Let $M$ be the midpoint of segment $BC$ of $\triangle ABC$. Let $D$ be a point such that $AD=AB$, $AD\perp AB$ and points $C$ and $D$ are on different sides of $AB$. Prove that: $$\sqrt{AB\cdot AC+BC\cdot AM}\geq\frac{\sqrt{2}}{2}CD.$$

A circle of radius $ 1$ is internally tangent to two circles of radius $ 2$ at points $ A$ and $ B$, where $ AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles? [asy]size(200);defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair B = (0,1); pair A = (0,-1); label("$B$",B,NW);label("$A$",A,2S); draw(Circle(A,2));draw(Circle(B,2)); fill((-sqrt(3),0)..B..(sqrt(3),0)--cycle,gray); fill((-sqrt(3),0)..A..(sqrt(3),0)--cycle,gray); draw((-sqrt(3),0)..B..(sqrt(3),0)); draw((-sqrt(3),0)..A..(sqrt(3),0)); path circ = Circle(origin,1); fill(circ,white); draw(circ); dot(A);dot(B); pair A1 = B + dir(45)*2; pair A2 = dir(45); pair A3 = dir(-135)*2 + A; draw(B--A1,EndArrow(HookHead,2)); draw(origin--A2,EndArrow(HookHead,2)); draw(A--A3,EndArrow(HookHead,2)); label("$2$",midpoint(B--A1),NW); label("$1$",midpoint(origin--A2),NW); label("$2$",midpoint(A--A3),NW);[/asy]$ \textbf{(A)}\ \frac {5}{3}\pi \minus{} 3\sqrt {2}\qquad \textbf{(B)}\ \frac {5}{3}\pi \minus{} 2\sqrt {3}\qquad \textbf{(C)}\ \frac {8}{3}\pi \minus{} 3\sqrt {3}\qquad\textbf{(D)}\ \frac {8}{3}\pi \minus{} 3\sqrt {2}$ $ \textbf{(E)}\ \frac {8}{3}\pi \minus{} 2\sqrt {3}$

It is known that the angles of the triangle $ABC$ are $1: 3: 5$. Find the angle between the bisector of the largest angle of the triangle and the line containing the altitude drawn to the smallest side of the triangle.

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N_{21}$ be the answer to question 21. Suppose a jar has $3N_{21}$ colored balls in it: $N_{21}$ red, $N_{21}$ green, and $N_{21}$ blue balls. Jonathan takes one ball at a time out of the jar uniformly at random without replacement until all the balls left in the jar are the same color. Compute the expected number of balls left in the jar after all balls are the same color. [b]p20.[/b] Let $N_{19}$ be the answer to question 19. For every non-negative integer $k$, define $$f_k(x) = x(x - 1) + (x + 1)(x - 2) + ...+ (x + k)(x - k - 1),$$ and let $r_k$ and $s_k$ be the two roots of $f_k(x)$. Compute the smallest positive integer $m$ such that $|r_m - s_m| > 10N_{19}$. [b]p21.[/b] Let $N_{20}$ be the answer to question 20. In isosceles trapezoid $ABCD$ (where $\overline{BC}$ and $\overline{AD}$ are parallel to each other), the angle bisectors of $A$ and $D$ intersect at $F$, and the angle bisectors of points $B$ and $C$ intersect at $H$. Let $\overline{BH}$ and $\overline{AF}$ intersect at $E$, and let $\overline{CH}$ and $\overline{DF}$ intersect at $G$. If $CG = 3$, $AE = 15$, and $EG = N_{20}$, compute the area of the quadrilateral formed by the four tangency points of the largest circle that can fit inside quadrilateral $EFGH$.

$ABC$ is an acute-angled triangle. The tangents to the circumcircle at $A$ and $C$ meet the tangent at $B$ at $M$ and $N$. The altitude from $B$ meets $AC$ at $P$. Show that $BP$ bisects the angle $MPN$

There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.

Let $ ABC$ be an equilateral triangle and $ \Gamma$ the semicircle drawn exteriorly to the triangle, having $ BC$ as diameter. Show that if a line passing through $ A$ trisects $ BC,$ it also trisects the arc $ \Gamma.$

Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.

Find the maximum possible volume of a tetrahedron which has three faces with area $1$.

Let $A_1B_3A_2B_1A_3B_2$ be a cyclic hexagon such that $A_1B_1,A_2B_2,A_3B_3$ intersect at one point. Let $C_1=A_1B_1\cap A_2A_3,C_2=A_2B_2\cap A_1A_3,C_3=A_3B_3\cap A_1A_2$. Let $D_1$ be the point on the circumcircle of the hexagon such that $C_1B_1D_1$ touches $A_2A_3$. Define $D_2,D_3$ analogously. Show that $A_1D_1,A_2D_2,A_3D_3$ meet at one point.

In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.

[asy] draw(unitsquare);draw((0,0)--(.25,sqrt(3)/4)--(.5,0)); label("Z",(0,1),NW);label("Y",(1,1),NE);label("A",(0,0),SW);label("X",(1,0),SE);label("B",(.5,0),S);label("P",(.25,sqrt(3)/4),N); //Credit to Zimbalono for the diagram[/asy] Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to $\textbf{(A) }20\pi/3\qquad\textbf{(B) }32\pi/3\qquad\textbf{(C) }12\pi\qquad\textbf{(D) }40\pi/3\qquad \textbf{(E) }15\pi$

Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.

The midpoints of alternative sides of a convex hexagon are connected by line segments. Prove that the intersection points of the medians of the two triangles obtained coincide.

In the interior of a cube of side $6$ there are $1001$ unit cubes with the faces parallel to the faces of the given cube. Prove that there are $2$ unit cubes with the property that the center of one of them lies in the interior or on one of the faces of the other cube. [i]Dinu Serbanescu[/i]

In the $xy$-plane, a circle of radius $4$ with center on the positive $x$-axis is tangent to the $y$-axis at the origin, and a circle with radius $10$ with center on the positive $y$-axis is tangent to the $x$-axis at the origin. What is the slope of the line passing through the two points at which these circles intersect? $\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

Triangle $ABC$ satisfying $AB<AC$ has circumcircle $\Omega$. $E, F$ lies on $AC, AB$, respectively, such that $BCEF$ is cyclic. $T$ lies on $EF$ such that $\odot(TEF)$ is tangent to $BC$ at $T$. $A'$ is the antipode of $A$ on $\Omega$. $TA', TA$ intersects $\Omega$ again at $X, Y$, respectively, and $EF$ intersects $\odot(TXY)$ again at $W$. Prove that $\measuredangle WBA=\measuredangle ACW$ [i]Proposed by BlessingOfHeaven[/i]

2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

Let $O$ be the center of a circle. Let $OU,OV$ be perpendicular radii of the circle. The chord $PQ$ passes through the midpoint $M$ of $UV$. Let $W$ be a point such that $PM = PW$, where $U, V,M,W$ are collinear. Let $R$ be a point such that $PR = MQ$, where $R$ lies on the line $PW$. Prove that $MR = UV$. [u]Alternative version:[/u] A circle $S$ is given with center $O$ and radius $r$. Let $M$ be a point whose distance from $O$ is $\frac{r}{\sqrt{2}}$. Let $PMQ$ be a chord of $S$. The point $N$ is defined by $\overrightarrow{PN} =\overrightarrow{MQ}$. Let $R$ be the reflection of $N$ by the line through $P$ that is parallel to $OM$. Prove that $MR =\sqrt{2}r$.

Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

All vertices of triangles $ABC$ and $A_1B_1C_1$ lie on the hyperbola $y=1/x$. It is known that $AB \parallel A_1B_1$ and $BC \parallel B_1C_1$. Prove that $AC_1 \parallel A_1C$. (I. Gorodnin)

Let $C$ be a variable interior point of a fixed segment $AB$. Equilateral triangles $ACB' $ and $CBA'$ are constructed on the same side and $ABC' $ on the other side of the line $AB$. (a) Prove that the lines $AA' ,BB'$ , and $CC'$ meet at some point $P$. (b) Find the locus of $P$ as $C$ varies. (c) Prove that the centers $A'' ,B'' ,C''$ of the three triangles form an equilateral triangle. (d) Prove that $A'' ,B'',C''$ , and $P$ lie on a circle.

Through the vertex of a trihedral angle in which no edge is perpendicular to the opposite face, a straight line is drawn in the plane of each face perpendicular to the opposite edge. Prove that the three straight lines obtained lie in one plane.

Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.

A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?