Find the area of the figure bounded by three curves $ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.

The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

Let $z_1,z_2,z_3,\dots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\displaystyle\sum_{j=1}^{12} w_j$ can be written as $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.

The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face? $\text{\textbf{(A) }red}\qquad \text{\textbf{(B) } white}\qquad\text{\textbf{(C) }green}\qquad\text{\textbf{(D) }brown }\qquad\text{\textbf{(E)} purple}$ [asy] unitsize(2 cm); pair x, y, z, trans; int i; x = dir(-5); y = (0.6,0.5); z = (0,1); trans = (2,0); for (i = 0; i <= 2; ++i) { draw(shift(i*trans)*((0,0)--x--(x + y)--(x + y + z)--(y + z)--z--cycle)); draw(shift(i*trans)*((x + z)--x)); draw(shift(i*trans)*((x + z)--(x + y + z))); draw(shift(i*trans)*((x + z)--z)); } label(rotate(-3)*"$R$", (x + z)/2); label(rotate(-5)*slant(0.5)*"$B$", ((x + z) + (y + z))/2); label(rotate(35)*slant(0.5)*"$G$", ((x + z) + (x + y))/2); label(rotate(-3)*"$W$", (x + z)/2 + trans); label(rotate(50)*slant(-1)*"$B$", ((x + z) + (y + z))/2 + trans); label(rotate(35)*slant(0.5)*"$R$", ((x + z) + (x + y))/2 + trans); label(rotate(-3)*"$P$", (x + z)/2 + 2*trans); label(rotate(-5)*slant(0.5)*"$R$", ((x + z) + (y + z))/2 + 2*trans); label(rotate(-85)*slant(-1)*"$G$", ((x + z) + (x + y))/2 + 2*trans); [/asy]

Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \ne A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T$, $E$, $B$, $Z$ are concyclic. Proposed by [i]Theoklitos Parayiou, Cyprus[/i]

Let $n\geq 3$ be a positive integer and a circle $\omega$ is given. A regular polygon(with $n$ sides) $P$ is drawn and your vertices are in the circle $\omega$ and these vertices are red. One operation is choose three red points $A,B,C$, such that $AB=BC$ and delete the point $B$. Prove that one can do some operations, such that only two red points remain in the circle.

Let $AH_A, BH_B$ and $CH_C$ be altitudes in $\triangle ABC$. Let $p_A,p_B,p_C$ be the perpendicular lines from vertices $A,B,C$ to $H_BH_C, H_CH_A, H_AH_B$ respectively. Prove that $p_A,p_B,p_C$ are concurrent lines.

In an isosceles trapezoid $ABCD$ with bases $AD$ and $BC$, diagonals intersect at point $P$, and lines $AB$ and $CD$ intersect at point $Q$. $O_1$ and $O_2$ are the centers of the circles circumscribed around the triangles $ABP$ and $CDP$, $r$ is the radius of these circles. Construct the trapezoid ABCD given the segments $O_1O_2$, $PQ$ and radius $r$.

In an acute-angled triangle $ABC$ with orthocenter $H$, the line $AH$ cuts $BC$ at point $A_1$. Let $\Gamma$ be a circle centered on side $AB$ tangent to $AA_1$ at point $H$. Prove that $\Gamma$ is tangent to the circumscribed circle of triangle $AMA_1$, where $M$ is the midpoint of $AC$.

Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.

Let ABCD be a parallelogram, and P a point such that $2 PDA=ABP$ and $2 PAD=PCD$ Show that $AB=BP=CP$

In the trapezoid $ABCD$ , the base $AB$ is smaller than the $CD$ base. The point $K$ is chosen such that $AK$ is parallel to BC and $BK$ is parallel to $AD$. The points $P$ and $Q$ are chosen on the $AK$ and $BK$ rays respectively, such that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$. (a) Show that the lines $AD, BC$ and $PQ$ go through the same point. (b) Assuming that the circumscribed circumferences of the $APD$ and $BCQ$ triangles intersect at two points, show that one of those points belongs to the line $PQ$.

In $\triangle ABC$, $AB = 360$, $BC = 507$, and $CA = 780$. Let $M$ be the midpoint of $\overline{CA}$, and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC$. Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}$. Suppose that $\overline{DF}$ meets $\overline{BM}$ at $E$. The ratio $DE: EF$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Two circles $O_1$ and $O_2$ intersect at $A,B$. Diameter $AC$ of $\odot O_1$ intersects $\odot O_2$ at $E$, Diameter $AD$ of $\odot O_2$ intersects $\odot O_1$ at $F$. $CF$ intersects $O_2$ at $H$, $DE$ intersects $O_1$ at $G,H$. $GH\cap O_1=P$. Prove that $PH=PK$.

For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions. (1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$. (2) Find the values of $ a,\ b$. (3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.

In a triangle $\vartriangle ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ$ is smaller than $AB$ if and only if $\angle B$ is obtuse.

Let $ a\ge 2 $ be a natural number. Show that the following relations are equivalent: $ \text{(i)} \ a $ is the hypothenuse of a right triangle whose sides are natural numbers. $ \text{(ii)}\quad $ there exists a natural number $ d $ for which the polynoms $ X^2-aX\pm d $ have integer roots.

On the extension of the largest side $AC$ of the triangle $ABC$ set aside the segment $CM$ such that $CM = BC$. Prove that the angle $ABM$ is obtuse or right.

Let $O$ and $H$ be the circumcenter and the orthocenter, respectively, of an acute triangle $ABC$. Points $D$ and $E$ are chosen from sides $AB$ and $AC$, respectively, such that $A$, $D$, $O$, $E$ are concyclic. Let $P$ be a point on the circumcircle of triangle $ABC$. The line passing $P$ and parallel to $OD$ intersects $AB$ at point $X$, while the line passing $P$ and parallel to $OE$ intersects $AC$ at $Y$. Suppose that the perpendicular bisector of $\overline{HP}$ does not coincide with $XY$, but intersect $XY$ at $Q$, and that points $A$, $Q$ lies on the different sides of $DE$. Prove that $\angle EQD = \angle BAC$. [i]Proposed by Shuang-Yen Lee[/i]

Let $n\ge2$ be an integer. A regular $(2n+1)-gon$ is divided in to $2n-1$ triangles by diagonals which do not meet except at the vertices. Prove that at least three of these triangles are isosceles.

Given is a triangle $ABC$ with median $BM$. The point $D$ lies on the line $AC$ after $C$, such that $BD=2CD$. The circle $(BMC)$ meets the segment $BD$ at $N$. Show that $AC+BM>2MN$.

An equilateral triangle $ABC$ with center $O$ and radius $OA = R$ is given, and consider the seven regions that the lines of the sides determine on the plane. It is asked to draw and describe the region of the plane transformed from the two shaded regions in the attached figure, by the inversion of center $O$ and power $R^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/bf1cb12c961467d216d54885f3387b328ce744.png[/img]

Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

The bisector of angle $CAB$ of triangle $ABC$ intersects the side $CB$ at $L$. The point $D$ is the foot of the perpendicular from $C$ to $AL$ and the point $E$ is the foot of perpendicular from $L$ to $AB$. The lines $CB$ and $DE$ meet at $F$. Prove that $AF$ is an altitude of triangle $ABC$.

Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$