Let $n$ be a positive integer and $S = \{ m \mid 2^n \le m < 2^{n+1} \}$. We call a pair of non-negative integers $(a, b)$ [i]fancy[/i] if $a + b$ is in $S$ and is a palindrome in binary. Find the number of [i]fancy[/i] pairs $(a, b)$.

Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?

Given positive real numbers $x_1, x_2, \ldots , x_n$ find the polygon $A_0A_1\ldots A_n$ with $A_iA_{i+1} = x_{i+1}$ and which has greatest area.

Let $n$ be an odd integer greater than $11$; $k\in \mathbb{N}$, $k \geq 6$, $n=2k-1$. We define \[d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |\] for $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$ and $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$. Show that $n=23$ if $T$ has a subset $S$ satisfying [list=i] [*]$|S|=2^k$ [*]For each $x \in T$, there exists exacly one $y\in S$ such that $d(x,y)\leq 3$[/list]

Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

Let $ M=\{1,2,...,2013\} $ and let $ \Gamma $ be a circle. For every nonempty subset $ B $ of the set $ M $, denote by $ S(B) $ sum of elements of the set $ B $, and define $ S(\varnothing)=0 $ ( $ \varnothing $ is the empty set ). Is it possible to join every subset $ B $ of $ M $ with some point $ A $ on the circle $ \Gamma $ so that following conditions are fulfilled: $ 1 $. Different subsets are joined with different points; $ 2 $. All joined points are vertices of a regular polygon; $ 3 $. If $ A_1,A_2,...,A_k $ are some of the joined points, $ k>2 $ , such that $ A_1A_2...A_k $ is a regular $ k-gon $, then $ 2014 $ divides $ S(B_1)+S(B_2)+...+S(B_k) $ ?

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real $x,y$: $$ f(x^2) + xf(y) = f(x) f(x + f(y)) \, . $$

Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient \[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\] is a rational number.

In isosceles triangle, the condition $AB=AC>BC$ is satisfied. Point $D$ is taken on the circumcircle of $ABC$ such that $\angle CAD=90^{\circ}$.A line parallel to $AC$ which passes from $D$ intersects $AB$ and $BC$ respectively at $E$ and $F$.Show that circumcircle of $ADE$ passes from circumcenter of $DFC$.

Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.

Let $ABCD$ a convex quadrilateral such that $AD$ and $BC$ are not parallel. Let $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The segment $MN$ intersects $AC$ and $BD$ in $K$ and $L$ respectively, Show that at least one point of the intersections of the circumcircles of $AKM$ and $BNL$ is in the line $AB$.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Let $A_1A_2A_3 \cdots A_{2013}$ be a cyclic $2013$-gon. Prove that for every point $P$ not the circumcenter of the $2013$-gon, there exists a point $Q\neq P$ such that $\frac{A_iP}{A_iQ}$ is constant for $i \in \{1, 2, 3, \cdots, 2013\}$. [i]Proposed by Robin Park[/i]

Chord $EF$ is the perpendicular bisector of chord $BC$, intersecting it in $M$. Between $B$ and $M$ point $U$ is taken, and $EU$ extended meets the circle in $A$. Then, for any selection of $U$, as described, triangle $EUM$ is similar to triangle: [asy] pair B = (-0.866, -0.5); pair C = (0.866, -0.5); pair E = (0, -1); pair F = (0, 1); pair M = midpoint(B--C); pair A = (-0.99, -0.141); pair U = intersectionpoints(A--E, B--C)[0]; draw(B--C); draw(F--E--A); draw(unitcircle); label("$B$", B, SW); label("$C$", C, SE); label("$A$", A, W); label("$E$", E, S); label("$U$", U, NE); label("$M$", M, NE); label("$F$", F, N); //Credit to MSTang for the asymptote [/asy] $\textbf{(A)}\ EFA \qquad \textbf{(B)}\ EFC \qquad \textbf{(C)}\ ABM \qquad \textbf{(D)}\ ABU \qquad \textbf{(E)}\ FMC$

The incircle $\odot (I)$ of $\triangle ABC$ touch $AC$ and $AB$ at $E$ and $F$ respectively. Let $H$ be the foot of the altitude from $A$, if $R \equiv IC \cap AH, \ \ Q \equiv BI \cap AH$ prove that the midpoint of $AH$ lies on the radical axis between $\odot (REC)$ and $\odot (QFB)$ I hope that this is not repost :)

Compute \[\sum_{k=1}^{1007}\left(\cos\left(\dfrac{\pi k}{1007}\right)\right)^{2014}.\]

Square $ ABCD$ has side length $ s$, a circle centered at $ E$ has radius $ r$, and $ r$ and $ s$ are both rational. The circle passes through $ D$, and $ D$ lies on $ \overline{BE}$. Point $ F$ lies on the circle, on the same side of $ \overline{BE}$ as $ A$. Segment $ AF$ is tangent to the circle, and $ AF \equal{} \sqrt {9 \plus{} 5\sqrt {2}}$. What is $ r/s$? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=3; pair B=(0,0), C=(3,0), D=(3,3), A=(0,3); pair Ep=(3+5*sqrt(2)/6,3+5*sqrt(2)/6); pair F=intersectionpoints(Circle(A,sqrt(9+5*sqrt(2))),Circle(Ep,5/3))[0]; pair[] dots={A,B,C,D,Ep,F}; draw(A--F); draw(Circle(Ep,5/3)); draw(A--B--C--D--cycle); dot(dots); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,SW); label("$E$",Ep,E); label("$F$",F,NW);[/asy]$ \textbf{(A) } \frac {1}{2}\qquad \textbf{(B) } \frac {5}{9}\qquad \textbf{(C) } \frac {3}{5}\qquad \textbf{(D) } \frac {5}{3}\qquad \textbf{(E) } \frac {9}{5}$

Let X be a dense set homeomorphic to $\mathbb R^n$ in the compact Hausdorff space Y. Prove that for $n\geq 2$ , $Y \setminus X$ is connected, and for n=1 it consists of at most two components.

An airline operates flights between any two capital cities in the European Union. Each flight has a fixed price which is the same in both directions. Furthermore, the flight prices from any given city are pairwise distinct. Anna and Bella wish to visit each city exactly once, not necessarily starting from the same city. While Anna always takes the cheapest flight from her current city to some city she hasn't visited yet, Bella always continues her tour with the most expensive flight available. Is it true that Bella's tour will surely cost at least as much as Anna's tour? [i](Based on a Soviet problem)[/i]

Points $C,E,D$ and $F$ lie on a circle with centre $O$. Two chords $CD$ and $EF$ intersect at a point $N$. The tangents at $C$ and $D$ intersect at $A$, and the tangents at $E$ and $F$ intersect at $B$. Prove that $ON\perp AB$.

Let $f(x)$ be a polynomial of nonzero degree. Can it happen that for any real number $a$, an even number of real numbers satisfy the equation $f(x) = a$?

If $ a > 1$, then the sum of the real solutions of \[\sqrt{a \minus{} \sqrt{a \plus{} x}} \equal{} x\] is equal to $ \textbf{(A)}\ \sqrt{a} \minus{} 1\qquad \textbf{(B)}\ \frac{\sqrt{a} \minus{} 1}{2}\qquad \textbf{(C)}\ \sqrt{a \minus{} 1}\qquad \textbf{(D)}\ \frac{\sqrt{a \minus{} 1}}{2}\qquad \textbf{(E)}\ \frac{\sqrt{4a \minus{} 3} \minus{} 1}{2}$

If the answer to this problem is $x$, then compute the value of $\tfrac{x^2}{8} +2$. [i]Proposed by Lewis Chen [/i]

Find max number $n$ of numbers of three digits such that : 1. Each has digit sum $9$ 2. No one contains digit $0$ 3. Each $2$ have different unit digits 4. Each $2$ have different decimal digits 5. Each $2$ have different hundreds digits

Consider a tetrahedron $SABC$. The incircle of the triangle $ABC$ has the center $I$ and touches its sides $BC$, $CA$, $AB$ at the points $E$, $F$, $D$, respectively. Let $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the points on the segments $SA$, $SB$, $SC$ such that $AA^{\prime}=AD$, $BB^{\prime}=BE$, $CC^{\prime}=CF$, and let $S^{\prime}$ be the point diametrically opposite to the point $S$ on the circumsphere of the tetrahedron $SABC$. Assume that the line $SI$ is an altitude of the tetrahedron $SABC$. Show that $S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}$.