There are $10$ points on a circle and all possible segments are drawn on the which two of these points are the endpoints. Determine the probability that selecting two segments randomly, they intersect at some point (it could be on the circumference).

There are $100$ glasses, with $101,102,...,200$ cents.Two players play next game. In every move they can take some cents from one glass, but after move should be different number of cents in every glass. Who will win with right strategy?

Tristan is eating his favorite cereal, Tiger Crunch, which has marshmallows of two colors, black and orange. He eats the marshmallows by randomly choosing from those remaining one at a time, and he starts out with $17$ orange and $5$ black marshmallows. If $\frac{p}{q}$ is the expected number of marshmallows remaining the instant that there is only one color left, and $p$ and $q$ are relatively prime positive integers, find $p + q$.

In the lower left corner of the square $7 \times 7$ board there is a king. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different ways can the king get to the upper right corner of the board if he is prohibited from visiting the central square?

Urn A contains $4$ white balls and $2$ red balls. Urn B contains $3$ red balls and $3$ black balls. An urn is randomly selected, and then a ball inside of that urn is removed. We then repeat the process of selecting an urn and drawing out a ball, without returning the first ball. What is the probability that the first ball drawn was red, given that the second ball drawn was black?

We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$. We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!

Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called [i]pretty [/i]if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different [i]pretty [/i]sets on such a board. (Poland)

A copsychus and a sparrow, each initially located at one of the vertex of a regular polygon with $103$ edges, fly clockwise to another vertex each. The copsychus moves across $\ell$ edges each time while the sparrow moves through$ d$ edges of the polygon, where $\ell \ne d$ are both integers less than $103$. Assume that, during their journeys, the copsychus has stopped at $m$ vertices while sparrow has stopped at $n$ vertices of the polygon, for $m \ge n \ge 3$. Determine the value of $m, n$ given that there is only one common single vertex of the polygon that both of birds have stopped at, and there is only one vertex that neither of the birds have reached. Vu Thi Khoi, Topo University, Hanoi Mathematics Institute, Vietnam, Hoang Qu6c Vietnam, Hanoi.

There are $12$ seats at a round table in a restaurant. A group of five women and seven men arrives at the table. How many ways are there for choosing the sitting order, provided that every woman ought to be surrounded by two men and two orders are regarded as different, if at least one person has a different neighbour on one's right side.

Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime.

Square $ABCD$ with side-length $n$ is divided into $n^2$ small (fundamental) squares by drawing lines parallel to its sides (the case $n=5$ is presented on the diagram).The squares' vertices that lie inside (or on the boundary) of the triangle $ABD$ are connected with each other with arcs.Starting from $A$,we move only upwards or to the right.Each movement takes place on the segments that are defined by the fundamental squares and the arcs of the circles.How many possible roots are there in order to reach $C$;

If $S=\{s_1,s_2,\dots,s_n\}$ is a set of integers with $s_1<s_2<\dots<s_n$, define $$f(S)=\sum_{k=1}^n (-1)^k k^2 s_k.$$ (If $S$ is empty, $f(S)=0$.) Compute the average value of $f(S)$ as $S$ ranges over all subsets of $\{1^2,2^2,\dots,100^2\}$. [i]Proposed by Connor Gordon and Nairit Sarkar[/i]

The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$. Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$. [i](I. Voronovich)[/i]

[b]p1.[/b] Consider the following function $f(x) = \left(\frac12 \right)^x - \left(\frac12 \right)^{x+1}$. Evaluate the infinite sum $f(1) + f(2) + f(3) + f(4) +...$ [b]p2.[/b] Find the area of the shape bounded by the following relations $$y \le |x| -2$$ $$y \ge |x| - 4$$ $$y \le 0$$ where |x| denotes the absolute value of $x$. [b]p3.[/b] An equilateral triangle with side length $6$ is inscribed inside a circle. What is the diameter of the largest circle that can fit in the circle but outside of the triangle? [b]p4.[/b] Given $\sin x - \tan x = \sin x \tan x$, solve for $x$ in the interval $(0, 2\pi)$, exclusive. [b]p5.[/b] How many rectangles are there in a $6$ by $6$ square grid? [b]p6.[/b] Find the lateral surface area of a cone with radius $3$ and height $4$. [b]p7.[/b] From $9$ positive integer scores on a $10$-point quiz, the mean is $ 8$, the median is $ 8$, and the mode is $7$. Determine the maximum number of perfect scores possible on this test. [b]p8.[/b] If $i =\sqrt{-1}$, evaluate the following continued fraction: $$2i +\frac{1}{2i +\frac{1}{2i+ \frac{1}{2i+...}}}$$ [b]p9.[/b] The cubic polynomial $x^3-px^2+px-6$ has roots $p, q$, and $r$. What is $(1-p)(1-q)(1-r)$? [b]p10.[/b] (Variant on a Classic.) Gilnor is a merchant from Cutlass, a town where $10\%$ of the merchants are thieves. The police utilize a lie detector that is $90\%$ accurate to see if Gilnor is one of the thieves. According to the device, Gilnor is a thief. What is the probability that he really is one? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $12$ mathematicians living in a village, each of whom belongs to the $\sqrt2$-clan or belong to the $\pi$-clan. Moreover every mathematician's birthday is in a different month and every mathematician has an odd number of friends among them the mathematicians. We agree that if mathematician $A$ is a friend of mathematician $B$, then so is $B$ is a friend of $A$. On his birthday, every mathematician looks at which clan the majority of his friends belong to, and decides to join that clan until his next birthday. Prove that the mathematicians no longer change clans after a certain point.

Marko lives on the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\tfrac{2}{9}$, $1$ unit right with probability $\tfrac{2}{9}$, $1$ unit up and $1$ unit right with probability $\tfrac{4}{9}$, and he doesn’t move with probability $\tfrac{1}{9}$. After $2019$ seconds, Marko ends up on the point $(A, B)$. What is the expected value of $A\cdot B$?

All the integers from $1$ to $100$ are arranged in a $10 \times 10$ table as shown below. Prove that if some ten numbers are removed from the table, the remaining $90$ numbers contain 10 numbers in Arithmetic Progression. $1 \,\,\,\,2\,\, \,\,3 \,\,\,\,... \,\,10$ $11 \,\,12 \,\,13 \,\,... \,\,20$ $\,\,.\,\,\,\,.\,\,\,.$ $\,\,.\,\,\,\,.\,\,\,\,.$ $91 \,\,92 \,\,93\,\, ... \,\,100$

Let $n$ be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves: (1) select a row and multiply each entry in this row by $n$; (2) select a column and subtract $n$ from each entry in this column. Find all possible values of $n$ for which the following statement is true: Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is $0$.

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

For natural numbers $n$ and $b$, let $V(n, b)$ denote the number of decompositions of $n$ into the product of integers each of which is greater than $b$: for example $$36 = 6\times 6 = 4\times 9 = 3\times 3\times 4 = 3\times 12,$$ i.e. $V(36,2) = 5$. Prove that $V(n, b) < n/b$ for all $n$ and $b$. (N.B. Vasiliev, Moscow)

The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

A fair coin is flipped every second and the results are recorded with $1$ meaning heads and $0$ meaning tails. What is the probability that the sequence $10101$ occurs before the first occurance of the sequence $010101$?

Shining tells Prajit a positive integer $n \ge 2025$. Prajit then tries to place n points such that no four points are concyclic and no $3$ points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so? [i](Prajit Adhikari, Nepal and Shining Sun, USA)[/i]

Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order). a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully. b) Prove that such a tasteful tiling is unique. [i]Victor Wang.[/i]