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

The quadrilateral $A_1A_2A_3A_4$ is cyclic, and its sides are $a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4$ and $a_4 = A_4A_1.$ The respective circles with centres $I_i$ and radii $r_i$ are tangent externally to each side $a_i$ and to the sides $a_{i+1}$ and $a_{i-1}$ extended. ($a_0 = a_4$). Show that \[ \prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2. \]

Sketch the graph of $u(x, 1)$, if $0 \le x\le1$, \[\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\;u|_{t=0}=x^2,\;u|_{x^2=x}=x^2\]

Let $AB CD$ be a rectangle with $AB=1$. If $m ( \angle BDC) = 82^o30'$, compute the length of$ BD$ and the cosine of $82^o30'$.

A projectile, thrown with initial velocity $v_0$ in a direction making angle $\alpha$ with the horizontal, is acted on by no force except gravity. Find the lenght of its path until it strikes a horizontal plane through the starting point. Show that the flight is longest when $$\sin \alpha \log(\sec \alpha+ \tan \alpha)=1.$$

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that \[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\] and find the conditions for equality.

Sines of three acute angles form an arithmetic progression, while the cosines of these angles form a geometric progression. Prove that all three angles are equal.

In triangle $ABC$ we have $\angle ABC = 2 \angle ACB$. Prove that a) $AC^2 = AB^2 + AB \cdot BC$; b) $AB+BC < 2 \cdot AC$.

Prove the equality: $$\sum_{k=1}^{n-1}\frac1{\sin^2\frac{(2k+1)\pi}{2n}}=n^2$$ where $n$ is a natural number. [i]H. Lesov[/i]

Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$

In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$. Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.

Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$

Calculate the following indefinite integrals. [1] $\int (4-5\tan x)\cos x dx$ [2] $\int \frac{dx}{\sqrt[3]{(1-3x)^2}}dx$ [3] $\int x^3\sqrt{4-x^2}dx$ [4] $\int e^{-x}\sin \left(x+\frac{\pi}{4}\right)dx$ [5] $\int (3x-4)^2 dx$

Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.

A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

In triangle $ABC$, we have $\angle ABC=60$. The line through $B$ perpendicular to side $AB$ intersects angle bisector of $\angle BAC$ in $D$ and the line through $C$ perpendicular $BC$ intersects angle bisector of $\angle ABC$ in $E$. prove that $\angle BED\le 30$.

A circle of radius $t$ is tangent to the hypotenuse, the incircle, and one leg of an isosceles right triangle with inradius $r=1+\sin \frac{\pi}{8}$. Find $rt$.

[b]p1.[/b] An $8$-inch by $11$-inch sheet of paper is laid flat so that the top and bottom edges are $8$ inches long. The paper is then folded so that the top left corner touches the right edge. What is the minimum possible length of the fold? [b]p2.[/b] Triangle $ABC$ is equilateral, with $AB = 6$. There are points $D$, $E$ on segment AB (in the order $A$, $D$, $E$, $B$), points $F$, $G$ on segment $BC$ (in the order $B$, $F$, $G$, $C$), and points $H$, $I$ on segment $CA$ (in the order $C$, $H$, $I$, $A$) such that $DE = F G = HI = 2$. Considering all such configurations of $D$, $E$, $F$, $G$, $H$, $I$, let $A_1$ be the maximum possible area of (possibly degenerate) hexagon $DEF GHI$ and let $A_2$ be the minimum possible area. Find $A_1 - A_2$. [b]p3.[/b] Find $$\tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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?

Let $O$ be the center of $\bigtriangleup ABC$'s circumcircle. $CO$ line intersect $AB$ at $D$ and $BO$ line intersect $AC$ at $E$. If $\angle A=\angle CDE=50$° then find $\angle ADE$

Let $ f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x.$ (1) Evaluate $ \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx$. (2) Find the area of the region bounded by two curves $ y\equal{}f(x)$ and $ y\equal{}g(x)\ (0\leq x\leq \pi).$

An equilateral triangle is given. A point lies on the incircle of this triangle. If the smallest two distances from the point to the sides of the triangle is $1$ and $4$, the sidelength of this equilateral triangle can be expressed as $\tfrac{a\sqrt b}c$ where $(a,c)=1$ and $b$ is not divisible by the square of an integer greater than $1$. Find $a+b+c$.

An integer-sided triangle has angles $ p\theta$ and $ q\theta$, where $ p$ and $ q$ are relatively prime integers. Prove that $ \cos\theta$ is irrational.

A cuboctahedron is the convex hull of (smallest convex set containing) the $12$ points $(\pm 1, \pm 1, 0), (\pm 1, 0, \pm 1), (0, \pm 1, \pm 1)$. Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the figure. The area of that set is the solid angle of the figure as viewed from the origin.)