Several paper-made disks (not necessarily equal) are put on the table so that there is some overlapping, but no disk is entirely inside another. The parts that overlap are cut off and removed. Show that the remaining parts cannot be assembled so as to form different disks.

Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.

For what positive integers $n\geq 4$ does there exist a set $S$ of $n$ points on the plane, not all collinear, such that for any three non-collinear points $A,B,C$ in $S$, either the incenter, $A$-excenter, $B$-excenter, or $C$-excenter of triangle $ABC$ is also contained in $S$?

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves? Proposed by [i]Nikola Velov, Macedonia[/i]

Let S be a set of sequences of length 15 formed by using the letters a and b such that every pair of sequences in S differ in at least 3 places. What is the maximum number of sequences in S?

Let N be a positive integer greater than 2. We number the vertices of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N + 1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way. In the first step we mark the vertex 1. If ni is the vertex marked in the i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away from vertex ni, counting clockwise if ni is positive and counter-clockwise if ni is negative. This procedure is repeated till we reach a vertex that has already been marked. Let $f(N)$ be the number of non-marked vertices. (a) If $f(N) = 0$, prove that 2N + 1 is a prime number. (b) Compute $f(1997)$.

A knight is placed on each square of the first column of a $2017 \times 2017$ board. A [i]move[/i] consists in choosing two different knights and moving each of them to a square which is one knight-step away. Find all integers $k$ with $1 \leq k \leq 2017$ such that it is possible for each square in the $k$-th column to contain one knight after a finite number of moves. Note: Two squares are a knight-step away if they are opposite corners of a $2 \times 3$ or $3 \times 2$ board.

p1. It is known that $f$ is a function such that $f(x)+2f\left(\frac{1}{x}\right)=3x$ for every $x\ne 0$. Find the value of $x$ that satisfies $f(x) = f(-x)$. p2. It is known that ABC is an acute triangle whose vertices lie at circle centered at point $O$. Point $P$ lies on side $BC$ so that $AP$ is the altitude of triangle ABC. If $\angle ABC + 30^o \le \angle ACB$, prove that $\angle COP + \angle CAB < 90^o$. p3. Find all natural numbers $a, b$, and $c$ that are greater than $1$ and different, and fulfills the property that $abc$ divides evenly $bc + ac + ab + 2$. p4. Let $A, B$, and $ P$ be the nails planted on the board $ABP$ . The length of $AP = a$ units and $BP = b$ units. The board $ABP$ is placed on the paths $x_1x_2$ and $y_1y_2$ so that $A$ only moves freely along path $x_1x_2$ and only moves freely along the path $y_1y_2$ as in following image. Let $x$ be the distance from point $P$ to the path $y_1y_2$ and y is with respect to the path $x_1x_2$ . Show that the equation for the path of the point $P$ is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. [img]https://cdn.artofproblemsolving.com/attachments/4/6/d88c337370e8c3bc5a1833bc9588d3fb047bd0.png[/img] p5. There are three boxes $A, B$, and $C$ each containing $3$ colored white balls and $2$ red balls. Next, take three ball with the following rules: 1. Step 1 Take one ball from box $A$. 2. Step 2 $\bullet$ If the ball drawn from box $A$ in step 1 is white, then the ball is put into box $B$. Next from box $B$ one ball is drawn, if it is a white ball, then the ball is put into box $C$, whereas if the one drawn is red ball, then the ball is put in box $A$. $\bullet$ If the ball drawn from box $A$ in step 1 is red, then the ball is put into box $C$. Next from box $C$ one ball is taken. If what is drawn is a white ball then the ball is put into box $A$, whereas if the ball drawn is red, the ball is placed in box $B$. 3. Step 3 Take one ball each from squares $A, B$, and $C$. What is the probability that all the balls drawn in step 3 are colored red?

A set of $n \ge 2$ light bulbs are arranged around a circle, and are consecutively numbered with $1, 2, . . . , n$. Each bulb can be in one of two states: either it is [b]on[/b] or [b]off[/b]. In the initial configuration, at least one bulb is turned on. On each one of $n$ days we change the current on/off configuration as follows: for $1 \le k \le n$, on the $k$-th day we start from the $k$-th bulb and moving in clockwise direction along the circle, we change the state of every traversed bulb until we switch on a bulb which was previously off. Prove that the final configuration, reached on the $n$-th day, coincides with the initial one.

Let $A$ be the set of $n$-digit integers whose digits are all from $\{ 1, 2, 3, 4, 5 \}$. $B$ is subset of $A$ such that it contains digit $5$, and there is no digit $3$ in front of digit $5$ (i.e. for $n = 2$, $35$ is not allowed, but $53$ is allowed). How many elements does set $B$ have?

You have $9$ colors of socks and $5$ socks of each type of color. Pick two socks randomly. What is the probability that they are the same color?

[b]Problem 1. [/b]In the cells of square table are written the numbers $1$, $0$ or $-1$ so that in every line there is exactly one $1$, amd exactly one $-1$. Each turn we change the places of two columns or two rows. Is it possible, from any such table, after finite number of turns to obtain its opposite table (two tables are opposite if the sum of the numbers written in any two corresponding squares is zero)? [i] Emil Kolev[/i]

All the points with integer coordinates in the $ xy$-Plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point $ (0,0)$ is red and the point $ (0,1)$ is blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.

A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly \(6\) friends.

If $ 1 \le r \le n$ are integers, prove the identity: $ \displaystyle\sum_{d\equal{}1}^{\infty}\binom {n\minus{}r\plus{}1}{d} \binom {r\minus{}1} {d\minus{}1}\equal{}\binom {n}{r}.$

Let $m, n$ be positive integers. There are $n$ piles of gold coins, so that pile $i$ has $a_i > 0$ coins in it $(i = 1, ..., n)$. Consider the following game: Step 1. Nadech picks sets $B_1, B_2, ... , B_n$, where each $B_i$ is a nonempty subset of $\{1, 2, . . . , m\}$, and gives them to Yaya. Step 2. Yaya picks a set $S$ which is also a nonempty subset of $\{1, 2, . . . , m\}$. Step 3. For each $i = 1, 2, . . . , n$, Nadech wins the coins in pile $i$ if $B_i \cap S$ has an even number of elements, and Yaya wins the coins in pile $i$ if $B_i \cap S$ has an odd number of elements. Show that, no matter how Nadech picks the sets $B_1, B_2, . . . , B_n$, Yaya can always pick $S$ so that she ends up with more gold coins than Nadech

A certain country wishes to interconnect $2021$ cities with flight routes, which are always two-way, in the following manner: • There is a way to travel between any two cities either via a direct flight or via a sequence of connecting flights. • For every pair $(A, B)$ of cities that are connected by a direct flight, there is another city $C$ such that $(A, C)$ and $(B, C)$ are connected by direct flights. Show that at least $3030$ flight routes are needed to satisfy the two requirements.

Determine all positive integers $n$ such that it is possible to fill the $n \times n$ table with numbers $1, 2$ and $-3$ so that the sum of the numbers in each row and each column is $0$.

Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?

Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.) (A Rubin)

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Let $n$ be an odd positive integer. There is a graph $G$ with $2n$ vertices such that if you partition the vertices into two groups $A$ and $B$ with $n$ vertices each, then the subgraph consisting of only vertices and edges within $A$ is connected and has a closed path containing all of its edges, starting and ending with the same vertex. The same condition is true for $B$ as well. Is $G$ necessarily a clique? [i](Proposed by Ho Janson)[/i]

Assume that the connected graph $G$ has $n$ vertices all with degree at least three. Prove that there exists a spanning tree of $G$ with more than $\frac{2}{9}n$ leaves.

On the plane $n$ points are chosen so that exactly $m$ of them lie on one straight line and no three points not included in these $m$ points lie on one straight line. What is the number of all lines, each of which contains at least two of these points?