcal S$ of finitely many smaller rectangles with sides parallel to the sides of $\mathcal R$ such that no three rectangles in $\mathcal S$ share a common corner. An ant is initially located at the bottom-left corner of $\mathcal R$. In one operation, we can choose a rectangle $r$ in $\mathcal S$ such that the ant is currently located at one of the corners of $r$, say $c$, and move the ant to one of the two corners of $r$ adjacent to $c$. Suppose that after a finite number of operations, the ant ends up at the top-right corner of $\mathcal R$. Prove that some rectangle $r$ in $\mathcal S$ was chosen in at least two operations.

Rosemonde is stacking spheres to make pyramids. She constructs two types of pyramids $S_n$ and $T_n$. The pyramid $S_n$ has $n$ layers, where the top layer is a single sphere and the $i^{th}$ layer is an $i\times $i square grid of spheres for each $2 \le i \le n$. Similarly, the pyramid $T_n$ has $n$ layers where the top layer is a single sphere and the $i^{th}$ layer is $\frac{i(i+1)}{2}$ spheres arranged into an equilateral triangle for each $2 \le i \le n$.

There are $100$ points on the circumference of a circle, arbitrarily labelled by $1,2,\ldots,100$. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Prove that it is impossible to have exactly $2017$ [b]clockwise[/b] triangles.

Consider a sequence of equilateral triangles $T_{n}$ as represented below: [asy] defaultpen(linewidth(0.8));size(350); real r=sqrt(3); path p=origin--(2,0)--(1,sqrt(3))--cycle; int i,j,k; for(i=1; i<5; i=i+1) { for(j=0; j<i; j=j+1) { for(k=0; k<j; k=k+1) { draw(shift(5*i-5+(i-2)*(i-1)*1,0)*shift(2(j-k)+k, k*r)*p); }}}[/asy] The length of the side of the smallest triangles is $1$. A triangle is called a delta if its vertex is at the top; for example, there are $10$ deltas in $T_{3}$. A delta is said to be perfect if the length of its side is even. How many perfect deltas are there in $T_{20}$?

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

An ant crawled a total distance of $1$ in the plane and returned to its original position (so that its path is a closed loop of length $1$; the width is considered to be $0$). Prove that there is a circle of radius $\frac{1}{4}$ containing the path. Illustration of an example path:

Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \times n$. We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \times n$ board, and such that each rectangle covers the top-right corner of the $n \times n$ board. Then, the $k$ rectangles are painted of black. This process leaves a white figure in the board. How many different white figures are possible to do with $k$ rectangles that can't be done with less than $k$ rectangles? Proposed by David Torres Flores

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?

[b]p1[/b]. Evaluate $1! + 2! + 3! + 4! + 5! $ (where $n!$ is the product of all integers from $1$ to $n$, inclusive). [b]p2.[/b] Harold opens a pack of Bertie Bott's Every Flavor Beans that contains $10$ blueberry, $10$ watermelon, $3$ spinach and $2$ earwax-flavored jelly beans. If he picks a jelly bean at random, then what is the probability that it is not spinach-flavored? [b]p3.[/b] Find the sum of the positive factors of $32$ (including $32$ itself). [b]p4.[/b] Carol stands at a flag pole that is $21$ feet tall. She begins to walk in the direction of the flag's shadow to say hi to her friends. When she has walked $10$ feet, her shadow passes the flag's shadow. Given that Carol is exactly $5$ feet tall, how long in feet is her shadow? [b]p5.[/b] A solid metal sphere of radius $7$ cm is melted and reshaped into four solid metal spheres with radii $1$, $5$, $6$, and $x$ cm. What is the value of $x$? [b]p6.[/b] Let $A = (2,-2)$ and $B = (-3, 3)$. If $(a,0)$ and $(0, b)$ are both equidistant from $A$ and $B$, then what is the value of $a + b$? [b]p7.[/b] For every flip, there is an $x^2$ percent chance of flipping heads, where $x$ is the number of flips that have already been made. What is the probability that my first three flips will all come up tails? [b]p8.[/b] Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order? [b]p9.[/b] A square and a rectangle overlap each other such that the area inside the square but outside the rectangle is equal to the area inside the rectangle but outside the square. If the area of the rectangle is $169$, then find the side length of the square. [b]p10.[/b] If $A = 50\sqrt3$, $B = 60\sqrt2$, and $C = 85$, then order $A$, $B$, and $C$ from least to greatest. [b]p11.[/b] How many ways are there to arrange the letters of the word $RACECAR$? (Identical letters are assumed to be indistinguishable.) [b]p12.[/b] A cube and a regular tetrahedron (which has four faces composed of equilateral triangles) have the same surface area. Let $r$ be the ratio of the edge length of the cube to the edge length of the tetrahedron. Find $r^2$. [b]p13.[/b] Given that $x^2 + x + \frac{1}{x} +\frac{1}{x^2} = 10$, find all possible values of $x +\frac{1}{x}$ . [b]p14.[/b] Astronaut Bob has a rope one unit long. He must attach one end to his spacesuit and one end to his stationary spacecraft, which assumes the shape of a box with dimensions $3\times 2\times 2$. If he can attach and re-attach the rope onto any point on the surface of his spacecraft, then what is the total volume of space outside of the spacecraft that Bob can reach? Assume that Bob's size is negligible. [b]p15.[/b] Triangle $ABC$ has $AB = 4$, $BC = 3$, and $AC = 5$. Point $B$ is reflected across $\overline{AC}$ to point $B'$. The lines that contain $AB'$ and $BC$ are then drawn to intersect at point $D$. Find $AD$. [b]p16.[/b] Consider a rectangle $ABCD$ with side lengths $5$ and $12$. If a circle tangent to all sides of $\vartriangle ABD$ and a circle tangent to all sides of $\vartriangle BCD$ are drawn, then how far apart are the centers of the circles? [b]p17.[/b] An increasing geometric sequence $a_0, a_1, a_2,...$ has a positive common ratio. Also, the value of $a_3 + a_2 - a_1 - a_0$ is equal to half the value of $a_4 - a_0$. What is the value of the common ratio? [b]p18.[/b] In triangle $ABC$, $AB = 9$, $BC = 11$, and $AC = 16$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{BC}$, respectively, such that $BE = BF = 4$. What is the area of triangle $CEF$? [b]p19.[/b] Xavier, Yuna, and Zach are running around a circular track. The three start at one point and run clockwise, each at a constant speed. After $8$ minutes, Zach passes Xavier for the first time. Xavier first passes Yuna for the first time in $12$ minutes. After how many seconds since the three began running did Zach first pass Yuna? [b]p20.[/b] How many unit fractions are there such that their decimal equivalent has a cycle of $6$ repeating integers? Exclude fractions that repeat in cycles of $1$, $2$, or $3$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$-hedron. On earth we don't have any other information about sides of planet tarator. We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money. On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money? [b]Note 1:[/b] Tourists spend no money during travel! [b]Note 2:[/b] The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables. [b]Note 3:[/b] The maximum must be among all possible polyhedras. Time allowed for this problem was 90 minutes.

There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either [i]on[/i] or [i]off[/i]. Every second, each lamp undergoes a change according to the following rule: (a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is [i]off[/i] right now. (Indices taken mod $n$.) (b) Otherwise, $L_i$ is [i]on[/i] right now. Initially, all the lamps are [i]off[/i], except for $L_1$ which is [i]on[/i]. Prove that for infinitely many integers $n$ all the lamps will be [i]off[/i] eventually, after a finite amount of time.