An isosceles triangle with $AB=AC$ has an inscribed circle $O$, which touches its sides $BC,CA,AB$ at $K,L,M$ respectively. The lines $OL$ and $KM$ intersect at $N$; the lines $BN$ and $CA$ intersect at $Q$. Let $P$ be the foot of the perpendicular from $A$ on $BQ$. Suppose that $BP=AP+2\cdot PQ$. Then, what values can the ratio $\frac{AB}{BC}$ assume?

A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.

Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the intersection point of the lines $AB$ and $CD$ and $N$ is the intersection point of the lines $BC$ and $AD$. It is known that $BM = DN$. Prove that $CM = CN$. (F Nazarov)

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

Let $ABC$ and $XYZ$ be two triangles. Define \[A_1=BC\cap ZX, A_2=BC\cap XY,\]\[B_1=CA\cap XY, B_2=CA\cap YZ,\]\[C_1=AB\cap YZ, C_2=AB\cap ZX.\] Hereby, the abbreviation $g\cap h$ means the point of intersection of two lines $g$ and $h$. Prove that $\frac{C_1C_2}{AB}=\frac{A_1A_2}{BC}=\frac{B_1B_2}{CA}$ holds if and only if $\frac{A_1C_2}{XZ}=\frac{C_1B_2}{ZY}=\frac{B_1A_2}{YX}$.

Let $ABC$ be an acute triangle and $D$ be a point of side $ BC$. Consider points $E,Z$ on line $AD$ such that $EB \perp AB$ and $ZC \perp AC$, and points $H, T $ on line $BC$ such that $EH \parallel AC$ and $ZT \parallel AB$. Circumcircle of triangle $BHE$ intersects for second time line $AB$ at point $M$ ($M \ne B$) and circumcircle of triangle $CTZ$ intersects for second time line $AC$ at point $N$ ($N \ne C$). Prove that lines $MH$, $NT$ and $AD$ concur.

The vertices of hexagon $ABCDEF$ lie on a circle. Sides $AB = CD = EF = 6$, and sides $BC = DE = F A = 10$. The area of the hexagon is $m\sqrt3$. Find $m$.

A particle $P$ moves in the plane in such a way that the angle between the two tangents drawn from $P$ to the curve $y^2=4ax$ is always $90^\circ$. The locus of $P$ is $\textbf{(A)}~\text{a parabola}$ $\textbf{(B)}~\text{a circle}$ $\textbf{(C)}~\text{an ellipse}$ $\textbf{(D)}~\text{a straight line}$

The area of a circle inscribed in an equilateral triangle is $ 48\pi$. The perimeter of this triangle is: $ \textbf{(A)}\ 72\sqrt{3} \qquad \textbf{(B)}\ 48\sqrt{3}\qquad \textbf{(C)}\ 36\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 72$

In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle? $ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $

An equilateral triangle $ABC$ is inscribed in a square with side 1 (each vertex of the triangle is on a side of the square and no two are on the same side). Determine the greatest and smallest value of the side of $\Delta ABC$.

Let $ABC$ be a triangle with $AB < AC,$ $I$ its incenter, and $M$ the midpoint of the side $BC$. If $IA=IM,$ determine the smallest possible value of the angle $AIM$.

Two circles $k_1$ and $k_2$ intersect at two points $A$ and $B$. Some line through the point $B$ meets the circle $k_1$ at a point $C$ (apart from $B$), and the circle $k_2$ at a point $E$ (apart from $B$). Another line through the point $B$ meets the circle $k_1$ at a point $D$ (apart from $B$), and the circle $k_2$ at a point $F$ (apart from $B$). Assume that the point $B$ lies between the points $C$ and $E$ and between the points $D$ and $F$. Finally, let $M$ and $N$ be the midpoints of the segments $CE$ and $DF$. Prove that the triangles $ACD$, $AEF$ and $AMN$ are similar to each other.

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the set of the first $2017$ positive integers. Find the number of elements $n \in S$ such that $\sum^n_{i=1} \left\lfloor \frac{n}{i} \right\rfloor$ is even. [b]p26.[/b] Let $\{x_n\}_{n \ge 0}$ be a sequence with $x_0 = 0$,$x_1 = \frac{1}{20}$ ,$x_2 = \frac{1}{17}$ ,$x_3 = \frac{1}{10}$ , and $x_n = \frac12 ((x_{n-2} +x_{n-4})$ for $n\ge 4$. Compute $$ \left\lfloor \frac{1}{x_{2017!} -x_{2017!-1}} \right\rfloor.$$ [b]p27.[/b] Let $ABCDE$ be be a cyclic pentagon. Given that $\angle CEB = 17^o$, find $\angle CDE + \angle EAB$, in degrees. [u]Round 10[/u] [b]p28.[/b] Let $S = \{1,2,4, ... ,2^{2016},2^{2017}\}$. For each $0 \le i \le 2017$, let $x_i$ be chosen uniformly at random from the subset of $S$ consisting of the divisors of $2^i$ . What is the expected number of distinct values in the set $\{x_0,x_1,x_2,... ,x_{2016},x_{2017}\}$? [b]p29.[/b] For positive real numbers $a$ and $b$, the points $(a, 0)$, $(20,17)$ and $(0,b)$ are collinear. Find the minimum possible value of $a+b$. [b]p30.[/b] Find the sum of the distinct prime factors of $2^{36}-1$. [u]Round 11[/u] [b]p31.[/b] There exist two angle bisectors of the lines $y = 20x$ and $y = 17x$ with slopes $m_1$ and $m_2$. Find the unordered pair $(m_1,m_2)$. [b]p32.[/b] Triangle 4ABC has sidelengths $AB = 13$, $BC = 14$, $C A =15$ and orthocenter $H$. Let $\Omega_1$ be the circle through $B$ and $H$, tangent to $BC$, and let $\Omega_2$ be the circle through $C$ and $H$, tangent to $BC$. Finally, let $R \ne H$ denote the second intersection of $\Omega_1$ and $\Omega_2$. Find the length $AR$. [b]p33.[/b] For a positive integer $n$, let $S_n = \{1,2,3, ...,n\}$ be the set of positive integers less than or equal to $n$. Additionally, let $$f (n) = |\{x \in S_n : x^{2017}\equiv x \,\, (mod \,\, n)\}|.$$ Find $f (2016)- f (2015)+ f (2014)- f (2013)$. [u]Round 12[/u] [b]p34. [/b] Estimate the value of $\sum^{2017}_{n=1} \phi (n)$, where $\phi (n)$ is the number of numbers less than or equal $n$ that are relatively prime to n. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be max $\max \left(0,\lfloor 15 - 75 \frac{|A-E|}{A} \rceil \right).$ [b]p35.[/b] An up-down permutation of order $n$ is a permutation $\sigma$ of $(1,2,3, ..., n)$ such that $\sigma(i ) <\sigma (i +1)$ if and only if $i$ is odd. Denote by $P_n$ the number of up-down permutations of order $n$. Estimate the value of $P_{20} +P_{17}$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, 16 -\lceil \max \left(\frac{E}{A}, 2- \frac{E}{A}\right) \rceil \right).$ [b]p36.[/b] For positive integers $n$, superfactorial of $n$, denoted $n\$ $, is defined as the product of the first $n$ factorials. In other words, we have $n\$ = \prod^n_{i=1}(i !)$. Estimate the number of digits in the product $(20\$)\cdot (17\$)$. If your estimate is $E$ and the correct answer is $A$, your score for this problem will be $\max \left(0, \lfloor 15 -\frac12 |A-E| \rfloor \right).$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158491p28715220]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $R$ be the radius of circumcircle of $\triangle ABC$. Let $A',B',C'$ be the points on $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ respectively such that $AH.AA'=R^2,BH.BB'=R^2,CH.CC'=R^2$. Prove that $O$ is incenter of $\triangle A'B'C'$.

Let $\gamma_1, \gamma_2, \gamma_3$ be mutually externally tangent circles and $\Gamma_1, \Gamma_2, \Gamma_3$ also be mutually externally tangent circles. For each $1 \le i \le 3$, $\gamma_i$ and $\Gamma_{i+1}$ are externally tangent at $A_i$, $\gamma_i$ and $\Gamma_{i+2}$ are externally tangent at $B_i$, and $\gamma_i$ and $\Gamma_i$ do not meet. Show that the six points $A_1, A_2, A_3, B_1, B_2, B_3$ lie on either a line or a circle.

We are given a disk $\mathcal D$ of radius $1$ in the plane. (a) Prove that $\mathcal D$ cannot be covered with two disks of radii $r<1$. (b) Prove that, for some $r<1$, $\mathcal D$ can be covered with three disks of radius $r$. What is the smallest such $r$?

$A_0A_1\ldots A_n$ is a regular polygon with $n+1$ vertices ($n&gt;2$). Initially $n$ stones are placed at vertex $A_0$. In each allowed operation, $2$ stones are moved simultaneously, at the player's choice: each stone is moved from the vertex where it is located to one of the adjacent $2$ vertices. Find all the values of $n$ for which it is possible to have, after a succession of permitted operations, a stone at each of the vertices $A_1,A_2,\ldots ,A_n$. Clarification: The two stones that move in an allowed operation can be at the same vertex or at different vertices.

$ ABCD$ is a $ 4\times 4$ square. $ E$ is the midpoint of $ \left[AB\right]$. $ M$ is an arbitrary point on $ \left[AC\right]$. How many different points $ M$ are there such that $ \left|EM\right|\plus{}\left|MB\right|$ is an integer? $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$

Triangle $ABC$ is such that $\angle BAC>\angle ABC>60^o$. The perpendicular bisector of $\overline{AB}$ intersects the segment $\overline {BC}$ at $O$. Suppose there exists a point $D$ on the segment $\overline{AC}$ such that $OD=AB$ and $\angle ODA=30^o$. Find $\angle BAC$. [i](Vlad Robu)[/i]

The incircle of a triangle $ABC$ touches $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Find the angles of $\triangle A_1B_1C_1$ in terms of the angles of $\triangle ABC$.

Given an acute triangle $ABC$ with $AB>AC$ that has circumcenter $O$. Line $BO$ and $CO$ meet the bisector of $\angle BAC$ at $P$ and $Q$, respectively. Moreover, line $BQ$ and $CP$ meet at $R$. Show that $AR$ is perpendicular to $BC$. [i]Proposer: Soewono and Fajar Yuliawan[/i]

Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.

The number of ordered pairs of integers $ (m,n)$ for which $ mn \ge 0$ and \[m^3 \plus{} n^3 \plus{} 99mn \equal{} 33^3\] is equal to $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 33\qquad \textbf{(D)}\ 35\qquad \textbf{(E)}\ 99$

(a) A radio lamp has a $7$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $7$ holes. Is it possible to number the contacts and the holes so that for any insertion at least one contact would match the hole with the same number? (b) A radio lamp has a $20$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $20$ holes. Let the contacts in the plug and the socket be already numbered. Is it always possible to insert the plug so that none of the contacts matches its socket?