Found problems: 487
Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.
$(BEL 5)$ 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.
$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$
A standard parabola is the graph of a quadratic polynomial $y = x^2 + ax + b$ with leading co\"efficient 1. Three standard parabolas with vertices $V1, V2, V3$ intersect pairwise at points $A1, A2, A3$. Let $A \mapsto s(A)$ be the reflection of the plane with respect to the $x$-axis.
Prove that standard parabolas with vertices $s (A1), s (A2), s (A3)$ intersect pairwise at the points $s (V1), s (V2), s (V3)$.
Let $p>0$ be a real constant. From any point $(a,b)$ in the cartesian plane, show that
i) Three normals, real or imaginary, can be drawn to the parabola $y^2=4px$.
ii) These are real and distinct if $4(2-p)^3 +27pb^2<0$.
iii) Two of them coincide if $(a,b)$ lies on the curve $27py^2=4(x-2p)^3$.
iv) All three coincide only if $a=2p$ and $b=0$.
All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 16$
There is a parabola $y = x^2$ drawn on the coordinate plane. The axes are deleted. Can you restore them with the help of compass and ruler?
An ellipse has foci $A$ and $B$ and has the property that there is some point $C$ on the ellipse such that the area of the circle passing through $A$, $B$, and, $C$ is equal to the area of the ellipse. Let $e$ be the largest possible eccentricity of the ellipse. One may write $e^2$ as $\frac{a+\sqrt{b}}{c}$ , where $a, b$, and $c$ are integers such that $a$ and $c$ are relatively prime, and b is not divisible by the square of any prime. Find $a^2 + b^2 + c^2$.
Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value.
Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$?
$ \textbf{(A)}\ \frac{11}{81} \qquad
\textbf{(B)}\ \frac{13}{81} \qquad
\textbf{(C)}\ \frac{5}{27} \qquad
\textbf{(D)}\ \frac{17}{81} \qquad
\textbf{(E)}\ \frac{19}{81}
$
Parabola $y^2=2px$, two fixed points $A(a,b),B(-a,0)(ab\neq0,b^2\neq 2pa)$. $M$ is a point on the parabola, $AM$ intersects the parabola at $M_1$, $BM$ intersects the parabola at $M_2$.
Prove: When $M$ changes, line $M_1M_2$ passes a fixed point, and find the fixed point.
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic.
[i]David Yang.[/i]
Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$.
The length of minor axis of ellipse $\rho-\frac{1}{2-\cos\theta}$ is________.
In a cyclic quadrilateral $ABCD$, let $I,J$ be the midpoints of diagonals $AC, BD$ respectively and let $O$ be the center of the circle inscribed in $ABCD.$ Prove that $I, J$ and $O$ are collinear.
Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.
Points $ A$ and $ B$ are on the parabola $ y \equal{} 4x^2 \plus{} 7x \minus{} 1$, and the origin is the midpoint of $ \overline{AB}$. What is the length of $ \overline{AB}$?
$ \textbf{(A)}\ 2\sqrt5 \qquad
\textbf{(B)}\ 5\plus{}\frac{\sqrt2}{2} \qquad
\textbf{(C)}\ 5\plus{}\sqrt2 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 5\sqrt2$
In rectangular coordinate system, if $m(x^2+y^2+2y+1)=(x-2y+3)^2$ refers to an ellipse, then the range value of $m$ is
$\text{(A)}(0,1)\qquad\text{(B)}(1,+\infty)\qquad\text{(C)}(0,5)\qquad\text{(D)}(5,+\infty)$
There exists complex numbers $z=x+yi$ such that the point $(x, y)$ lies on the ellipse with equation $\frac{x^2}9+\frac{y^2}{16}=1$. If $\frac{z-1-i}{z-i}$ is real, compute $z$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 3)[/i]
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]
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic.
[i]David Yang.[/i]
Let $T$ be the triangle in the $xy$-plane with vertices $(0, 0)$, $(3, 0)$, and $\left(0, \frac32\right)$. Let $E$ be the ellipse inscribed in $T$ which meets each side of $T$ at its midpoint. Find the distance from the center of $E$ to $(0, 0)$.
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.
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$.
Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$.
(2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.
Let $p$ be a real constant. The parabola $y^2=-4px$ rolls without slipping around the parabola $y^2=4px$. Find the equation of the locus of the vertex of the rolling parabola.