[b]p1.[/b] In the correct identity $(x^2 - 1)(x + ...) = (x + 3)(x- 1)(x +...)$ two numbers were replaced with dots. What were these numbers? [b]p2.[/b] A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take $10\%$ of the amount currently available, on the other - $20\%$, etc. . How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will he bring to B? [img]https://cdn.artofproblemsolving.com/attachments/f/5/ab62ce8fce3d482bc52b89463c953f4271b45e.png[/img] [b]p3.[/b] Find the angle between the hour and minute hands at $7$ hours $38$ minutes. [b]p4.[/b] The lottery game is played as follows. A random number from $1$ to $1000$ is selected. If it is divisible by $2$, they pay a ruble, if it is divisible by $10$ - two rubles, by $12$ - four rubles, by $20$ - eight, if it is divisible by several of these numbers, then they pay the sum. How much can you win (at one time) in such a game? List all options. [b]p5.[/b]The sum of the digits of a positive integer $x$ is equal to $n$. Prove that between $x$ and $10x$ you can find an integer whose sum of digits is $ n + 5$. [b]p6.[/b] $9$ people took part in the campaign, which lasted $12$ days. There were $3$ people on duty every day. At the same time, the duty officers quarreled with each other and no two of them wanted to be on duty together ever again. Nevertheless, the participants of the campaign claim that for all $12$ days they were able to appoint three people on duty, taking into account this requirement. Could this be so? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle? [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((12,0)--(12,1)--(11,1)--(11,0)--cycle); fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black); draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle); draw((15,0)--(15,1)--(16,1)--(16,0)--cycle); fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black); [/asy]

Let $ \mathcal{F}$ be a surface which is simply covered by two systems of geodesics such that any two lines belonging to different systems form angles of the same opening. Prove that $ \mathcal{F}$ can be developed (that is, isometrically mapped) into the plane.

Evaluate: \[\sum^{10}_{x=2} \dfrac{2}{x(x^2-1)}\] .

Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.

There is a closed polyline with $n$ edges on the plane. We build a new polyline which edges connect the midpoints of two adjacent edges of the previous polyline. Then we erase previous polyline and start over and over. Also we know that each polyline satisfy that all vertices are different and not all of them are collinear. For which $n$ we can get a polyline that is a сonvex polygon?

On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible? $ \text{(A)}\ 90\qquad\text{(B)}\ 91\qquad\text{(C)}\ 92\qquad\text{(D)}\ 95\qquad\text{(E)}\ 97 $

[b]p1.[/b] Anne purchased yesterday at WalMart in Puerto Rico $6$ identical notebooks, $8$ identical pens and $7$ identical erasers. Anne remembers that each eraser costs $73$ cents. She did not buy anything else. Anne told her mother that she spent $12$ dollars and $76$ cents at Walmart. Can she be right? Note that in Puerto Rico there is no sales tax. [b]p2.[/b] Two men ski one after the other first in a flat field and then uphill. In the field the men run with the same velocity $12$ kilometers/hour. Uphill their velocity drops to $8$ kilometers/hour. When both skiers enter the uphill trail segment the distance between them is $300$ meters less than the initial distance in the field. What was the initial distance between skiers? (There are $1000$ meters in 1 kilometer.) [b]p3.[/b] In the equality $** + **** = ****$ all the digits are replaced by $*$. Restore the equality if it is known that any numbers in the equality does not change if we write all its digits in the opposite order. [b]p4.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started: ”-None of you has $8$ legs. Only I have 8 legs!” Which polyleg has exactly $8$ legs? [b]p5.[/b] Cut the figure shown below in two equal pieces. (Both the area and the form of the pieces must be the same.) [img]https://cdn.artofproblemsolving.com/attachments/e/4/778678c1e8748e213ffc94ba71b1f3cc26c028.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For which $n$, $ n \in \mathbb{N}$ is the number $1^n+2^n+3^n$ divisible by $7$?

Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of ​​this quadrilateral does not depend on the choice of the point $M$.

Sydney the squirrel is at $(0, 0)$ and is trying to get to $(2021, 2022)$. She can move only by reflecting her position over any line that can be formed by connecting two lattice points, provided that the reflection puts her on another lattice point. Is it possible for Sydney to reach $(2021, 2022)$?

The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.

Sam and Jonathan play a game where they take turns flipping a weighted coin, and the game ends when one of them wins. The coin has a $\frac89$ chance of landing heads and a $\frac19$ chance of landing tails. Sam wins when he flips heads, and Jonathan wins when he flips tails. Find the probability that Samwins, given that he takes the first turn. [i]Proposed by Samuel Tsui[/i]

In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$. [i](Proposed by Anton Trygub)[/i]

Let $a,b,x,y$ be positive numbers with $a+b+x+y < 2$. Given that $$\begin{cases} a+b^2 = x+y^2 \\ a^2 +b = x^2 +y\end {cases} $$ show that $a = x$ and $b = y$

A word of length $n$ is an ordered sequence $x_1x_2\ldots x_n$ where $x_i$ is a letter from the set $\{ a,b,c \}$. Denote by $A_n$ the set of words of length $n$ which do not contain any block $x_ix_{i+1}, i=1,2,\ldots ,n-1,$ of the form $aa$ or $bb$ and by $B_n$ the set of words of length $n$ in which none of the subsequences $x_ix_{i+1}x_{i+2}, i=1,2,\ldots n-2,$ contains all the letters $a,b,c$. Prove that $|B_{n+1}|=3|A_n|$. [i]Vasile Pop[/i]

How many ten-digit whole numbers satisfy the following property: they have only $2$ and $5$ as digits, and there are no consecutive $2$'s in the number (i.e. any two $2$'s are separated by at least one $5$)?

In the acute-angled triangle $ABC$, with $AB \ne AC$, $D$ is the foot of the angle bisector of angle $A$, and $E, F$ are the feet of the altitudes from $B$ and $C$, respectively. The circumcircles of triangles $DBF$ and $DCE$ intersect for the second time at $M$. Prove that $ME = MF$. Leonard Giugiuc

Let $\Gamma$ be the circumcircle of a triangle $ABC,$ and let $D$ and $E$ be two points different from the vertices on the sides $AB$ and $AC,$ respectively. Let $A'$ be the second point where $\Gamma$ intersects the bisector of the angle $BAC,$ and let $P$ and $Q$ be the second points where $\Gamma$ intersects the lines $A'D$ and $A'E,$ respectively. Let $R$ and $S$ be the second points of intersection of the lines $AA'$ and the circumcircles of the triangles $APD$ and $AQE,$ respectively. Show that the lines $DS, \: ER$ and the tangent line to $\Gamma$ through $A$ are concurrent.

In a $5 \times 5$ grid of squares, how many nonintersecting pairs rectangles of rectangles are there? (Note sharing a vertex or edge still means the rectangles intersect.)

Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.

Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?

A point is chosen inside a regular $k$-gon in such a way that its orthogonal projections on to the sides all meet the respective sides at interior points. These points divide the sides into $2k$ segments. Let these segments be enumerated consecutively by the numbers $1,2, 3, ... ,2k$. Prove that the sum of the lengths of the segments having even numbers equals the sum of the segments having odd numbers. (A Andjans, Riga)

Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.

In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if there are two people $A$ and $ B$ such that A took $B$ and $ B $ took $A$. For each positive integer n, let $f (n)$ be the number of hidden friends with n people where there is no “marmalade”, i.e. $f (n)$ is equal to the number of permutations $\sigma$ of {$1, 2,. . . , n$} such that: *$\sigma (i) \neq i$ for all $i=1,2,...,n$ * there are no $ 1 \leq i <j \leq n $ such that $ \sigma (i) = j$ and $\sigma (j) = i. $ Determine the limit $\lim_{n \to + \infty} \frac{f(n)}{n!}$