Consider (in the plane) three concentric circles with radii $1, 2$ and $3$ and equilateral triangle $\Delta$ such that on each of the three circles is one vertex of $\Delta$ . Calculate the length of the side of $\Delta$ . [img]https://1.bp.blogspot.com/-q40dl3TSQqE/Xy1QAcno_9I/AAAAAAAAMR8/11nsSA0syNAaGb3W7weTHsNpBeGQZXnHACLcBGAsYHQ/s0/flanders%2B2013%2Bp4.png[/img]

A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent modulo p to the product of the numbers in the second set.

Let $1=d_1<d_2<\ldots<d_k=n$ be all divisors of $n$. It turned out that numbers $d_2-d_1,\ldots,d_k-d_{k-1}$ are $1,3,\ldots,2k-3$ in some order. Find all possible values of $n$ [i]M. Zorka[/i]

Let $G$ be the complete graph on $2015$ vertices. Each edge of $G$ is dyed red, blue or white. For a subset $V$ of vertices of $G$, and a pair of vertices $(u,v)$, define \[ L(u,v) = \{ u,v \} \cup \{ w | w \in V \ni \triangle{uvw} \text{ has exactly 2 red sides} \}\]Prove that, for any choice of $V$, there exist at least $120$ distinct values of $L(u,v)$.

Let $A_0BC_0D$ be a convex quadrilateral inscribed in a circle $\omega$. For all integers $i\ge0$, let $P_i$ be the intersection of lines $A_iB$ and $C_iD$, let $Q_i$ be the intersection of lines $A_iD$ and $BC_i$, let $M_i$ be the midpoint of segment $P_iQ_i$, and let lines $M_iA_i$ and $M_iC_i$ intersect $\omega$ again at $A_{i+1}$ and $C_{i+1}$, respectively. The circumcircles of $\triangle A_3M_3C_3$ and $\triangle A_4M_4C_4$ intersect at two points $U$ and $V$. If $A_0B=3$, $BC_0=4$, $C_0D=6$, $DA_0=7$, then $UV$ can be expressed in the form $\tfrac{a\sqrt b}c$ for positive integers $a$, $b$, $c$ such that $\gcd(a,c)=1$ and $b$ is squarefree. Compute $100a+10b+c $. [i]Proposed by Eric Shen[/i]

Consider the functions $f,g:\mathbb{R}\to\mathbb{R}$ such that $f{}$ is continous. For any real numbers $a<b<c$ there exists a sequence $(x_n)_{n\geqslant 1}$ which converges to $b{}$ and for which the limit of $g(x_n)$ as $n{}$ tends to infinity exists and satisfies \[f(a)<\lim_{n\to\infty}g(x_n)<f(c).\][list=a] [*]Give an example of a pair of such functions $f,g$ for which $g{}$ is discontinous at every point. [*]Prove that if $g{}$ is monotonous, then $f=g.$ [/list]

6. Circles $W_{1}$ and $W_{2}$ with equal radii are given. Let $P$,$Q$ be the intersection of the circles. points $B$ and $C$ are on $W_{1}$ and $W_{2}$ such that they are inside $W_{2}$ and $W_{1}$ respectively. Points $X$,$Y$ $\neq$ $P$ are on $W_{1}$ and $W_{2}$ respectively, such that $\angle{BPQ}=\angle{BYQ}$ and $\angle{CPQ}=\angle{CXQ}$.Denote by $S$ as the other intersection of $(YPB)$ and $(XPC)$. Prove that $QS,BC,XY$ are concurrent.

Colin has a peculiar $12$-sided dice: it is made up of two regular hexagonal pyramids. Colin wants to paint each face one of three colors so that no two adjacent faces on the same pyramid have the same color. How many ways can he do this? Two paintings are considered identical if there is a way to rotate or flip the dice to go from one to the other. Faces are adjacent if they share an edge. [center][img]https://cdn.artofproblemsolving.com/attachments/b/2/074e9a4bc404d45546661a5ae269248d20ed5a.png[/img][/center]

The circumcentre of a triangle $ABC$ is called $O$. The points $A',B'$ and $C'$ are the reflections of $O$ in $BC, CA$, and $AB$, respectively. Show that the three lines $AA' , BB'$, and $CC'$ meet in a common point.

The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2=40$, first quartile $Q_1=33$, and third quartile $Q_3=43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartile ($Q_1$) or more than $1.5$ times the interquartile range above the third quartile ($Q_3$), where the interquartile range is defined as $Q_3-Q_1$. How many outliers does this data set have? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Let $k$ be incircle of acute triangle $ABC$, $AC\neq BC$, and $l$ be excircle that touches $AB$. Line $p$ through the $C$ is orthogonal to $AB$, $p\cap k = \{X, Y\}$ , $p\cap l = \{Z, T\}$ and the point arrangement is $X-Y-Z-T$. Circle $m$ through $X$ and $Z$ intersects $AB$ at $D$ and $E$. Prove that points $D,Y,E,T$ are concyclic.

Find all $a$ real number such that $x_n=n\{an! \}$ is convergeant Gabriel Dospinescu

Point $E$ is marked on side $BC$ of parallelogram $ABCD$, and on the side $AD$ - point $F$ so that the circumscribed circle of $ABE$ is tangent to line segment $CF$. Prove that the circumcircle of triangle $CDF$ is tangent to line $AE$.

The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.

Calculate the following indefinite integrals. [1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$ [2] $\int x\log_{10} x dx$ [3] $\int \frac{x}{\sqrt{2x-1}}dx$ [4] $\int (x^2+1)\ln x dx$ [5] $\int e^x\cos x dx$

At Euler Middle School, $198$ students voted on two issues in a school referendum with the following results: $149$ voted in favor of the first issue and $119$ voted in favor of the second issue. If there were exactly $29$ students who voted against both issues, how many students voted in favor of both issues? $\textbf{(A) }49\qquad\textbf{(B) }70\qquad\textbf{(C) }79\qquad\textbf{(D) }99\qquad \textbf{(E) }149$

Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$, find the smallest possible value for $a$.

Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0, y_1+y_2+y_3=0$. Prove that \[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\] [i]Ray Li, Max Schindler.[/i]

How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a>b>c>d>e>f$ and $a+f=b+e=c+d=30$ are simultaneously true?

$$\int\frac{\ln 2}{1+2^{-x}}\,dx$$ [i]Proposed by Connor Gordon[/i]

Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times? $\textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{1}{7} \qquad \textbf{(C)}\ \frac{1}{6} \qquad \textbf{(D)}\ \frac{1}{4} \qquad \textbf{(E)}\ \frac{1}{3}$

It is given that $r=\left(3\left(\sqrt6-1\right)-4\left(\sqrt3+1\right)+5\sqrt2\right)R$ where $r$ and $R$ are the radii of the inscribed and circumscribed spheres in a regular $n$-angled pyramid. If it is known that the centers of the spheres given coincide, (a) find $n$; (b) if $n=3$ and the lengths of all edges are equal to a find the volumes of the parts from the pyramid after drawing a plane $\mu$, which intersects two of the edges passing through point $A$ respectively in the points $E$ and $F$ in such a way that $|AE|=p$ and $|AF|=q$ $(p<a,q<a)$, intersects the extension of the third edge behind opposite of the vertex $A$ wall in the point $G$ in such a way that $|AG|=t$ $(t>a)$.

Each of the two opposite sides of a convex quadrilateral is divided into seven equal parts, and corresponding division points are connected by a segment, thus dividing the quadrilateral into seven smaller quadrilaterals. Prove that the area of at least one of the small quadrilaterals equals $1\slash 7$ slash of the area of the large quadrilateral.

Consider the set of continuous functions $f$, whose $n^{\text{th}}$ derivative exists for all positive integer $n$, satisfying $f(x)=\tfrac{\text{d}^3}{\text{dx}^3}f(x)$, $f(0)+f'(0)+f''(0)=0$, and $f(0)=f'(0)$. For each such function $f$, let $m(f)$ be the smallest nonnegative $x$ satisfying $f(x)=0$. Compute all possible values of $m(f)$.

Billy let his herd freely. Enjoying their time the horses started to jump on the squares of a lattice of meadow that is infinite in both directions. Each horse can jump as follows: horizontally or vertically moves three, then turn to left and moves two. Naturally, under the jump a horse don’t touch the ground. The horses are standing on squares that no two can meet by such a jump. How many horses does Billy have if their number is the maximum possible? [i]The figure below shows where a horse can jump to. Notice that there 4 places and not 8 like in chess.[/i] [img]https://cdn.artofproblemsolving.com/attachments/c/6/8b6f9ca4e0aad46a13e133d87bcd4dd4384e7a.png[/img]