A set $S$ of $100$ points, no four in a plane, is given in space. Prove that there are no more than $4 .101^2$ tetrahedra with the vertices in $S$, such that any two of them have at most two vertices in common.

Let $G = (V, E)$ be a finite simple graph on $n$ vertices. An edge $e$ of $G$ is called a [i]bottleneck[/i] if one can partition $V$ into two disjoint sets $A$ and $B$ such that [list] [*] at most $100$ edges of $G$ have one endpoint in $A$ and one endpoint in $B$; and [*] the edge $e$ is one such edge (meaning the edge $e$ also has one endpoint in $A$ and one endpoint in $B$). [/list] Prove that at most $100n$ edges of $G$ are bottlenecks. [i]Proposed by Yang Liu[/i]

Jesse and Tjeerd are playing a game. Jesse has access to $n\ge 2$ stones. There are two boxes: in the black box there is room for half of the stones (rounded down) and in the white box there is room for half of the stones (rounded up). Jesse and Tjeerd take turns, with Jesse starting. Jesse grabs in his turn, always one new stone, writes a positive real number on the stone and places put him in one of the boxes that isn't full yet. Tjeerd sees all these numbers on the stones in the boxes and on his turn may move any stone from one box to the other box if it is not yet full, but he may also choose to do nothing. The game stops when both boxes are full. If then the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise win Tjeerd. For every $n \ge 2$, determine who can definitely win (and give a corresponding winning strategy).

[b]p1.[/b] One day, Granny Smith bought a certain number of apples at Horock’s Farm Market. When she returned the next day she found that the price of the apples was reduced by $20\%$. She could therefore buy more apples while spending the same amount as the previous day. How many percent more? [b]p2.[/b] You are asked to move several boxes. You know nothing about the boxes except that each box weighs no more than $10$ tons and their total weight is $100$ tons. You can rent several trucks, each of which can carry no more than $30$ tons. What is the minimal number of trucks you can rent and be sure you will be able to carry all the boxes at once? [b]p3.[/b] The five numbers $1, 2, 3, 4, 5$ are written on a piece of paper. You can select two numbers and increase them by $1$. Then you can again select two numbers and increase those by $1$. You can repeat this operation as many times as you wish. Is it possible to make all numbers equal? [b]p4.[/b] There are $15$ people in the room. Some of them are friends with others. Prove that there is a person who has an even number of friends in the room. [u]Bonus Problem [/u] [b]p5.[/b] Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We are given $n>1$ piles of coins. There are two different types of coins: real and fake coins; they all look alike, but coins of the same type have the same mass, while the coins from different types have different masses. Coins that belong to the same pile are of the same type. We know the mass of real coin. Find the minimal number of weightings on digital scale that we need in order to conclude: which piles consists of which type of coins and also the mass of fake coin. (We assume that every pile consists from infinite number of coins.)

Alex, Bob, Charlie, Daniel, and Ethan are five classmates. Some pairs of them are friends. How many possible ways are there for them to be friends such that everyone has at least one friend, and such that there is exactly one loop of friends among the five classmates? Note: friendship is two-way, so if person x is friends with person y then person y is friends with person $x$.

Let be the points $ K,L,M $ on the sides $ BC,CA,AB, $ respectively, of a triangle $ ABC. $ Show that at least one of the areas of the triangles $ MAL,KBM,LCK $ doesn't surpass a fourth of the area of $ ABC. $

Positive integers $1, 2, 3, ..., n$ are written on a blackboard ($n > 2$). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number $97$ remains. Find the least $n$ for which it is possible.

The set of nonempty integers $A$ is said to be "elegant" if it is for any $a\in A,$ $1\leq k\leq 2023,$ $$\left| \left\{ b\in A:\left\lfloor\frac b{3^k}\right\rfloor =\left\lfloor\frac a{3^k}\right\rfloor\right\}\right| =2^k.$$ Prove that if the intersection of the integer set $S$ and any "elegant" set is not empty$,$ then $S$ contains an "elegant" set.

In the plane are given $100$ points, such that no three of them are on the same line. The points are arranged in $10$ groups, any group containing at least $3$ points. Any two points in the same group are joined by a segment. a) Determine which of the possible arrangements in $10$ such groups is the one giving the minimal numbers of triangles. b) Prove that there exists an arrangement in such groups where each segment can be coloured with one of three given colours and no triangle has all edges of the same colour. [i]Vasile Pop[/i]

[b]p1.[/b] What is $\sqrt[2015]{2^01^5}$? [b]p2.[/b] What is the ratio of the area of square $ABCD$ to the area of square $ACEF$? [b]p3.[/b] $2015$ in binary is $11111011111$, which is a palindrome. What is the last year which also had this property? [b]p4.[/b] What is the next number in the following geometric series: $1020100$, $10303010$, $104060401$? [b]p5.[/b] A circle has radius $A$ and area $r$. If $A = r^2\pi$, then what is the diameter, $C$, of the circle? [b]p6.[/b] If $$O + N + E = 1$$ $$T + H + R + E + E = 3$$ $$N + I + N + E = 9$$ $$T + E + N = 10$$ $$T + H + I + R + T + E + E + N = 13$$ Then what is the value of $O$? [b]p7.[/b] By shifting the initial digit, which is $6$, of the positive integer $N$ to the end (for example, $65$ becomes $56$), we obtain a number equal to $\frac{N}{4}$ . What is the smallest such $N$? [b]p8.[/b] What is $\sqrt[3]{\frac{2015!(2013!)+2014!(2012!)}{2013!(2012!)}}$ ? [b]p9.[/b] How many permutations of the digits of $1234$ are divisible by $11$? [b]p10.[/b] If you choose $4$ cards from a normal $52$ card deck (with replacement), what is the probability that you will get exactly one of each suit (there are $4$ suits)? [b]p11.[/b] If $LMT$ is an equilateral triangle, and $MATH$ is a square, such that point $A$ is in the triangle, then what is $HL/AL$? [b]p12.[/b] If $$\begin{tabular}{cccccccc} & & & & & L & H & S\\ + & & & & H & I & G & H \\ + & & S & C & H & O & O & L \\ \hline = & & S & O & C & O & O & L \\ \end{tabular}$$ and $\{M, A, T,H, S, L,O, G, I,C\} = \{0, 1, 2, 3,4, 5, 6, 7, 8, 9\} $, then what is the ordered pair $(M + A +T + H, [T + e + A +M])$ where $e$ is $2.718...$and $[n]$ is the greatest integer less than or equal to $n$ ? [b]p13.[/b] There are $5$ marbles in a bag. One is red, one is blue, one is green, one is yellow, and the last is white. There are $4$ people who take turns reaching into the bag and drawing out a marble without replacement. If the marble they draw out is green, they get to draw another marble out of the bag. What is the probability that the $3$rd person to draw a marble gets the white marble? [b]p14.[/b] Let a "palindromic product" be a product of numbers which is written the same when written back to front, including the multiplication signs. For example, $234 * 545 * 432$, $2 * 2 *2 *2$, and $14 * 41$ are palindromic products whereas $2 *14 * 4 * 12$, $567 * 567$, and $2* 2 * 3* 3 *2$ are not. 2015 can be written as a "palindromic product" in two ways, namely $13 * 5 * 31$ and $31 * 5 * 13$. How many ways can you write $2016$ as a palindromic product without using 1 as a factor? [b]p15.[/b] Let a sequence be defined as $S_n = S_{n-1} + 2S_{n-2}$, and $S_1 = 3$ and $S_2 = 4$. What is $\sum_{n=1}^{\infty}\frac{S_n}{3^n}$ ? [b]p16.[/b] Put the numbers $0-9$ in some order so that every $2$-digit substring creates a number which is either a multiple of $7$, or a power of $2$. [b]p17.[/b] Evaluate $\dfrac{8+ \dfrac{8+ \dfrac{8+...}{3+...}}{3+ \dfrac{8+...}{3+...}}}{3+\dfrac{8+ \dfrac{8+...}{3+...}}{ 3+ \dfrac{8+...}{3+...}}}$, assuming that it is a positive real number. [b]p18.[/b] $4$ non-overlapping triangles, each of area $A$, are placed in a unit circle. What is the maximum value of $A$? [b]p19.[/b] What is the sum of the reciprocals of all the (positive integer) factors of $120$ (including $1$ and $120$ itself). [b]p20.[/b] How many ways can you choose $3$ distinct elements of $\{1, 2, 3,...,4000\}$ to make an increasing arithmetic series? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A one-player game is played as follows: There is a bowl at each integer on the $Ox$-axis. All the bowls are initially empty, except for that at the origin, which contains $n \geq 2$ stones. A move is either (A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or (B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left. Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for a given $n$).

On a square of a chessboard there is a pawn . Two players take turns to move it to another square, subject to the rule that , at each move the distance moved is strictly greater than that of the previous move. A player loses when unable to make a move on his turn. Who wins if the players always choose the best strategy? (The pawn is always placed in the centre of its square. ) ( F . L . Nazarov)

A pile of $n$ pebbles is placed in a vertical column. This configuration is modified according to the following rules. A pebble can be moved if it is at the top of a column which contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as a column with 0 pebbles.) At each stage, choose a pebble from among those that can be moved (if there are any) and place it at the top of the column to its right. If no pebbles can be moved, the configuration is called a [i]final configuration[/i]. For each $n$, show that, no matter what choices are made at each stage, the final configuration obtained is unique. Describe that configuration in terms of $n$. [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=119189]IMO ShortList 2001, combinatorics problem 7, alternative[/url]

p1. Given $N = 9 + 99 + 999 + ... +\underbrace{\hbox{9999...9}}_{\hbox{121\,\,numbers}}$. Determine the value of N. p2. The triangle $ABC$ in the following picture is isosceles, with $AB = AC =90$ cm and $BC = 108$ cm. The points $P$ and $Q$ are located on $BC$, respectively such that $BP: PQ: QC = 1: 2: 1$. Points $S$ and $R$ are the midpoints of $AB$ and $AC$ respectively. From these two points draw a line perpendicular to $PR$ so that it intersects at $PR$ at points $M$ and $N$ respectively. Determine the length of $MN$. [img]https://cdn.artofproblemsolving.com/attachments/7/1/e1d1c4e6f067df7efb69af264f5c8de5061a56.png[/img] p3. If eight equilateral triangles with side $ 12$ cm are arranged as shown in the picture on the side, we get a octahedral net. Define the volume of the octahedron. [img]https://cdn.artofproblemsolving.com/attachments/4/8/18cdb8b15aaf4d92f9732880784facf9348a84.png[/img] p4. It is known that $a^2 + b^2 = 1$ and $x^2 + y^2 = 1$. Continue with the following algebraic process. $(a^2 + b^2)(x^2 + y^2) – (ax + by)^2 = ...$ a. What relationship can be concluded between $ax + by$ and $1$? b. Why? p5. A set of questions consists of $3$ questions with a choice of answers True ($T$) or False ($F$), as well as $3$ multiple choice questions with answers $A, B, C$, or $D$. Someone answer all questions randomly. What is the probability that he is correct in only $2$ questions?

A solid pyramid $TABCD$, with a quadrilateral base $ABCD$ is to be coloured on each of the five faces such that no two faces with a common edge will have the same colour. If five different colours are available, what is the number of ways to colour the pyramid?

$S$ is an infinite set of points in the plane. The distance between any two points of $S$ is integral. Prove that $S$ is a subset of a straight line.

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$.

Set $T$ consists of $66$ points in plane, and $P$ consists of $16$ lines in plane. Pair $(A,l)$ is [i]good[/i] if $A \in T$, $l \in P$ and $A \in l$. Prove that maximum number of good pairs is no greater than $159$, and prove that there exits configuration with exactly $159$ good pairs.

Given a regular $ 2n$-gon, show that each of its sides and diagonals can be assigned in such a way that the sum of the obtained vectors equals zero.

p1. Given a set of $n$ the first natural number. If one of the numbers is removed, then the average number remaining is $21\frac14$ . Specify the number which is deleted. p2. Ipin and Upin play a game of Tic Tac Toe with a board measuring $3 \times 3$. Ipin gets first turn by playing $X$. Upin plays $O$. They must fill in the $X$ or $O$ mark on the board chess in turn. The winner of this game was the first person to successfully compose a sign horizontally, vertically, or diagonally. Determine as many final positions as possible, if Ipin wins in the $4$th step. For example, one of the positions the end is like the picture on the side. [img]https://cdn.artofproblemsolving.com/attachments/6/a/a8946f24f583ca5e7c3d4ce32c9aa347c7e083.png[/img] p3. Numbers $ 1$ to $10$ are arranged in pentagons so that the sum of three numbers on each side is the same. For example, in the picture next to the number the three numbers are $16$. For all possible arrangements, determine the largest and smallest values ​​of the sum of the three numbers. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3dd629361715b4edebc7803e2734e4f91ca3dc.png[/img] p4. Define $$S(n)=\sum_{k=1}^{n}(-1)^{k+1}\,\, , \,\, k=(-1)^{1+1}1+(-1)^{2+1}2+...+(-1)^{n+1}n$$ Investigate whether there are positive integers $m$ and $n$ that satisfy $S(m) + S(n) + S(m + n) = 2011$ p5. Consider the cube $ABCD.EFGH$ with side length $2$ units. Point $A, B, C$, and $D$ lie in the lower side plane. Point $I$ is intersection point of the diagonal lines on the plane of the upper side. Next, make a pyramid $I.ABCD$. If the pyramid $I.ABCD$ is cut by a diagonal plane connecting the points $A, B, G$, and $H$, determine the volume of the truncated pyramid low part.

An [i]$ n\minus{}m$ society[/i] is a group of $ n$ girls and $ m$ boys. Prove that there exists numbers $ n_0$ and $ m_0$ such that every [i]$ n_0\minus{}m_0$ society[/i] contains a subgroup of five boys and five girls with the following property: either all of the boys know all of the girls or none of the boys knows none of the girls.

There are 1995 segments such that a triangle can be formed from any three of them. Prove that using these $1995 $ segments, it is possible to assemble $664$ acute-angled triangles so that each segment is part of no more than one triangle.

A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018\leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?

Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.