For any positive real $r$, let $d(r)$ be the distance of the nearest lattice point from the circle center the origin and radius $r$. Show that $d(r)$ tends to zero as $r$ tends to infinity.

Let $f(n)=\sum_{k=0}^{n-1}x^ky^{n-1-k}$ with, $x$, $y$ real numbers. If $f(n)$, $f(n+1)$, $f(n+2)$, $f(n+3)$, are integers for some $n$, prove $f(n)$ is integer for all $n$.

Given that $a, b, c, d \in \mathbb{R}$, prove that $(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)$.

Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$.

Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$.

Is there a positive integer $n$ such that the fractional part of \[ \left(3+\sqrt{5}\right)^n >0.99 ? \]

Let $\triangle ABC$ be an acute triangle with $AB>AC$, let $I$ be the center of the incircle. Let $M,N$ be the midpoint of $AC$ and $AB$ respectively. $D,E$ are on $AC$ and $AB$ respectively such that $BD\parallel IM$ and $CE\parallel IN$. A line through $I$ parallel to $DE$ intersects $BC$ in $P$. Let $Q$ be the projection of $P$ on line $AI$. Prove that $Q$ is on the circumcircle of $\triangle ABC$.

Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

Find all prime number pairs $(p,q)$ such that $p(p^2-p-1)=q(2q+3).$

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that \[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \] [i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$. Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$. [i]Proposed by Dirk Laurie, South Africa[/i]

Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$. [i](Karl Czakler)[/i]

Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

Find the largest positive integer $N$ satisfying the following properties: [list] [*]$N$ is divisible by $7$; [*]Swapping the $i^{\text{th}}$ and $j^{\text{th}}$ digits of $N$ (for any $i$ and $j$ with $i\neq j$) gives an integer which is $\textit{not}$ divisible by $7$. [/list]

Let $ A\equal{}\{A_1,\dots,A_m\}$ be a family distinct subsets of $ \{1,2,\dots,n\}$ with at most $ \frac n2$ elements. Assume that $ A_i\not\subset A_j$ and $ A_i\cap A_j\neq\emptyset$ for each $ i,j$. Prove that: \[ \sum_{i\equal{}1}^m\frac1{\binom{n\minus{}1}{|A_i|\minus{}1}}\leq1\]

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.