Found problems: 8
VI Soros Olympiad 1999 - 2000 (Russia), 11.3
Three spheres $s_1$, $s_2$, $s_3$ intersect along one circle $\omega$. Let $A $be an arbitrary point lying on the circle $\omega$. Ray $AB$ intersects spheres $s_1$, $s_2$, $s_3$ at points $B_1$, $B_2$, $B_3$, respectively, ray $AC$ intersects spheres $s_1$, $s_2$, $s_3$ at points $C_1$, $C_2$, $C_3$, respectively ($B_i \ne A_i$, $C_i \ne A_i$, $i=1,2,3$). It is known that $B_2$ is the midpoint of the segment $B_1B_3$. Prove that $C_2$ is the midpoint of the segment $C_1C_3$.
2013 Chile National Olympiad, 5
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]
2022 239 Open Mathematical Olympiad, 2
Five edges of a tetrahedron are tangent to a sphere. Prove that there are another five edges from this tetrahedron that are also tangent to a $($not necessarily the same$)$ sphere.
2018 Iran MO (1st Round), 25
Astrophysicists have discovered a minor planet of radius $30$ kilometers whose surface is completely covered in water. A spherical meteor hits this planet and is submerged in the water. This incidence causes an increase of $1$ centimeters to the height of the water on this planet. What is the radius of the meteor in meters?
2000 All-Russian Olympiad Regional Round, 11.2
The height and radius of the base of the cylinder are equal to $1$. What is the smallest number of balls of radius $1$ that can cover the entire cylinder?
2004 All-Russian Olympiad Regional Round, 11.8
Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.
1985 Bundeswettbewerb Mathematik, 2
The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$
1994 Chile National Olympiad, 4
Consider a box of dimensions $10$ cm $\times 16$ cm $\times 1$ cm. Determine the maximum number of balls of diameter $ 1$ cm that the box can contain.