Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.

Let be given two parallel lines $k$ and $l$, and a circle not intersecting $k$. Consider a variable point $A$ on the line $k$. The two tangents from this point $A$ to the circle intersect the line $l$ at $B$ and $C$. Let $m$ be the line through the point $A$ and the midpoint of the segment $BC$. Prove that all the lines $m$ (as $A$ varies) have a common point.

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\dfrac{1}{\sqrt[4]{3}}\,$. What is the value of $900p$, rounded to the nearest integer?

Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant. (1) Find the value of $ a$ and the coordinate of the point $ P$. (2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$. [color=green][Edited.][/color]

$(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$ $(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas. $(b)$ Find the locus of the centers of these hyperbolas.

$M$ is midpoint of $BC$.$P$ is an arbitary point on $BC$. $C_{1}$ is tangent to big circle.Suppose radius of $C_{1}$ is $r_{1}$ Radius of $C_{4}$ is equal to radius of $C_{1}$ and $C_{4}$ is tangent to $BC$ at P. $C_{2}$ and $C_{3}$ are tangent to big circle and line $BC$ and circle $C_{4}$. [img]http://aycu01.webshots.com/image/4120/2005120338156776027_rs.jpg[/img] Prove : \[r_{1}+r_{2}+r_{3}=R\] ($R$ radius of big circle)

An ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ passes point $(2,1)$, then all points $(x,y)$ on the ellipse that $|y|>1$ are (shown as shadow) [img]https://graph.baidu.com/resource/122481219e60931bb707101582696834.jpg[/img]

The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is $\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$ $\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$ $\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$ $\textbf{(D) }\text{an elllipse having positive area}\qquad$ $\textbf{(E) }\text{a parabola}$

The lines $t$ and $ t'$, tangent to the parabola $y = x^2$ at points $A$ and $B$ respectively, intersect at point $C$. The median of triangle $ABC$ from $C$ has length $m$. Find the area of $\triangle ABC$ in terms of $m$.

A triangle $ABC$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $BC$ meets segments $AB$, $AC$ at points $D$, $E$ respectively, and meets $\omega$ at points $K$, $L$ (where $D$ lies between $K$ and $E$). Circle $\gamma_1$ is tangent to the segments $KD$ and $BD$ and also tangent to $\omega$, while circle $\gamma_2$ is tangent to the segments $LE$ and $CE$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_1$ and $\gamma_2$. [i](Russia) Vasily Mokin and Fedor Ivlev[/i]

Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

Given two perpendicular diameters $AB$ and $CD$ of an ellipse, we say that the diameter $A'B'$ is conjugate to $AB$ if $A'B'$ is parallel to the tangent to the ellipse at $A$. Let $A'B'$ be conjugate to $AB$ and $C'D'$ be conjugate to $CD$. Prove that the rectangular hyperbola through $A', B', C'$ and $D'$ passes through the foci of the ellipse.

When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into regions whose areas are in the ratio $2$ to $1$. How many integer values of $k$ are there such that $0<k\leq 2007$ and the area between the parabola $y=k-x^2$ and the $x$-axis is an integer? [asy] import graph; size(300); defaultpen(linewidth(0.8)+fontsize(10)); real k=1.5; real endp=sqrt(k); real f(real x) { return k-x^2; } path parabola=graph(f,-endp,endp)--cycle; filldraw(parabola, lightgray); draw((endp,0)--(endp,k)--(-endp,k)--(-endp,0)); label("Region I", (0,2*k/5)); label("Box II", (51/64*endp,13/16*k)); label("area(I) = $\frac23$\,area(II)",(5/3*endp,k/2)); [/asy]

(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis. (2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of $a$.

Find the shortest distance between the point $(6,12)$ and the parabola given by the equation $x=\frac{y^2}{2}$

Let $F$ be the focus of parabola $y^2=2px(p>0)$, with directrix $l$ and two points $A,B$ on it. Knowing that $\angle AFB=\frac{\pi}{3}$, find the maximal value of $\frac{|MN|}{|AB|}$, where $M$ is the midpoint of $AB$ and $N$ is the projection of $M$ to $l$.

Find the slope of a line passing through the point $(0,\ 1)$ with which the area of the part bounded by the line and the parabola $y=x^2$ is $\frac{5\sqrt{5}}{6}.$

A parabola has focus $F$ and vertex $V$ , where $VF = 1$0. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\vartriangle VAB$.

A conical surface $C$ is cut by a plane $T$ as shown in the figure on the back of this sheet. Show that $C \cap T$ is an ellipse. You can use as an aid the fact that if you consider the two spheres tangent to $C$ and $T$ as shown in the figure, they intersect $T$ in the bulbs. [asy] // calculate intersection of line and plane // p = point on line // d = direction of line // q = point in plane // n = normal to plane triple lineintersectplan(triple p, triple d, triple q, triple n) { return (p + dot(n,q - p)/dot(n,d)*d); } // projection of point A onto line BC triple projectionofpointontoline(triple A, triple B, triple C) { return lineintersectplan(B, B - C, A, B - C); } // calculate area of space triangle with vertices A, B, and C real trianglearea(triple A, triple B, triple C) { return abs(cross(A - C, B - C)/2); } // calculate incentre of space triangle ABC triple triangleincentre(triple A, triple B, triple C) { return (abs(B - C) * A + abs(C - A) * B + abs(A - B) * C)/(abs(B - C) + abs(C - A) + abs(A - B)); } // calculate inradius of space triangle ABC real triangleinradius(triple A, triple B, triple C) { return 2*trianglearea(A,B,C)/(abs(B - C) + abs(C - A) + abs(A - B)); } // calculate excentre of space triangle ABC triple triangleexcentre(triple A, triple B, triple C) { return (-abs(B - C) * A + abs(C - A) * B + abs(A - B) * C)/(-abs(B - C) + abs(C - A) + abs(A - B)); } // calculate exradius of space triangle ABC real triangleexradius(triple A, triple B, triple C) { return 2*trianglearea(A,B,C)/(-abs(B - C) + abs(C - A) + abs(A - B)); } unitsize(2 cm); pair project (triple A, real t) { return((A.x, A.y*Sin(t) + A.z*Cos(t))); } real alpha, beta, theta, t; real coneradius = 1, coneheight = 3; real a, b, c; real[] m, r; triple A, B, V; triple ellipsecenter, ellipsex, ellipsey; triple[] F, O, P, R, W; path[] ellipse, spherering; theta = 15; V = (0,0,-coneheight); m[1] = sqrt(Cos(theta)^2*coneheight^2 - Sin(theta)^2*coneradius^2)/coneradius; m[2] = -m[1]; alpha = -aTan(Sin(theta)/m[1]); beta = -aTan(Sin(theta)/m[2]) + 180; A = (coneradius*Cos(alpha), coneradius*Sin(alpha), 0); B = (coneradius*Cos(beta), coneradius*Sin(beta), 0); W[1] = interp(V,(coneradius,0,0),0.6); W[2] = interp(V,(-coneradius,0,0),0.4); O[1] = triangleexcentre(V,W[1],W[2]); O[2] = triangleincentre(V,W[1],W[2]); r[1] = triangleexradius(V,W[1],W[2]); r[2] = triangleinradius(V,W[1],W[2]); F[1] = projectionofpointontoline(O[1],W[1],W[2]); F[2] = projectionofpointontoline(O[2],W[1],W[2]); P[1] = O[1] - (0,0,r[1]*coneradius/sqrt(coneradius^2 + coneheight^2)); P[2] = O[2] - (0,0,r[2]*coneradius/sqrt(coneradius^2 + coneheight^2)); spherering[11] = shift(project(P[1],theta))*yscale(Sin(theta))*arc((0,0),r[1]*coneheight/sqrt(coneradius^2 + coneheight^2),alpha,beta); spherering[12] = shift(project(P[1],theta))*yscale(Sin(theta))*arc((0,0),r[1]*coneheight/sqrt(coneradius^2 + coneheight^2),beta,alpha + 360); spherering[21] = shift(project(P[2],theta))*yscale(Sin(theta))*arc((0,0),r[2]*coneheight/sqrt(coneradius^2 + coneheight^2),alpha,beta); spherering[22] = shift(project(P[2],theta))*yscale(Sin(theta))*arc((0,0),r[2]*coneheight/sqrt(coneradius^2 + coneheight^2),beta,alpha + 360); ellipsecenter = (W[1] + W[2])/2; a = abs(W[1] - ellipsecenter); c = abs(F[1] - ellipsecenter); b = sqrt(a^2 - c^2); ellipsex = (W[1] - W[2])/abs(W[1] - W[2]); ellipsey = (0,1,0); ellipse[1] = project(ellipsecenter + a*ellipsex, theta); for (t = 0; t <= 180; t = t + 5) { ellipse[1] = ellipse[1]--project(ellipsecenter + a*Cos(t)*ellipsex + b*Sin(t)*ellipsey, theta); } ellipse[2] = project(ellipsecenter - a*ellipsex, theta); for (t = 180; t <= 360; t = t + 5) { ellipse[2] = ellipse[2]--project(ellipsecenter + a*Cos(t)*ellipsex + b*Sin(t)*ellipsey, theta); } R[1] = ellipsecenter + 1*ellipsex + ellipsey; R[2] = ellipsecenter - 1.2*ellipsex + ellipsey; R[3] = ellipsecenter - 1*ellipsex - ellipsey; R[4] = ellipsecenter + 1.2*ellipsex - ellipsey; fill(ellipse[1]--ellipse[2]--cycle, gray(0.9)); draw(yscale(Sin(theta))*Circle((0,0),coneradius)); draw(project(V,theta)--project(A,theta)); draw(project(V,theta)--project(B,theta)); draw(Circle(project(O[1],theta),r[1])); draw(Circle(project(O[2],theta),r[2])); draw(spherering[11], dashed); draw(spherering[12]); draw(spherering[21], dashed); draw(spherering[22]); draw(ellipse[1], dashed); draw(ellipse[2]); draw(project(R[1],theta)--interp(project(R[1],theta),project(R[2],theta),0.13)); draw(interp(project(R[1],theta),project(R[2],theta),0.13)--interp(project(R[1],theta),project(R[2],theta),0.76), dashed); draw(interp(project(R[1],theta),project(R[2],theta),0.76)--project(R[2],theta)); draw(project(R[2],theta)--project(R[3],theta)--project(R[4],theta)--project(R[1],theta)); label("$C$", (-1,0.3)); label("$T$", (1.2,-0.8)); dot(project(F[1],theta)); dot(project(F[2],theta)); //dot("$F_1$", project(F[1],theta)); //dot("$F_2$", project(F[2],theta)); //dot("$O_1$", project(O[1],theta)); //dot("$O_2$", project(O[2],theta)); //dot("$P_1$", project(P[1],theta)); //dot("$V$", project(V,theta)); //dot("$W_1$", project(W[1],theta)); //dot("$W_2$", project(W[2],theta)); [/asy]

In the interior of an ellipse with major axis 2 and minor axis 1, there are more than 6 segments with total length larger than 15. Prove that there exists a line passing through all of the segments.

Determine all monic polynomials $p(x)$ having real coefficients and satisfying the following two conditions: $\bullet$ $p(x)$ is nonconstant, and all of its roots are distinct reals $\bullet$ If $a $and $b$ are roots of $p(x)$ then $a + b + ab$ is also a root of $p(x)$.

A triangle $ ABC$ is given in the plane. Let $ M$ be a point inside the triangle and $ A'$, $ B'$, $ C'$ be its projections on the sides $ BC$, $ CA$, $ AB$, respectively. Find the locus of $ M$ for which $ MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'$.

Suppose you have a sphere tangent to the $xy$-plane with its center having positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\tfrac pq$ where $p,q$ are integers, find $p+q$.

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 1 Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$. Find the maximum value of $3p+2q+r-4|s|$.