Given a point $O$ inside triangle $ABC$ . Prove that $$S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}$$ where $S_A, S_B, S_C$ denote areas of triangles $BOC, COA, AOB$ respectively.

On the map, the Flower City has the form of a right triangle $ABC$ (see Fig.1). The length of each leg is $6$ meters. All the streets of the city run parallel to one of the legs at a distance of $1$ meter from each other. A river flows along the hypotenuse. From their houses that are located at points $V$ and $S$, at the same time get the Cog and Tab. Each short moves to rivers according to the following rule: tosses his coin, and if the [b]heads[/b] falls, he passes $1$ meter parallel to the leg $AB$ to the north (up), and if tails, then passes $1$ meter parallel to the leg $AC$ on east (right). If the Cog and the Tab meet at the same point, then they move together, tossing a coin. a) Which is more likely: Cog and Tab will meet on the way to the river, or will they come to different points on the shore? b) At what point near the river should the Stranger sit, if he wants the most did Gvintik and Shpuntik come to him together? [img]https://cdn.artofproblemsolving.com/attachments/d/c/5d6f75d039e8f2dd6a0ddfe6c4cb046b83f24c.png[/img] [hide=original wording] На мапi Квiткове мiсто має вигляд прямокутного трикутника ABC (див. рисунок 1). Довжина кожного катету – 6 метрiв. Всi вулицi мiста проходять паралельно одному за катетiв на вiдстанi 1 метра одна вiд одної. Вздовж гiпотенузи тече рiка. Зi своїх будиночкiв, що знаходяться в точках V та S, одночасно виходять Гвинтик та Шпунтик. Кожен коротулька рухається до рiчки за таким правилом: пiдкидає свою монетку, та якщо випадає Орел, вiн проходить 1 метр паралельно катету AB на пiвнiч (вгору), а якщо Решка, то проходить 1 метр паралельно катету AC на схiд (вправо). Якщо Гвинтик та Шпунтик зустрiчаються в однiй точцi, то далi вони рушають разом, пiдкидаючи монетку Гвинтика. 1. Що бiльш ймовiрно: Гвинтик та Шпунтик зустрiнуться на шляху до рiки, або вони прийдуть у рiзнi точки берега? 2. В якiй точцi бiля рiки має сидiти Незнайка, якщо вiн хоче, щоб найбiльш ймовiрно до нього прийшли Гвинтик та Шпунтик разом?[/hide]

For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$. Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$. [i]Proposed by Georgia[/i]

Find the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to an integer number of degrees.

On the AMC 12, each correct answer is worth $ 6$ points, each incorrect answer is worth $ 0$ points, and each problem left unanswered is worth $ 2.5$ points. If Charlyn leaves $ 8$ of the $ 25$ problems unanswered, how many of the remaining problems must she answer correctly in order to score at least $ 100$? $ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 17$

Given two triangles with the same perimeter. Both triangles have integer side lengths. The first triangle is an equilateral triangle. The second triangle has a side with length 1 and a side with length $d$. Prove that when $d$ is divided by 3, the remainder is 1.

Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Because Bubble Boy loves Bubble Girl, he wants to reach her as quickly as possible, but he needs to bring a gift; luckily, there are plenty of gifts along the $x$-axis. Assuming that Bubble Girl remains stationary, find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then reach Bubble Girl (the bubble is a solid boundary, and anything the bubble can touch, Bubble Boy can touch too)

A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get $-1$ mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)

Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$

We call a positive integer $n$ [i]delightful[/i] if there exists an integer $k$, $1 < k < n$, such that \[1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n\] Does there exist a delightful number $N$ satisfying the inequalities \[2013^{2013}<\dfrac{N}{2013^{2013}}<2013^{2013}+4 ?\]

Let triangle $\triangle{MNP}$ have side lengths $MN=13$, $NP=89$, and $PM=100$. Define points $S$, $R$, and $B$ as the midpoints of $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ respectively. A line $\ell$ cuts lines $\overline{MN}$, $\overline{NP}$, and $\overline{PM}$ at points $I$, $J$, and $A$ respectively. Find the minimum value of $(SI+RJ+BA)^2.$

Positive integers $a$, $b$, and $c$ satisfy $a^2 +b^2 = c^3 -1$ where $c \le 40$. Find the sum of all distinct possible values of $c$.

Let $\mathcal{F}$ be a family of subsets of $\{1,2,\ldots, 2017\}$ with the following property: if $S_1$ and $S_2$ are two elements of $\mathcal{F}$ with $S_1\subsetneq S_2$, then $|S_2\setminus S_1|$ is odd. Compute the largest number of subsets $\mathcal{F}$ may contain.

Let $\omega$ be the circumscribed circle of the triangle $ABC$, in which $AC< AB$, $K$ is the center of the arc $BAC$, $KW$ is the diameter of the circle $\omega$. The circle $\gamma$ is inscribed in the curvilinear triangle formed by the segments $BC$, $AB$ and the arc $AC$ of the circle $\omega$. It turned out that circle $\gamma$ also touches $KW$ at point $F$. Let $I$ be the center of the triangle $ABC$, $M$ is the midpoint of the smaller arc $AK$, and $T$ is the second intersection point of $MI$ with the circle $\omega$. Prove that lines $FI$, $TW$ and $BC$ intersect at one point. (Mykhailo Sydorenko)

We say that a prime $p$ is $n$-$\textit{rephinado}$ if $n | p - 1$ and all $1, 2, \ldots , \lfloor \sqrt[\delta]{p}\rfloor$ are $n$-th residuals modulo $p$, where $\delta = \varphi+1$. Are there infinitely many $n$ for which there are infinitely many $n$-$\textit{rephinado}$ primes? Notes: $\varphi =\frac{1+\sqrt{5}}{2}$. We say that an integer $a$ is a $n$-th residue modulo $p$ if there is an integer $x$ such that $$x^n \equiv a \text{ (mod } p\text{)}.$$

In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Points $C,D$ are on side $BE$ of triangle $ABE$, such that $BC=CD=DE$. Points $X,Y,Z,T$ are circumcenters of $ABE,ABC,ADE,ACD$. Prove, that $T$ - centroid of $XYZ$

Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)

A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to $(1,0)$. Then it makes a $90^\circ$ counterclockwise and travels $\frac 12$ a unit to $\left(1, \frac 12 \right)$. If it continues in this fashion, each time making a $90^\circ$ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest? $\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)$

Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

The quadrilateral $ABCD$ is inscribed in circle $\Omega$ with center $O$, not lying on either of the diagonals. Suppose that the circumcircle of triangle $AOC$ passes through the midpoint of the diagonal $BD$. Prove that the circumcircle of triangle $BOD$ passes through the midpoint of diagonal $AC$. [i](A. Zaslavsky)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

In the convex quadrilateral of the $ABCD$, the diagonals of $AC$ and $BD$ are mutually perpendicular and intersect at point $E$. On the side of $AD$, a point $P$ is chosen such that $PE = EC$. The circumscribed circle of the triangle $BCD$ intersects the segment $AD$ at the point $Q$. The circle passing through point $A$ and tangent to the line $EP$ at point $P$ intersects the segment $AC$ at point $R$. It turns out that points $B, Q, R$ are collinear. Prove that $\angle BCD = 90^o$.

$ \int \frac{1+2x^3}{1+x^2-2x^3+x^6} dx $ [i]Ion Nedelcu[/i] and [i]Lucian Tutescu[/i]

a. Let $ABC$ be a triangle with altitude $AD$ and $P$ a variable point on $AD$. Lines $PB$ and $AC$ intersect each other at $E$, lines $PC$ and $AB$ intersect each other at $F.$ Suppose $AEDF$ is a quadrilateral inscribed . Prove that \[\frac{PA}{PD}=(\tan B+\tan C)\cot \frac{A}{2}.\] b. Let $ABC$ be a triangle with orthocentre $H$ and $P$ a variable point on $AH$. The line through $C$ perpendicular to $AC$ meets $BP$ at $M$, The line through $B$ perpendicular to $AB$ meets $CP$ at $N.$ $K$ is the projection of $A$on $MN$. Prove that $\angle BKC+\angle MAN$ is invariant .