Let $\mathcal{K}$ be a circle centered in $A$. Let $p$ be a line tangent to $\mathcal{K}$ in $B$ and let a line parallel to $p$ intersect $\mathcal{K}$ in $C$ and $D$. Let the line $AD$ intersect $p$ in $E$ and let $F$ be the intersection of the lines $CE$ and $AB$. Prove that the line through $D$, parallel to the tangent through $A$ to the circumcircle of $AFD$ intersects the line $CF$ on $\mathcal{K}$.

In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal. [i]H. Lesov[/i]

Suppose 2017 points in a plane are given such that no three points are collinear. Among the triangles formed by any three of these 2017 points, those triangles having the largest area are said to be [i]good[/i]. Prove that there cannot be more than 2017 good triangles.

Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.

Let $ M,M'$ be two conjugates point in triangle $ ABC$ (in the sense that $ \angle MAB\equal{}\angle M'AC,\dots$). Let $ P,Q,R,P',Q',R'$ be foots of perpendiculars from $ M$ and $ M'$ to $ BC,CA,AB$. Let $ E\equal{}QR\cap Q'R'$, $ F\equal{}RP\cap R'P'$ and $ G\equal{}PQ\cap P'Q'$. Prove that the lines $ AG, BF, CE$ are parallel.

$ABCDE$ is a cyclic pentagon, with circumcentre $O$. $AB=AE=CD$. $I$ midpoint of $BC$. $J$ midpoint of $DE$. $F$ is the orthocentre of $\triangle ABE$, and $G$ the centroid of $\triangle AIJ$.$CE$ intersects $BD$ at $H$, $OG$ intersects $FH$ at $M$. Show that $AM\perp CD$.

Given are a circle $\mathcal{C}$ with center $K$ and radius $r,$ point $A$ on the circle and point $R$ in its exterior. Consider a variable line $e$ through $R$ that intersects the circle at two points $B$ and $C.$ Let $H$ be the orthocenter of triangle $ABC.$ Show that there is a unique point $T$ in the plane of circle $\mathcal{C}$ such that the sum $HA^2 + HT^2$ remains constant (as $e$ varies.)

We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

Shreyas has a rectangular piece of paper $ABCD$ such that $AB = 20$ and $AD = 21$. Given that Shreyas can make exactly one straight-line cut to split the paper into two pieces, compute the maximum total perimeter of the two pieces

[u]Round 1[/u] [b]p1.[/b] Find all pairs $(a,b)$ of positive integers with $a > b$ and $a^2 -b^2 =111$. [b]p2.[/b] Alice drives at a constant rate of $2017$ miles per hour. Find all positive values of $x$ such that she can drive a distance of $x^2$ miles in a time of $x$ minutes. [b]p3.[/b] $ABC$ is a right triangle with right angle at $B$ and altitude $BH$ to hypotenuse $AC$. If $AB = 20$ and $BH = 12$, find the area of triangle $\vartriangle ABC$. [u]Round 2[/u] [b]p4.[/b] Regular polygons $P_1$ and $P_2$ have $n_1$ and $n_2$ sides and interior angles $x_1$ and $x_2$, respectively. If $\frac{n_1}{n_2}= \frac75$ and $\frac{x_1}{x_2}=\frac{15}{14}$ , find the ratio of the sum of the interior angles of $P_1$ to the sum of the interior angles of $P_2$. [b]p5.[/b] Joey starts out with a polynomial $f (x) = x^2 +x +1$. Every turn, he either adds or subtracts $1$ from $f$ . What is the probability that after $2017$ turns, $f$ has a real root? [b]p6.[/b] Find the difference between the greatest and least positive integer values $x$ such that $\sqrt[20]{\lfloor \sqrt[17]{x}\rfloor}=1$. [u]Round 3[/u] [b]p7.[/b] Let $ABCD$ be a square and suppose $P$ and $Q$ are points on sides $AB$ and $CD$ respectively such that $\frac{AP}{PB} = \frac{20}{17}$ and $\frac{CQ}{QD}=\frac{17}{20}$ . Suppose that $PQ = 1$. Find the area of square $ABCD$. [b]p8.[/b] If $$\frac{\sum_{n \ge 0} r^n}{\sum_{n \ge 0} r^{2n}}=\frac{1+r +r^2 +r^3 +...}{1+r^2 +r^4 +r^6 +...}=\frac{20}{17},$$ find $r$ . [b]p9.[/b] Let $\overline{abc}$ denote the $3$ digit number with digits $a,b$ and $c$. If $\overline{abc}_{10}$ is divisible by $9$, what is the probability that $\overline{abc}_{40}$ is divisible by $9$? [u]Round 4[/u] [b]p10.[/b] Find the number of factors of $20^{17}$ that are perfect cubes but not perfect squares. [b]p11.[/b] Find the sum of all positive integers $x \le 100$ such that $x^2$ leaves the same remainder as $x$ does upon division by $100$. [b]p12.[/b] Find all $b$ for which the base-$b$ representation of $217$ contains only ones and zeros. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $D$ be a point inside $\triangle ABC$ such that $\angle CDA + \angle CBA = 180^{\circ}.$ The line $CD$ meets the circle $\odot ABC$ at the point $E$ for the second time. Let $G$ be the common point of the circle centered at $C$ with radius $CD$ and the arc $\overset{\LARGE \frown}{AC}$ of $\odot ABC$ which does not contain the point $B$. The circle centered at $A$ with radius $AD$ meets $\odot BCD$ for the second time at $F$. Prove that the lines $GE, FD, CB$ are concurrent or parallel.

Let $R$ be a convex region symmetrical about the origin with area greater than $4$. Show that $R$ must contain a lattice point different from the origin.

Given a segment $AB$ of length $2003$ in a coordinate plane, determine the maximal number of unit squares with vertices in the lattice points whose intersection with the given segment is non-empty.

Let there be a given parabola $y^2=4ax$ in the coordinate plane. Consider all chords of the parabola that are visible at a right angle from the origin of the coordinate system. Prove that all these chords pass through a fixed point.

There is a stone at each vertex of a given regular $13$-gon, and the color of each stone is black or white. Prove that we may exchange the position of two stones such that the coloring of these stones are symmetric with respect to some symmetric axis of the $13$-gon.

Given triangle $\triangle ABC$ and let $D,E,F$ be the foot of angle bisectors of $A,B,C$ ,respectively. $M,N$ lie on $EF$ such that $AM=AN$. Let $H$ be the foot of $A$-altitude on $BC$. Points $K,L$ lie on $EF$ such that triangles $\triangle AKL, \triangle HMN$ are correspondingly similiar (with the given order of vertices) such that $AK \not\parallel HM$ and $AK \not\parallel HN$. Show that: $DK=DL$

Acute triangle $ABC$ is inscribed in circle $\omega$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Rays $MH$ and $NH$ meet $\omega$ at $P$ and $Q$, respectively. Lines $MN$ and $PQ$ meet at $R$. Prove that $OA\perp RA$.

Given a convex polygon with $n$ sides and perimeter $S$, which has an incircle $\omega$ with radius $R$. A regular polygon with $n$ sides, whose vertices lie on $\omega$, has a perimeter $s$. Determine whether the following inequality holds: \[ S \ge \frac{2sRn}{\sqrt{4n^2R^2-s^2}}. \]

Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior? $ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $

Let $D$ be an interior point of the side $BC$ of a triangle $ABC$. Let $I_1$ and $I_2$ be the incentres of triangles $ABD$ and $ACD$ respectively. Let $AI_1$ and $AI_2$ meet $BC$ in $E$ and $F$ respectively. If $\angle BI_1E = 60^o$, what is the measure of $\angle CI_2F$ in degrees?

Given a quadrilateral $ABCD$ with $AB = BD = DC$ and $AC = BC$. On $BC$ lies point $E$ such that $AE = AB$. Prove that $ED = EB$.

Let triangle $\vartriangle ABC$ have circumcenter $O$ and circumradius $r$, and let $\omega$ be the circumcircle of ntriangle $\vartriangle BOC$. Let $F$ be the intersection of $\overleftrightarrow{AO}$ and $\omega$ not equal to $O$. Let $E$ be on line $\overleftrightarrow{AB}$ such that $\overline{EF} \perp \overline{AE}$, and let $G$ be on line $\overleftrightarrow{AC}$ such that $\overline{GF} \perp \overline{AG}$. If $AC =\frac{65}{63}$ , $BC = \frac{24}{13}r$, and $AB = \frac{126}{65}r$, compute $AF \cdot EG$.

In a triangle $ABC$ angle $\angle BAC = 60^{\circ}$. Point $M$ is the midpoint of $BC$, and $D$ is the foot of altitude from point $A$. Points $T$ and $P$ are marked such that $TBC$ is equilateral, and $\angle BPD=\angle DPC = 30^{\circ}$ and this points lie in the same half-plane with respect to $BC$, not in the same as $A$. Prove that the circumcircles of $ADP$ and $AMT$ are tangent. [i]Ivan Korshunau[/i]

Let $A_1C_2B_1B_2C_1A_2$ be a cyclic convex hexagon inscribed in circle $\Omega$, centered at $O$. Let $\{ P \} = A_2B_2 \cap A_1B_1$ and $\{ Q \} = A_2C_2 \cap A_1C_1$. Let $\Gamma_1$ be a circle tangent to $OB_1$ and $OC_1$ at $B_1,C_1$ respectively. Similarly, define $\Gamma_2$ to be the circle tangent to $OB_2,OC_2$ at $B_2, C_2$ respectively. Prove that there is a homothety that sends $\Gamma_1$ to $\Gamma_2$, whose center lies on $PQ$

Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which: (i) $f(x)\ge 0$ for all real $x,$ and (ii) $a_n=0$ whenever $n$ is a multiple of $3.$ Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained.