Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

$(GBR 3)$ A smooth solid consists of a right circular cylinder of height $h$ and base-radius $r$, surmounted by a hemisphere of radius $r$ and center $O.$ The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point $P$ on the hemisphere such that $OP$ makes an angle $\alpha$ with the horizontal. Show that if $\alpha$ is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through $P$, show that it will cross the common circular section of the hemisphere and cylinder at a point $Q$ such that $\angle SOQ = \phi$, $S$ being where it initially crossed this section, and $\sin \phi = \frac{r \tan \alpha}{h}$.

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\] $\textbf{(A) }3 + 2 \sqrt 2\qquad \textbf{(B) } 2 + \sqrt 3\qquad \textbf{(C ) }3 \sqrt 3\qquad \textbf{(D) }6\qquad \textbf{(E) }6 + 2 \sqrt 3$

A turtle runs away from an UFO with a speed of $0.2 \ m/s$. The UFO flies $5$ meters above the ground, with a speed of $20 \ m/s$. The UFO's path is a broken line, where after flying in a straight path of length $\ell$ (in meters) it may turn through for any acute angle $\alpha$ such that $\tan \alpha < \frac{\ell}{1000}$. When the UFO's center approaches within $13$ meters of the turtle, it catches the turtle. Prove that for any initial position the UFO can catch the turtle.

a. Let $ABC$ be a triangle with altitude $AD$ and $P$ a variable point on $AD$. Lines $PB$ and $AC$ intersect each other at $E$, lines $PC$ and $AB$ intersect each other at $F.$ Suppose $AEDF$ is a quadrilateral inscribed . Prove that \[\frac{PA}{PD}=(\tan B+\tan C)\cot \frac{A}{2}.\] b. Let $ABC$ be a triangle with orthocentre $H$ and $P$ a variable point on $AH$. The line through $C$ perpendicular to $AC$ meets $BP$ at $M$, The line through $B$ perpendicular to $AB$ meets $CP$ at $N.$ $K$ is the projection of $A$on $MN$. Prove that $\angle BKC+\angle MAN$ is invariant .

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.

Evaluate \[\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx\]

Prove the following inequality. \[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]

Consider the triangle $ ABC$ with $ m(\angle BAC) \equal{} 90^\circ$ and $ AB < AC$.Let a point $ D$ on the side $ AC$ such that: $ m(\angle ACB) \equal{} m(\angle DBA)$.Let $ E$ be a point on the side $ BC$ such that $ DE\perp BC$.It is known that $ BD \plus{} DE \equal{} AC$. Find the measures of the angles in the triangle $ ABC$.

Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of: $ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$

Given a parallelogram ABCD, let M and N be the midpoints of the sides BC and CD. Can the lines AM, AN divide the angle BAD into three equal angles?

Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$. Find $24\cot^2{x}$.

In a certain circle, the chord of a $d$-degree arc is 22 centimeters long, and the chord of a $2d$-degree arc is 20 centimeters longer than the chord of a $3d$-degree arc, where $d<120.$ The length of the chord of a $3d$-degree arc is $-m+\sqrt{n}$ centimeters, where $m$ and $n$ are positive integers. Find $m+n.$

Two circles of equal radius can tightly fit inside right triangle $ABC$, which has $AB=13$, $BC=12$, and $CA=5$, in the three positions illustrated below. Determine the radii of the circles in each case. [asy] size(400); defaultpen(linewidth(0.7)+fontsize(12)); picture p = new picture; pair s1 = (20,0), s2 = (40,0); real r1 = 1.5, r2 = 10/9, r3 = 26/7; pair A=(12,5), B=(0,0), C=(12,0); draw(p,A--B--C--cycle); label(p,"$B$",B,SW); label(p,"$A$",A,NE); label(p,"$C$",C,SE); add(p); add(shift(s1)*p); add(shift(s2)*p); draw(circle(C+(-r1,r1),r1)); draw(circle(C+(-3*r1,r1),r1)); draw(circle(s1+C+(-r2,r2),r2)); draw(circle(s1+C+(-r2,3*r2),r2)); pair D=s2+156/17*(A-B)/abs(A-B), E=s2+(169/17,0), F=extension(D,E,s2+A,s2+C); draw(incircle(s2+B,D,E)); draw(incircle(s2+A,D,F)); label("Case (i)",(6,-3)); label("Case (ii)",s1+(6,-3)); label("Case (iii)",s2+(6,-3));[/asy]

$AB$ is a chord of a circle with center $O$, $M$ is the midpoint of $AB$. A non-diameter chord is drawn through $M$ and intersects the circle at $C$ and $D$. The tangents of the circle from points $C$ and $D$ intersect line $AB$ at $P$ and $Q$, respectively. Prove that $PA$ = $QB$.

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]

In triangle $ABC$ medians from $B$ and $C$ are perpendicular. Prove that $\frac{\sin(B+C)}{\sin B \cdot \sin C} \geq \frac{2}{3}.$

An ellipse, whose semi-axes have length $a$ and $b$, rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?

Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.

Let $f$ be a function such that \[f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.\] Find the limit of $f$ in the point $x_0=1.$

Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcenter of triangle $BCR$. Prove that $AO \perp PQ.$

We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$. a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN. b) The circle is not CN. Which one of these sets are CN? 1) $A=\{x\in\mathbb R^3| |x|=1\}$ 2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$ 3) Graph of the function $f:[0,1]\to \mathbb R$ defined by \[f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0\]

In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals [asy] size(230); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=14*dir(36), C=intersectionpoint(B--(9001,0), Circle(A,19)), M=midpoint(B--C), b=A+14*dir(A--C), N=foot(A, B, b); draw(N--B--A--N--M--C--A^^B--M); markscalefactor=0.1; draw(rightanglemark(B,N,A)); pair point=N; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, S); label("$N$", N, dir(30)); label("$19$", (A+C)/2, dir(A--C)*dir(90)); label("$14$", (A+B)/2, dir(A--B)*dir(270)); [/asy] $\displaystyle \text{(A)} \ 2 \qquad \text{(B)} \ \frac{5}{2} \qquad \text{(C)} \ \frac{5}{2} - \sin \theta \qquad \text{(D)} \ \frac{5}{2} - \frac{1}{2} \sin \theta \qquad \text{(E)} \ \frac{5}{2} - \frac{1}{2} \sin \left(\frac{1}{2} \theta\right)$