A small chunk of ice falls from rest down a frictionless parabolic ice sheet shown in the figure. At the point labeled $\mathbf{A}$ in the diagram, the ice sheet becomes a steady, rough incline of angle $30^\circ$ with respect to the horizontal and friction coefficient $\mu_k$. This incline is of length $\frac{3}{2}h$ and ends at a cliff. The chunk of ice comes to a rest precisely at the end of the incline. What is the coefficient of friction $\mu_k$? [asy] size(200); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(sqrt(3),0)--(0,1)); draw(anglemark((0,1),(sqrt(3),0),(0,0))); label("$30^\circ$",(1.5,0.03),NW); label("A", (0,1),NE); dot((0,1)); label("rough incline",(0.4,0.4)); draw((0.4,0.5)--(0.5,0.6),EndArrow); dot((-0.2,4/3)); label("parabolic ice sheet",(0.6,4/3)); draw((0.05,1.3)--(-0.05,1.2),EndArrow); label("ice chunk",(-0.5,1.6)); draw((-0.3,1.5)--(-0.25,1.4),EndArrow); draw((-0.2,4/3)--(-0.19, 1.30083)--(-0.18,1.27)--(-0.17,1.240833)--(-0.16,1.21333)--(-0.15,1.1875)--(-0.14,1.16333)--(-0.13,1.140833)--(-0.12,1.12)--(-0.11,1.100833)--(-0.10,1.08333)--(-0.09,1.0675)--(-0.08,1.05333)--(-0.07,1.040833)--(-0.06,1.03)--(-0.05,1.020833)--(-0.04,1.01333)--(-0.03,1.0075)--(-0.02,1.00333)--(-0.01,1.000833)--(0,1)); draw((-0.6,0)--(-0.6,4/3),dashed,EndArrow,BeginArrow); label("$h$",(-0.6,2/3),W); draw((0.2,1.2)--(sqrt(3)+0.2,0.2),dashed,EndArrow,BeginArrow); label("$\frac{3}{2}h$",(sqrt(3)/2+0.2,0.7),NE); [/asy] $ \textbf{(A)}\ 0.866\qquad\textbf{(B)}\ 0.770\qquad\textbf{(C)}\ 0.667\qquad\textbf{(D)}\ 0.385\qquad\textbf{(E)}\ 0.333 $

If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$

Let $A$ be a $8\times 8$ array with entries from the set $\{-1,1\}$ such that any $2\times 2$ sub-square of the array has the absolute value of the sum of its element equal with 2. Prove that the array must have at least two identical lines.

Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by James Lin[/i]

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$. Let $P$ be a point on side $BC$, and let $\omega$ be the circle with diameter $CP$. Suppose that $\omega$ intersects $AC $and $AP$ again at $Q$ and $R$, respectively. Show that $CP^2 = AC \cdot CQ - AP \cdot P R$.

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

Let $n$ be a positive integer prove that $$6\nmid \lfloor (\sqrt[3]{28}-3)^{-n} \rfloor.$$

Find $$\prod^{2017}_{k=1} e^{\pi ik/2017}2 cos \left( \frac{\pi k}{2017} \right)$$

Let $ABCD$ be a cyclic quadrilateral such that $AB=AD$. Let $E$ and $F$ be points on the sides $BC$ and $CD$, respectively, such that $BE+DF=EF$. Prove that $\angle BAD = 2 \angle EAF$.

Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.

Show that there is a perfect square (a number which is a square of an integer) such that sum of its digits is $2011.$

Let $ABC$ be an equilateral triangle and $M$ be a point inside the triangle. We denote by $A'$, $B'$, $C'$ the projections of the point $M$ on the sides $BC$, $CA$ and $AB$ respectively. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent if and only if $M$ belongs to an altitude of the triangle.

$A_1,A_2,\cdots,A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side.

Compute $1+2\cdot3^4$. [i]Proposed by Evan Chen[/i]

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

Matt has somewhere between $1000$ and $2000$ pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles but ends up with one sheet left over each time. How many piles does he need?

Find a real function $f : [0,\infty)\to \mathbb R$ such that $f(2x+1) = 3f(x)+5$, for all $x$ in $[0,\infty)$.

What is the sum of distinct real numbers $x$ such that $(2x^2+5x+9)^2=56(x^3+1)$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ \dfrac{7}{4} \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \dfrac{9}{2} \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ABC$ be an acute triangle with orthocenter $H$ and let $M$ be the midpoint of $\overline{BC}$. (The [i]orthocenter[/i] is the point at the intersection of the three altitudes.) Denote by $\omega_B$ the circle passing through $B$, $H$, and $M$, and denote by $\omega_C$ the circle passing through $C$, $H$, and $M$. Lines $AB$ and $AC$ meet $\omega_B$ and $\omega_C$ again at $P$ and $Q$, respectively. Rays $PH$ and $QH$ meet $\omega_C$ and $\omega_B$ again at $R$ and $S$, respectively. Show that $\triangle BRS$ and $\triangle CRS$ have the same area. [i]Proposed by Aaron Lin[/i]

Find the median value of $m$ over all integers $m$ where $|m^2 + 8m - 65|$ is a perfect power. A perfect power is any integer at least $2$ which can be written as $a^b$, where $a$, $b$ are integers and $b \ge 2$. [i]Individual #9[/i]

There are $36$ gangsters bands.And there are war between some bands. Every gangster can belongs to several bands and every 2 gangsters belongs to different set of bands. Gangster can not be in feuding bands. Also for every gangster is true, that every band, where this gangster is not in, is in war with some band, where this gangster is in. What is maximum number of gangsters in city?

Assume that real numbers $x$, $y$ and $z$ satisfy $x + y + z = 3$. Prove that $xy + xz + yz \leq 3$.

Let $I$ be a point on the bisector of angle $BAC$ of a triangle $ABC$. Points $M,N$ are taken on the respective sides $AB$ and $AC$ so that $\angle ABI=\angle NIC$ and $\angle ACI=\angle MIB$. Show that $I$ is the incenter of triangle $ABC$ if and only if points $M,N$ and $I$ are collinear.

Show that if $n\ge 2$ is an integer, then there exist invertible matrices $A_1, A_2, \ldots, A_n \in \mathcal{M}_2(\mathbb{R})$ with non-zero entries such that: \[A_1^{-1} + A_2^{-1} + \ldots + A_n^{-1} = (A_1 + A_2 + \ldots + A_n)^{-1}.\] [hide=Edit.] The $77777^{\text{th}}$ topic in College Math :coolspeak: [/hide]

Does there exist an infinite set $ M$ of straight lines on the coordinate plane such that (i) no two lines are parallel, and (ii) for any integer point there is a line from $ M$ containing it?