Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap?

Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \in X$.

[b](a)[/b] Sketch the diagram of the function $f$ if \[f(x)=4x(1-|x|) , \quad |x| \leq 1.\] [b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$ [b](c)[/b] Let $g$ be a function such that \[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\] Is the function $g$ continuous in the point $x=0 \ ?$ [b](d)[/b] Sketch the diagram of $g.$

A regular pentagon is inscribed in a circle of radius $r$. $P$ is any point inside the pentagon. Perpendiculars are dropped from $P$ to the sides, or the sides produced, of the pentagon. a) Prove that the sum of the lengths of these perpendiculars is constant. b) Express this constant in terms of the radius $r$.

The graph of $ x^2\minus{}4y^2\equal{}0$: $ \textbf{(A)}\ \text{is a hyperbola intersecting only the }x\text{ \minus{}axis} \\ \textbf{(B)}\ \text{is a hyperbola intersecting only the }y\text{ \minus{}axis} \\ \textbf{(C)}\ \text{is a hyperbola intersecting neither axis} \\ \textbf{(D)}\ \text{is a pair of straight lines} \\ \textbf{(E)}\ \text{does not exist}$

A number of $N$ children are at a party and they sit in a circle to play a game of Pass and Parcel. Because the host has no other form of entertainment, the parcel has infinitely many layers. On turn $i$, starting with $i=1$, the following two things happen in order: [b]$(1)$[/b] The parcel is passed $i^2$ positions clockwise; and [b]$(2)$[/b] The child currently holding the parcel unwraps a layer and claims the prize inside. For what values of $N$ will every chidren receive a prize? $Patrick \ Winter \, United \ Kingdom$

Let $x$ and $y$ be two rational numbers such that $$x^5 + y^5 = 2x^2y^2.$$ Prove that $\sqrt{1-xy}$ is also a rational number.

Let $(a_n)$ be a sequence of non-negative real numbers satisfying $a_{n+m}\le a_n+a_m$ for all non-negative integers $m,n$. Prove that if $n\ge m$ then $a_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m$ holds.

We have a scalene acute triangle $ABC$ (triangle with no two equal sides) and a point $D$ on side $BC$. Pick a point $E$ on side $AB$ and a point $F$ on side $AC$ such that $\angle DEB=\angle DFC$. Lines $DF,\, DE$ intersect $AB,\, AC$ at points $M,\, N$, respectively. Denote $(I_1),\, (I_2)$ by the circumcircles of triangles $DEM,\, DFN$ in that order. The circle $(J_1)$ touches $(I_1)$ internally at $D$ and touches $AB$ at $K$, circle $(J_2)$ touches $(I_2)$ internally at $D$ and touches $AC$ at $H$. $P$ is the intersection of $(I_1),\, (I_2)$ different from $D$. $Q$ is the intersection of $(J_1),\, (J_2)$ different from $D$. a. Prove that all points $D,\, P,\, Q$ lie on the same line. b. The circumcircles of triangles $AEF,\, AHK$ intersect at $A,\, G$. $(AEF)$ also cut $AQ$ at $A,\, L$. Prove that the tangent at $D$ of $(DQG)$ cuts $EF$ at a point on $(DLG)$.

For values of $ x$ less than $ 1$ but greater than $ \minus{}4$, the expression \[ \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2} \] has: $ \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ \textbf{(B)}\ \text{a minimum value of }{\plus{}1}\qquad \\ \textbf{(C)}\ \text{a maximum value of }{\plus{}1}\qquad \\ \textbf{(D)}\ \text{a minimum value of }{\minus{}1}\qquad \\ \textbf{(E)}\ \text{a maximum value of }{\minus{}1}$

The squares of an infinite chessboard are numbered $1,2,\ldots $ along a spiral, as shown in the picture. A [i]rightline[/i] is the sequence of the numbers in the squares obtained by starting at one square at going to the right. a) Prove that exists a rightline without multiples of $3$. b) Prove that there are infinitely many pairwise disjoint rightlines not containing multiples of $3$.

Let $ABCD$ be a rectangle with $AB = a$ and $BC = b$. Suppose $r_1$ is the radius of the circle passing through $A$ and $B$ touching $CD$; and similarly $r_2$ is the radius of the circle passing through $B$ and $C$ and touching $AD$. Show that \[ r_1 + r_2 \geq \frac{5}{8} ( a + b) . \]

i) A transformation of the plane into itself preserves all rational distances. Prove that it preserves all distances. ii) Show that the corresponding statement for the line is false.

A convex 51-gon is given. For each of its vertices and each diagonal that does not contain this vertex, we mark in red a point symmetrical to the vertex relative to the middle of the diagonal. Prove that strictly inside the polygon there are no more than 20400 red dots. [i]Proposed by P. Kozhevnikov[/i]

There are $ 63$ points arbitrarily on the circle $ \mathcal{C}$ with its diameter being $ 20$. Let $ S$ denote the number of triangles whose vertices are three of the $ 63$ points and the length of its sides is no less than $ 9$. Fine the maximum of $ S$.

Let $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be distinct circles such that $\Gamma_1,\Gamma_3$ are externally tangent at $P$, and $\Gamma_2,\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A,B,C,D,$ respectively, and that all of these points are different from $P$. Prove that $$\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}$$

Can four equal polygons be placed on the plane in such a way that any two of them don't have common interior points, but have a common boundary segment? (S.Markelov)

$\boxed{N5}$Let $a,b,c,p,q,r$ be positive integers such that $a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q.$ Prove that $a=b=c$ or $p=q=r.$

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)