Prove the inequality $$\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R$$ in triangle $ABC$ where $a$, $b$ and $c$ are sides of triangle and $R$ radius of circumcircle of $ABC$

Let $a, b$ be two lines intersecting each other at $O$. Point $M$ is not on either $a$ or $b$. A variable circle $(C)$ passes through $O,M$ intersecting $a, b$ at $A,B$ respectively, distinct from $O$. Prove that the midpoint of $AB$ is on a fixed line. Hạ Vũ Anh

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.

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 $DFC$.

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 $A_1A_2A_3 \cdots A_{2013}$ 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] $\textbf{(A)}\ EFA \qquad \textbf{(B)}\ EFC \qquad \textbf{(C)}\ ABM \qquad \textbf{(D)}\ ABU \qquad \textbf{(E)}\ FMC$

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}$

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.

$ABCD$ is a scalene tetrahedron and let $G$ be its baricentre. A plane $\alpha$ passes through $G$ such that it intersects neither the interior of $\Delta BCD$ nor its perimeter. Prove that $$\textnormal{dist}(A,\alpha)=\textnormal{dist}(B,\alpha)+\textnormal{dist}(C,\alpha)+\textnormal{dist}(D,\alpha).$$ [i] (Adapted from folklore)[/i]

Consider a charged capacitor made with two square plates of side length $L$, uniformly charged, and separated by a very small distance $d$. The EMF across the capacitor is $\xi$. One of the plates is now rotated by a very small angle $\theta$ to the original axis of the capacitor. Find an expression for the difference in charge between the two plates of the capacitor, in terms of (if necessary) $d$, $\theta$, $\xi$, and $L$. Also, approximate your expression by transforming it to algebraic form: i.e. without any non-algebraic functions. For example, logarithms and trigonometric functions are considered non-algebraic. Assume $ d << L $ and $ \theta \approx 0 $. $\emph{Hint}$: You may assume that $ \frac {\theta L}{d} $ is also very small. [i]Problem proposed by Trung Phan[/i] [hide="Clarification"] There are two possible ways to rotate the capacitor. Both were equally scored but this is what was meant: [asy]size(6cm); real h = 7; real w = 2; draw((-w,0)--(-w,h)); draw((0,0)--(0,h), dashed); draw((0,0)--h*dir(64)); draw(arc((0,0),2,64,90)); label("$\theta$", 2*dir(77), dir(77)); [/asy] [/hide]

Let $ABCD$ be a quadrilateral, $E$ the midpoint of side $BC$, and $F$ the midpoint of side $AD$. Segment $AC$ intersects segment $BF$ at $M$ and segment $DE$ at $N$. If quadrilateral $MENF$ is also known to be a parallelogram, prove that $ABCD$ is also a parallelogram.

On the trapezoid $ABCD$ , side $DA$ is perpendicular to the bases $AB$ and $CD$ . The base $AB$ measures $45$, the base $CD$ measures $20$ and the $BC$ side measures $65$. Let $P$ on the $BC$ side such that $BP$ measures $45$ and $M$ is the midpoint of $DA$. Calculate the measure of the $PM$ segment.

$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$. A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$. Find the area of the square.

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?

The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

Let $ABC$ be a triangle and $D$ is a point inside of the triangle, such that $\angle DBC=60^{\circ}$ and $\angle DCB=\angle DAB=30^{\circ}$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. Prove that $\angle DMN=90^{\circ}$.

Let $ABC$ be a scalene acute triangle with incentre $I{}$ and circumcentre $O{}$. Let $AI$ cross $BC$ at $D$. On circle $ABC$, let $X$ and $Y$ be the mid-arc points of $ABC$ and $BCA$, respectively. Let $DX{}$ cross $CI{}$ at $E$ and let $DY{}$ cross $BI{}$ at $F{}$. Prove that the lines $FX, EY$ and $IO$ are concurrent on the external bisector of $\angle BAC$. [i]David-Andrei Anghel[/i]

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.

For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)

Prove that in any convex polygon with $4n+2$ sides ($n\geq 1$) there exist two consecutive sides which form a triangle of area at most $\frac 1{6n}$ of the area of the polygon.

The three points $A, B, C$ form a triangle. $AB=4, BC=5, AC=6$. Let the angle bisector of $\angle A$ intersect side $BC$ at $D$. Let the foot of the perpendicular from $B$ to the angle bisector of $\angle A$ be $E$. Let the line through $E$ parallel to $AC$ meet $BC$ at $F$. Compute $DF$.

[b]p1.[/b] A dog on a $10$ meter long leash is tied to a $10$ meter long, infinitely thin section of fence. What is the minimum area over which the dog will be able to roam freely on the leash, given that we can fix the position of the leash anywhere on the fence? [b]p2.[/b] Suppose that the equation $$\begin{tabular}{cccccc} &\underline{C} &\underline{H} &\underline{M}& \underline{M}& \underline{C}\\ +& &\underline{H}& \underline{M}& \underline{M} & \underline{T}\\ \hline &\underline{P} &\underline{U} &\underline{M} &\underline{A} &\underline{C}\\ \end{tabular}$$ holds true, where each letter represents a single nonnegative digit, and distinct letters represent different digits (so that $\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ and $ \underline{P}\, \underline{U}\, \underline{M}\, \underline{A}\, \underline{C}$ are both five digit positive integers, and the number $\underline{H }\, \underline{M}\, \underline{M}\, \underline{T}$ is a four digit positive integer). What is the largest possible value of the five digit positive integer$\underline{C}\, \underline{H}\, \underline{ M}\, \underline{ M}\, \underline{ C}$ ? [b]p3.[/b] Square $ABCD$ has side length $4$, and $E$ is a point on segment $BC$ such that $CE = 1$. Let $C_1$ be the circle tangent to segments $AB$, $BE$, and $EA$, and $C_2$ be the circle tangent to segments $CD$, $DA$, and $AE$. What is the sum of the radii of circles $C_1$ and $C_2$? [b]p4.[/b] A finite set $S$ of points in the plane is called tri-separable if for every subset $A \subseteq S$ of the points in the given set, we can find a triangle $T$ such that (i) every point of $A$ is inside $T$ , and (ii) every point of $S$ that is not in $A$ is outside$ T$ . What is the smallest positive integer $n$ such that no set of $n$ distinct points is tri-separable? [b]p5.[/b] The unit $100$-dimensional hypercube $H$ is the set of points $(x_1, x_2,..., x_{100})$ in $R^{100}$ such that $x_i \in \{0, 1\}$ for $i = 1$, $2$, $...$, $100$. We say that the center of $H$ is the point $$\left( \frac12,\frac12, ..., \frac12 \right)$$ in $R^{100}$, all of whose coordinates are equal to $1/2$. For any point $P \in R^{100}$ and positive real number $r$, the hypersphere centered at $P$ with radius $r$ is defined to be the set of all points in $R^{100}$ that are a distance $r$ away from $P$. Suppose we place hyperspheres of radius $1/2$ at each of the vertices of the $100$-dimensional unit hypercube $H$. What is the smallest real number $R$, such that a hypersphere of radius $R$ placed at the center of $H$ will intersect the hyperspheres at the corners of $H$? [b]p6.[/b] Greg has a $9\times 9$ grid of unit squares. In each square of the grid, he writes down a single nonzero digit. Let $N$ be the number of ways Greg can write down these digits, so that each of the nine nine-digit numbers formed by the rows of the grid (reading the digits in a row left to right) and each of the nine nine-digit numbers formed by the columns (reading the digits in a column top to bottom) are multiples of $3$. What is the number of positive integer divisors of $N$? [b]p7.[/b] Find the largest positive integer $n$ for which there exists positive integers $x$, $y$, and $z$ satisfying $$n \cdot gcd(x, y, z) = gcd(x + 2y, y + 2z, z + 2x).$$ [b]p8.[/b] Suppose $ABCDEFGH$ is a cube of side length $1$, one of whose faces is the unit square $ABCD$. Point $X$ is the center of square $ABCD$, and $P$ and $Q$ are two other points allowed to range on the surface of cube $ABCDEFHG$. Find the largest possible volume of tetrahedron $AXPQ$. [b]p9.[/b] Deep writes down the numbers $1, 2, 3, ... , 8$ on a blackboard. Each minute after writing down the numbers, he uniformly at random picks some number $m$ written on the blackboard, erases that number from the blackboard, and increases the values of all the other numbers on the blackboard by $m$. After seven minutes, Deep is left with only one number on the black board. What is the expected value of the number Deep ends up with after seven minutes? [b]p10.[/b] Find the number of ordered tuples $(x_1, x_2, x_3, x_4, x_5)$ of positive integers such that $x_k \le 6$ for each index $k = 1$, $2$, $... $,$ 5$, and the sum $$x_1 + x_2 +... + x_5$$ is $1$ more than an integer multiple of $7$. [b]p11.[/b] The equation $$\left( x- \sqrt[3]{13}\right)\left( x- \sqrt[3]{53}\right)\left( x- \sqrt[3]{103}\right)=\frac13$$ has three distinct real solutions $r$, $s$, and $t$ for $x$. Calculate the value of $$r^3 + s^3 + t^3.$$ [b]p12.[/b] Suppose $a$, $b$, and $c$ are real numbers such that $$\frac{ac}{a + b}+\frac{ba}{b + c}+\frac{cb}{c + a}= -9$$ and $$\frac{bc}{a + b}+\frac{ca}{b+c}+\frac{ab}{c + a}= 10.$$ Compute the value of $$\frac{b}{a + b}+\frac{c}{b + c}+\frac{a}{c + a}.$$ [b]p13.[/b] The complex numbers $w$ and $z$ satisfy the equations $|w| = 5$, $|z| = 13$, and $$52w - 20z = 3(4 + 7i).$$ Find the value of the product $wz$. [b]p14.[/b] For $i = 1, 2, 3, 4$, we choose a real number $x_i$ uniformly at random from the closed interval $[0, i]$. What is the probability that $x_1 < x_2 < x_3 < x_4$ ? [b]p15.[/b] The terms of the infinite sequence of rational numbers $a_0$, $a_1$, $a_2$, $...$ satisfy the equation $$a_{n+1} + a_{n-2} = a_na_{n-1}$$ for all integers $n\ge 2$. Moreover, the values of the initial terms of the sequence are $a_0 =\frac52$, $a_1 = 2$ and} $a_2 =\frac52.$ Call a nonnegative integer $m$ lucky if when we write $a_m =\frac{p}{q}$ for some relatively prime positive integers $p$ and $q$, the integer $p + q$ is divisible by $13$. What is the $101^{st}$ smallest lucky number? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A convex quadrilateral $ABCD$ is inscribed in a circle. Show that the line connecting the midpoints of the arcs $AB$ and $CD$ and the line connecting the midpoints of the arcs $BC$ and $DA$ are perpendicular.