The quadrangle $ABCD$ is circumscribed around the circle with the centre $O$. Prove that $$\angle AOB+ \angle COD=180^o. $$

Show that there exists a positive integer $N$ such that the decimal representation of $2000^N$ starts with the digits $200120012001.$

Initially, on the board, all integers from $1$ to $400$ are written. Two players play a game alternating their moves. In one move it is allowed to erase from the board any 3 integers, which form a triangle. The player, who can not perform a move loses. Who has a winning strategy?

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function with the property that $ (f\circ f) (x) =[x], $ for any real number $ x. $ Show that there exist two distinct real numbers $ a,b $ so that $ |f(a)-f(b)|\ge |a-b|. $ $ [] $ denotes the integer part.

[b]p1.[/b] A very large lucky number $N$ consists of eighty-eight $8$s in a row. Find the remainder when this number $N$ is divided by $6$. [b]p2.[/b] If $3$ chickens can lay $9$ eggs in $4$ days, how many chickens does it take to lay $180$ eggs in $ 8$ days? [b]p3.[/b] Find the ordered pair $(x, y)$ of real numbers satisfying the conditions $x > y$, $x+y = 10$, and $xy = -119$. [b]p4.[/b] There is pair of similar triangles. One triangle has side lengths $4, 6$, and $9$. The other triangle has side lengths $ 8$, $12$ and $x$. Find the sum of two possible values of $x$. [b]p5.[/b] If $x^2 +\frac{1}{x^2} = 3$, there are two possible values of $x +\frac{1}{x}$. What is the smaller of the two values? [b]p6.[/b] Three flavors (chocolate strawberry, vanilla) of ice cream are sold at Brian’s ice cream shop. Brian’s friend Zerg gets a coupon for $10$ free scoops of ice cream. If the coupon requires Zerg to choose an even number of scoops of each flavor of ice cream, how many ways can he choose his ice cream scoops? (For example, he could have $6$ scoops of vanilla and $4$ scoops of chocolate. The order in which Zerg eats the scoops does not matter.) [b]p7.[/b] David decides he wants to join the West African Drumming Ensemble, and thus he goes to the store and buys three large cylindrical drums. In order to ensure none of the drums drop on the way home, he ties a rope around all of the drums at their mid sections so that each drum is next to the other two. Suppose that each drum has a diameter of $3.5$ feet. David needs $m$ feet of rope. Given that $m = a\pi + b$, where $a$ and $b$ are rational numbers, find sum $a + b$. [b]p8.[/b] Segment $AB$ is the diameter of a semicircle of radius $24$. A beam of light is shot from a point $12\sqrt3$ from the center of the semicircle, and perpendicular to $AB$. How many times does it reflect off the semicircle before hitting $AB$ again? [b]p9.[/b] A cube is inscribed in a sphere of radius $ 8$. A smaller sphere is inscribed in the same sphere such that it is externally tangent to one face of the cube and internally tangent to the larger sphere. The maximum value of the ratio of the volume of the smaller sphere to the volume of the larger sphere can be written in the form $\frac{a-\sqrt{b}}{36}$ , where $a$ and $b$ are positive integers. Find the product $ab$. [b]p10.[/b] How many ordered pairs $(x, y)$ of integers are there such that $2xy + x + y = 52$? [b]p11.[/b] Three musketeers looted a caravan and walked off with a chest full of coins. During the night, the first musketeer divided the coins into three equal piles, with one coin left over. He threw it into the ocean and took one of the piles for himself, then went back to sleep. The second musketeer woke up an hour later. He divided the remaining coins into three equal piles, and threw out the one coin that was left over. He took one of the piles and went back to sleep. The third musketeer woke up and divided the remaining coins into three equal piles, threw out the extra coin, and took one pile for himself. The next morning, the three musketeers gathered around to divide the coins into three equal piles. Strangely enough, they had one coin left over this time as well. What is the minimum number of coins that were originally in the chest? [b]p12.[/b] The diagram shows a rectangle that has been divided into ten squares of different sizes. The smallest square is $2 \times 2$ (marked with *). What is the area of the rectangle (which looks rather like a square itself)? [img]https://cdn.artofproblemsolving.com/attachments/4/a/7b8ebc1a9e3808096539154f0107f3e23d168b.png[/img] [b]p13.[/b] Let $A = (3, 2)$, $B = (0, 1)$, and $P$ be on the line $x + y = 0$. What is the minimum possible value of $AP + BP$? [b]p14.[/b] Mr. Mustafa the number man got a $6 \times x$ rectangular chess board for his birthday. Because he was bored, he wrote the numbers $1$ to $6x$ starting in the upper left corner and moving across row by row (so the number $x + 1$ is in the $2$nd row, $1$st column). Then, he wrote the same numbers starting in the upper left corner and moving down each column (so the number $7$ appears in the $1$st row, $2$nd column). He then added up the two numbers in each of the cells and found that some of the sums were repeated. Given that $x$ is less than or equal to $100$, how many possibilities are there for $x$? [b]p15.[/b] Six congruent equilateral triangles are arranged in the plane so that every triangle shares at least one whole edge with some other triangle. Find the number of distinct arrangements. (Two arrangements are considered the same if one can be rotated and/or reflected onto another.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is an integer in each cell of a $2m\times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a necessary and sufficient condition, in terms of $m$, $n$ and the initial numbers on the table, for which there exists a sequence of operations that makes all the numbers on the table equal.

In the square, the midpoints of the two sides were marked and the segments shown in the figure on the left were drawn. Which of the shaded quadrilaterals has the largest area? [img]https://cdn.artofproblemsolving.com/attachments/d/f/2be7bcda3fa04943687de9e043bd8baf40c98c.png[/img]

Find all positive integer $n$ such that for all positive integers $ x $, $ y $, $ n \mid x^n-y^n \Rightarrow n^2 \mid x^n-y^n $.

A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer $x$ with another positive integer $y$, as long as $|x^2 - y^2|$ is a perfect square. For example, if the number on the blackboard is $17$, Carmela can replace it with $15$, because $|17^2 - 15^2| = 8^2$, then replace it with $9$, because $|15^2 - 9^2| = 12^2$. If the number on the blackboard is initially $3$, determine all integers that Carmela can write on the blackboard after finitely many operations.

Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

The line $l$ passing through the orthocenter $H$ of a triangle $ABC$ $(BC>AB)$ and parallel to $AC$ meets $AB$ and $BC$ at points $D$ and $E$ respectively. The line passing through the circumcenter of the triangle and parallel to the median $BM$ meets $l$ at point $F$. Prove that the length of segment $HF$ is three times greater than the difference of $FE$ and $DH$ Proposed by: A.Mardanov, K.Mardanova

In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$. [img]https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png[/img] If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.

Let $ABCD$ be a convex cyclic quadrilateral satisfying $AB\cdot CD=AD\cdot BC$. Let the inscribed circle $\omega$ of triangle $ABC$ be tangent to sides $BC$, $CA$ and $AB$ at points $A', B'$ and $C'$, respectively. Let point $K$ be the intersection of line $ID$ and the nine-point circle of triangle $A'B'C'$ that is inside line segment $ID$. Let $S$ denote the centroid of triangle $A'B'C'$. Prove that lines $SK$ and $BB'$ intersect each other on circle $\omega$. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Let $ABC$ be a triangle. Points $P$ and $Q$ lie on side $BC$ and satisfy $|BP| =|PQ| = |QC| = \frac13 |BC|$. Points $R$ and $S$ lie on side $CA$ and satisfy $|CR| =|RS| = |SA| = 1 3 |CA|$. Finally, points $T$ and $U$ lie on side $AB$ and satisfy $|AT| = |TU| = |UB| =\frac13 |AB|$. Points $P, Q,R, S, T$ and $U$ turn out to lie on a common circle. Prove that $ABC$ is an equilateral triangle.

Prove that for every positive integer $ n,$ there is a sequence of integers $ a_0,a_1,\dots,a_{2009}$ with $ a_0\equal{}0$ and $ a_{2009}\equal{}n$ such that each term after $ a_0$ is either an earlier term plus $ 2^k$ for some nonnnegative integer $ k,$ or of the form $ b\mod{c}$ for some earlier positive terms $ b$ and $ c.$ [Here $ b\mod{c}$ denotes the remainder when $ b$ is divided by $ c,$ so $ 0\le(b\mod{c})<c.$]

A set of $n$ dominoes, each colored with one white square and one black square, is used to cover a $2 \times n$ board of squares. For $n = 6$, how many different patterns of colors can the board have? (For $n = 2$, this number is $6$.)

In a triangle $ABC$ with $\angle BC A = 90^o$, the perpendicular bisector of $AB$ intersects segments $AB$ and $AC$ at $X$ and $Y$, respectively. If the ratio of the area of quadrilateral $BXYC$ to the area of triangle $ABC$ is $13 : 18$ and $BC = 12$ then what is the length of $AC$?

Find all $ n > 1$ such that the inequality \[ \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i\] holds for all real numbers $ x_1$, $ x_2$, $ \ldots$, $ x_n$.

Prove that there is no function $f: \mathbb{Z}\to\mathbb{Z}$ such that $f(f(x))=x+1$, for all $x\in\mathbb{Z}$.

Let $n$ be a positive integer. Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$

In a school, \( n \) different languages are taught. It is known that for any subset of these languages (including the empty set), there is exactly one student who knows these and only these languages (there are \( 2^n \) students in total). Each day, the students are divided into pairs and teach each other the languages that only one of them knows. If students are not allowed to be in the same pair twice, what is the minimum number of days the school administration needs to guarantee that all their students know all \( n \) languages? [i]Proposed by Oleksii Masalitin[/i]

Two circles $\omega_1$ and $\omega_2$ intersecting at points $X{}$ and $Y{}$ are inside the circle $\Omega$ and touch it at points $A{}$ and $B{}$, respectively; the segments $AB$ and $XY$ intersect. The line $AB$ intersects the circles $\omega_1$ and $\omega_2$ again at points $C{}$ and $D{}$, respectively. The circle inscribed in the curved triangle $CDX$ touches the side $CD$ at the point $Z{}$. Prove that $XZ$ is a bisector of $\angle AXB{}$.

Let $A$ be a commutative ring with $0 \neq 1$ such that for any $x \in A \setminus \{0\}$ there exist positive integers $m,n$ such that $(x^m+1)^n=x.$ Prove that any endomorphism of $A$ is an automorphism.

Let $\triangle ABC$ be a right triangle with right angle $C$. Let $I$ be the incenter of $ABC$, and let $M$ lie on $AC$ and $N$ on $BC$, respectively, such that $M,I,N$ are collinear and $\overline{MN}$ is parallel to $AB$. If $AB=36$ and the perimeter of $CMN$ is $48$, find the area of $ABC$.

The symbolism $ \lfloor x\rfloor$ denotes the largest integer not exceeding $ x$. For example. $ \lfloor3\rfloor\equal{}3$, and $ \lfloor 9/2\rfloor\equal{}4$. Compute \[ \lfloor\sqrt1\rfloor\plus{}\lfloor\sqrt2\rfloor\plus{}\lfloor\sqrt3\rfloor\plus{}\cdots\plus{}\lfloor\sqrt{16}\rfloor. \]$ \textbf{(A)}\ 35 \qquad \textbf{(B)}\ 38 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 136$