Found problems: 2265
2014 AMC 10, 23
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
[asy]
real r=(3+sqrt(5))/2;
real s=sqrt(r);
real Brad=r;
real brad=1;
real Fht = 2*s;
import graph3;
import solids;
currentprojection=orthographic(1,0,.2);
currentlight=(10,10,5);
revolution sph=sphere((0,0,Fht/2),Fht/2);
//draw(surface(sph),green+white+opacity(0.5));
//triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));}
triple f(pair t) {
triple v0 = Brad*(cos(t.x),sin(t.x),0);
triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht);
return (v0 + t.y*(v1-v0));
}
triple g(pair t) {
return (t.y*cos(t.x),t.y*sin(t.x),0);
}
surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2);
surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2);
surface base = surface(g,(0,0),(2pi,Brad),80,2);
draw(sback,rgb(0,1,0));
draw(sfront,rgb(.3,1,.3));
draw(base,rgb(.4,1,.4));
draw(surface(sph),rgb(.3,1,.3));
[/asy]
$ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $
1980 Austrian-Polish Competition, 3
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
1986 IMO Shortlist, 21
Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that
\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]
where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.
1961 Poland - Second Round, 2
Prove that all the heights of a tetrahedron intersect at one point if and only if the sums of the squares of the opposite edges are equal.
2011 Federal Competition For Advanced Students, Part 1, 4
Inside or on the faces of a tetrahedron with five edges of length $2$ and one edge of lenght $1$, there is a point $P$ having distances $a, b, c, d$ to the four faces of the tetrahedron. Determine the locus of all points $P$ such that $a+b+c+d$ is minimal and the locus of all points $P$ such that $a+b+c+d$ is maximal.
1999 AMC 12/AHSME, 29
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 0.1\qquad
\textbf{(C)}\ 0.2\qquad
\textbf{(D)}\ 0.3\qquad
\textbf{(E)}\ 0.4$
2022 Assara - South Russian Girl's MO, 7
In a $7\times 7\times 7$ cube, the unit cubes are colored white, black and gray colors so that for any two colors the number of cubes of these two colors are different. In this case, $N$ parallel rows of $7$ cubes were found, each of which there are more white cubes than gray and than black. Likewise, there were $N$ parallel rows of $7$ cubes, each of which contained gray there are more cubes than white and than black, and there are also N parallel rows of $7$ cubes, each of which contains more black cubes than white ones and than gray ones. What is the largest $N$ for which this is possible?
1967 IMO Shortlist, 2
Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?
2003 All-Russian Olympiad Regional Round, 11.7
Given a tetrahedron $ABCD.$ The sphere $\omega$ inscribed in it touches the face $ABC$ at point $T$. Sphere $\omega' $ touches face $ABC$ at point $T'$ and extensions of faces $ABD$, $BCD$, $CAD$. Prove that the lines $AT$ and $AT'$ are symmetric wrt bisector of angle $\angle BAC$
2006 Austrian-Polish Competition, 10
Let $ABCDS$ be a (not neccessarily straight) pyramid with a rectangular base $ABCD$ and acute triangular faces $ABS,BCS,CDS,DAS$. We consider all cuboids which are inscribed inside the pyramid with its base being in the plane $ABCD$ and its upper vertexes are in the triangular faces (one in each).
Find the locus of the midpoints of these cuboids.
2011 USAMO, 3
In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$. Furthermore $AB=DE$, $BC=EF$, and $CD=FA$. Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.
2001 National High School Mathematics League, 9
The length of edge of cube $ABCD-A_1B_1C_1D_1$ is $1$, then the distance between lines $A_1C_1$ and $BD_1$ is________.
1949 Putnam, A1
Answer either (i) or (ii):
(i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line.
(II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?
1981 Bulgaria National Olympiad, Problem 6
Planes $\alpha,\beta,\gamma,\delta$ are tangent to the circumsphere of a tetrahedron $ABCD$ at points $A,B,C,D$, respectively. Line $p$ is the intersection of $\alpha$ and $\beta$, and line $q$ is the intersection of $\gamma$ and $\delta$. Prove that if lines $p$ and $CD$ meet, then lines $q$ and $AB$ lie on a plane.
1982 Miklós Schweitzer, 9
Suppose that $ K$ is a compact Hausdorff space and $ K\equal{} \cup_{n\equal{}0}^{\infty}A_n$, where $ A_n$ is metrizable and $ A_n \subset A_m$ for $ n<m$. Prove that $ K$ is metrizable.
[i]Z. Balogh[/i]
2014 Harvard-MIT Mathematics Tournament, 6
[5] Find all integers $n$ for which $\frac{n^3+8}{n^2-4}$ is an integer.
1971 IMO Longlists, 15
Let $ABCD$ be a convex quadrilateral whose diagonals intersect at $O$ at an angle $\theta$. Let us set $OA = a, OB = b, OC = c$, and $OD = d, c > a > 0$, and $d > b > 0.$
Show that if there exists a right circular cone with vertex $V$, with the properties:
[b](1)[/b] its axis passes through $O$, and
[b](2)[/b] its curved surface passes through $A,B,C$ and $D,$ then
\[OV^2=\frac{d^2b^2(c + a)^2 - c^2a^2(d + b)^2}{ca(d - b)^2 - db(c - a)^2}.\]
Show also that if $\frac{c+a}{d+b}$ lies between $\frac{ca}{db}$ and $\sqrt{\frac{ca}{db}},$ and $\frac{c-a}{d-b}=\frac{ca}{db},$ then for a suitable choice of $\theta$, a right circular cone exists with properties [b](1) [/b]and [b](2).[/b]
1983 AIME Problems, 5
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?
2006 IMO Shortlist, 7
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron [i]antipodal[/i] if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces.
[i]Proposed by Kei Irei, Japan[/i]
1998 Romania National Olympiad, 3
Let $ABCD$ be a tetrahedron and $A'$, $B'$, $C'$ be arbitrary points on the edges $[DA]$, $[DB]$, $[DC]$, respectively. One considers the points $P_c \in [AB]$, $P_a \in [BC]$, $P_b \in [AC]$ and $P'_c \in [A'B']$, $P'_a \in [B'C']$, $P'_b \in [A'C']$ such that
$$\frac{P_cA}{P_cB}= \frac{P'_cA'}{P'_cB'}=\frac{AA'}{BB'}\,\,\, , \,\,\,\frac{P_aB}{P_aC}= \frac{P'_aB'}{P'_aC'}=\frac{BB'}{CC'}\,\,\, , \,\,\, \frac{P_bC}{P_bA}= \frac{P'_bC'}{P'_bA'}=\frac{CC'}{AA'}$$
Prove that:
a) the lines $AP_a,$ $BP_b$, $CP_c$ have a common point $P$ and the lines $A'P'_a$, $B'P'_b$ , $C'P'_c$ have a common point $P'$
b) $\frac{PC}{PP_c}=\frac{P'C'}{P'P'_c} $
c) if $A', B', C'$ are variable points on the edges $[DA]$, $[DB]$, $[DC]$, then the line $PP'$ is always parallel to a fixed line.
1967 IMO Longlists, 26
Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?
2012 Online Math Open Problems, 50
In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be the largest possible value of the circumradius of face $ABC$. Given that $R$ can be expressed in the form $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Author: Alex Zhu[/i]
1959 Poland - Second Round, 6
From a point $ M $ on the surface of a sphere, three mutually perpendicular chords $ MA $, $ MB $, $ MC $ are drawn. Prove that the segment joining the point $ M $ with the center of the sphere intersects the plane of the triangle $ ABC $ at the center of gravity of this triangle.
1977 IMO Shortlist, 14
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:
(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$
(ii) some plane contains exactly three points from $E.$
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.