To a sheet of paper, we glue $2011$ “handles” that do not intersect, that is, strips of paper glued to the sheet in your ends. No handle can be twisted. Prove that the surface boundary thus formed has at least two cycles (closed curves). That is, an ant that only walks along the edge of the paper never runs through the entire surface boundary. For example, the configuration represented in the figure has three cycles: one in dashed lines, one in lines dotted lines and another in a continuous line (this cycle passes under a tab twice). [img]https://cdn.artofproblemsolving.com/attachments/f/e/121146a240215f241278b3aabde13a67544e7a.png[/img]

The company has $100$ people. For any $k$, we can find a group of $k$ people such that there are two (different from them) strangers, each of them knows all of these $k$ people. At what maximum $k$ is this possible?

Find, with proof, all irrational numbers $x$ such that both $x^3-6x$ and $x^4-8x^2$ are rational.

Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$. (a) Prove that $$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$ (b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence: $$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$

There are seven rays emanating from a point $A$ on a plane, such that the angle between the two consecutive rays is $30 ^{\circ}$. A point $A_1$ is located on the first ray. The projection of $A_1$ onto the second ray is denoted as $A_2$. Similarly, the projection of $A_2$ onto the third ray is $A_3$, and this process continues until the projection of $A_6$ onto the seventh ray is $A_7$. Find the ratio $\frac{A_7A}{A_1A}$. [img]https://i.imgur.com/oxixe5q.png[/img] [i]Proposed by Giorgi Arabidze, Georgia[/i]

Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$(excenters corresponding vertex $P$). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k\sqrt{-1}}\]

A sequence $a_1, a_2,\dots, a_n$ of positive integers is [i]alagoana[/i], if for every $n$ positive integer, one have these two conditions I- $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$ II- The number $a_n$ is the $n$-power of a positive integer. Find all the sequence(s) [i]alagoana[/i].

[b]p1.[/b] I have $6$ envelopes full of money. The amounts (in dollars) in the $6$ envelopes are six consecutive integers. I give you one of the envelopes. The total amount in the remaining $5$ envelopes is $\$2018$. How much money did I give you? [b]p2. [/b]Two tangents $AB$ and $AC$ are drawn to a circle from an exterior point $A$. Let $D$ and $E$ be the midpoints of the line segments $AB$ and $AC$. Prove that the line DE does not intersect the circle. [b]p3.[/b] Let $n \ge 2$ be an integer. A subset $S$ of {0, 1, . . . , n − 2} is said to be closed whenever it satisfies all of the following properties: • $0 \in S$ • If $x \in S$ then $n - 2 - x \in S$ • If $x \in S$, $y \ge 0$, and $y + 1$ divides $x + 1$ then $y \in S$. Prove that $\{0, 1, . . . , n - 2\}$ is the only closed subset if and only if $n$ is prime. (Note: “$\in$” means “belongs to”.) [b]p4.[/b] Consider the $3 \times 3$ grid shown below $\begin{tabular}{|l|l|l|l|} \hline A & B & C \\ \hline D & E & F \\ \hline G & H & I \\ \hline \end{tabular}$ A knight move is a pair of elements $(s, t)$ from $\{A, B, C, D, E, F, G, H, I\}$ such that $s$ can be reached from $t$ by moving either two spaces horizontally and one space vertically, or by moving one space horizontally and two spaces vertically. (For example, $(B, I)$ is a knight move, but $(G, E)$ is not.) A knight path of length $n$ is a sequence $s_0$, $s_1$, $s_2$, $. . . $, $s_n$ drawn from the set $\{A, B, C, D, E, F, G, H, I\}$ (with repetitions allowed) such that each pair $(s_i , s_{i+1})$ is a knight move. Let $N$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $A$. Let $M$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $I$. Compute the value $(N- M)$, with proof. (Your answer must be in simplified form and may not involve any summations.) [b]p5.[/b] A strip is defined to be the region of the plane lying on or between two parallel lines. The width of the strip is the distance between the two lines. Consider a finite number of strips whose widths sum to a number $d < 1$, and let $D$ be a circular closed disk of diameter $1$. Prove or disprove: no matter how the strips are placed in the plane, they cannot entirely cover the disk $D$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$

Let $P(z) = z^8 + (4\sqrt{3} + 6) z^4 - (4\sqrt{3}+7)$. What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$? $ \textbf{(A)}\ 4\sqrt{3}+4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2}+3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2}+4\sqrt{3} \qquad $ $\textbf{(E)}\ 4\sqrt{3}+6 $

8. Sebuah lantai luasnya 3 meter persegi ditutupi lima buah karpet dengan ukuran masing-masing 1 meter persegi. Buktikan bahwa ada dua karpet yang tumpang tindih dengan luas tumpang tindih minimal 0,2 meter persegi. A floor of a certain room has a $ 3 \ m^2$ area. Then the floor is covered by 5 rugs, each has an area of $ 1 \ m^2$. Prove that there exists 2 overlapping rugs, with at least $ 0.2 \ m^2$ covered by both rugs.

Every cell of a $3\times 3$ board is coloured either by red or blue. Find the number of all colorings in which there are no $2\times 2$ squares in which all cells are red.

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

What is the $100^\text{th}$ digit to the right of the decimal point in the decimal form of $4/37$? $\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$

Jorge chooses $6$ different positive integers and writes one on each face of a cube. He threw his bucket three times. The first time his cube showed the number $5$ facing up and also the sum of the numbers on the faces sides was $20$. The second time his cube showed the number $7$ facing up and also the sum of the numbers on the faces sides was $17$. The third time his cube showed the number $4$ up, plus all the numbers on the side faces. They turned out to be primes. What are the numbers that Jorge chose and how did he distribute them on the faces of the cube? Analyze all odds. Remember that $1$ is not prime.

We say that a function $f: \mathbb R \to \mathbb R$ has the property $(P)$ if, for any real numbers $x$, \[ \sup_{t\leq x} f(x) = x. \] a) Give an example of a function with property $(P)$ which has a discontinuity in every real point. b) Prove that if $f$ is continuous and satisfies $(P)$ then $f(x) = x$, for all $x\in \mathbb R$.

Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?

Which positive integers satisfy the following three conditions? a) The number consists of at least two digits. b) The last digit is not zero. c) Inserting a zero between the last two digits yields a number divisible by the original number.

Prove that there exist two powers of $7$ whose difference is divisible by $2021$.

Polynomials $F$ and $G$ satisfy: $$F(F(x))>G(F(x))>G(G(x))$$ for all real $x$.Prove that $F(x)>G(x)$ for all real $x$.

$110$ teams participate in a volleyball tournament. Every team has played every other team exactly once (there are no ties in volleyball). Turns out that in any set of $55$ teams, there is one which has lost to no more than $4$ of the remaining $54$ teams. Prove that in the entire tournament, there is a team that has lost to no more than $4$ of the remaining $109$ teams.

Let $ n(\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying: (1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} $ or $ x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$ (2)For every $1\le i \le n$, there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$ (3)For every $1\le j \le n$, there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$

Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a function such that $f(ab)$ divides $\max \{f(a),b\}$ for any positive integers $a,b$. Must there exist infinitely many positive integers $k$ such that $f(k)=1$?