Found problems: 2265
2018 CCA Math Bonanza, T8
A rectangular prism with positive integer side lengths formed by stacking unit cubes is called [i]bipartisan[/i] if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there?
[i]2018 CCA Math Bonanza Team Round #8[/i]
2014 Math Prize for Girls Olympiad, 3
Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.
1969 IMO, 3
For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.
2007 Hungary-Israel Binational, 2
Given is an ellipse $ e$ in the plane. Find the locus of all points $ P$ in space such that the cone of apex $ P$ and directrix $ e$ is a right circular cone.
2011 Purple Comet Problems, 30
Four congruent spheres are stacked so that each is tangent to the other three. A larger sphere, $R$, contains the four congruent spheres so that all four are internally tangent to $R$. A smaller sphere, $S$, sits in the space between the four congruent spheres so that all four are externally tangent to $S$. The ratio of the surface area of $R$ to the surface area of $S$ can be written $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
2003 Tournament Of Towns, 6
Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.
2001 Polish MO Finals, 2
Given a regular tetrahedron $ABCD$ with edge length $1$ and a point $P$ inside it.
What is the maximum value of $\left|PA\right|+\left|PB\right|+\left|PC\right|+\left|PD\right|$.
2024-25 IOQM India, 7
Determine the sum of all possible surface area of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.
1990 AMC 12/AHSME, 25
Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere?
$ \textbf{(A)}\ 1-\frac{\sqrt{3}}{2} \qquad\textbf{(B)}\ \frac{2\sqrt{3}-3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{2}}{6} \qquad\textbf{(D)}\ \frac{1}{4} \qquad\textbf{(E)}\ \frac{\sqrt{3}(2-\sqrt{2})}{4} $
2018 PUMaC Geometry B, 2
Let a right cone of the base radius $r=3$ and height greater than $6$ be inscribed in a sphere of radius $R=6$. The volume of the cone can be expressed as $\pi(a\sqrt{b}+c)$, where $b$ is square free. Find $a+b+c$.
2009 National Olympiad First Round, 22
$ (a_n)_{n \equal{} 0}^\infty$ is a sequence on integers. For every $ n \ge 0$, $ a_{n \plus{} 1} \equal{} a_n^3 \plus{} a_n^2$. The number of distinct residues of $ a_i$ in $ \pmod {11}$ can be at most?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$
1976 Czech and Slovak Olympiad III A, 6
Consider two non-parallel half-planes $\pi,\pi'$ with the common boundary line $p.$ Four different points $A,B,C,D$ are given in the half-plane $\pi.$ Similarly, four points $A',B',C',D'\in\pi'$ are given such that $AA'\parallel BB'\parallel CC'\parallel DD'$. Moreover, none of these points lie on $p$ and the points $A,B,C,D'$ form a tetrahedron. Show that the points $A',B',C',D$ also form a tetrahedron with the same volume as $ABCD'.$
2022 AMC 8 -, 24
The figure below shows a polygon $ABCDEFGH$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$. What is the volume of the prism?
[asy]
// djmathman diagram
unitsize(1cm);
defaultpen(linewidth(0.7)+fontsize(11));
real r = 2, s = 2.5, theta = 14;
pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta);
pair N = (B+G)/2, J = N + s/2 * dir(180+theta);
pair E = F + r * dir(- 45 - theta/2), D = I+E-F;
pair H = J + r * dir(135 + theta/2), A = B+H-J;
draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B));
draw(J--B--G^^C--F--I,linetype ("4 4"));
dot("$A$",A,N);
dot("$B$",B,1.2*N);
dot("$C$",C,N);
dot("$D$",D,dir(0));
dot("$E$",E,S);
dot("$F$",F,1.5*S);
dot("$G$",G,S);
dot("$H$",H,W);
dot("$I$",I,NE);
dot("$J$",J,1.5*S);
[/asy]
$\textbf{(A)} ~112\qquad\textbf{(B)} ~128\qquad\textbf{(C)} ~192\qquad\textbf{(D)} ~240\qquad\textbf{(E)} ~288\qquad$
2008 AMC 8, 24
Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
$\textbf{(A)}\ \frac{1}{10}\qquad
\textbf{(B)}\ \frac{1}{6}\qquad
\textbf{(C)}\ \frac{11}{60}\qquad
\textbf{(D)}\ \frac{1}{5}\qquad
\textbf{(E)}\ \frac{7}{30}$
2005 Sharygin Geometry Olympiad, 24
A triangle is given, all the angles of which are smaller than $\phi$, where $\phi <2\pi / 3$. Prove that in space there is a point from which all sides of the triangle are visible at an angle $\phi$.
1968 Putnam, A4
Let $S^{2}\subset \mathbb{R}^{3}$ be the unit sphere. Show that for any $n$ points on $ S^{2}$, the sum of the squares of the $\frac{n(n-1)}{2}$ distances between them is at most $n^{2}$.
2014 AMC 12/AHSME, 17
A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$?
[asy]
import graph3;
import solids;
real h=2+2*sqrt(7);
currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2));
currentlight=light(4,-4,4);
draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0));
draw(shift((1,3,1))*unitsphere,gray(0.85));
draw(shift((3,3,1))*unitsphere,gray(0.85));
draw(shift((3,1,1))*unitsphere,gray(0.85));
draw(shift((1,1,1))*unitsphere,gray(0.85));
draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85));
draw(shift((1,3,h-1))*unitsphere,gray(0.85));
draw(shift((3,3,h-1))*unitsphere,gray(0.85));
draw(shift((3,1,h-1))*unitsphere,gray(0.85));
draw(shift((1,1,h-1))*unitsphere,gray(0.85));
draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h));
[/asy]
$\textbf{(A) }2+2\sqrt 7\qquad
\textbf{(B) }3+2\sqrt 5\qquad
\textbf{(C) }4+2\sqrt 7\qquad
\textbf{(D) }4\sqrt 5\qquad
\textbf{(E) }4\sqrt 7\qquad$
2019 Oral Moscow Geometry Olympiad, 6
The sum of the cosines of the flat angles of the trihedral angle is $-1$. Find the sum of its dihedral angles.
Durer Math Competition CD Finals - geometry, 2010.D5
Prove that we can put in any arbitrary triangle with sidelengths $a,b,c$ such that $0\le a,b,c \le \sqrt2$ into a unit cube.
1980 IMO, 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°.
1996 AMC 12/AHSME, 10
How many line segments have both their endpoints located at the vertices of a given cube?
$\text{(A)}\ 12 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 24 \qquad \text{(D)}\ 28\qquad \text{(E)}\ 56$
1998 AMC 12/AHSME, 27
A $ 9\times9\times9$ cube is composed of twenty-seven $ 3\times3\times3$ cubes. The big cube is 'tunneled' as follows: First, the six $ 3\times3\times3$ cubes which make up the center of each face as well as the center of $ 3\times3\times3$ cube are removed. Second, each of the twenty remaining $ 3\times3\times3$ cubes is diminished in the same way. That is, the central facial unit cubes as well as each center cube are removed.
[asy]
import three;
size(4.5cm);
triple eye = (6, 9, 5);
currentprojection = perspective(eye);
real eps = 0.001;
for(int i = 0; i < 3; ++i){
for(int j = 0; j < 3; ++j){
for(int k = 0; k < 3; ++k){
if(i == 1 && j == 1) continue;
if(j == 1 && k == 1) continue;
if(k == 1 && i == 1) continue;
draw(shift(i, j, k) * scale(1 - eps, 1 - eps, 1 - eps) * unitcube, gray(0.9), nolight);
draw(shift(i, j, k) * (X--(X + Y)--Y--(Y+Z)--Z--(Z + X)--cycle));
draw(shift(i, j, k) * (X + Y + Z--X + Y));
draw(shift(i, j, k) * (X + Y + Z--Y + Z));
draw(shift(i, j, k) * (X + Y + Z--Z + X));
}
}
}
[/asy]
The surface area of the final figure is
$ \textbf{(A)}\ 384\qquad
\textbf{(B)}\ 729\qquad
\textbf{(C)}\ 864\qquad
\textbf{(D)}\ 1024\qquad
\textbf{(E)}\ 1056$
1966 Czech and Slovak Olympiad III A, 4
Two triangles $ABC,ABD$ (with the common side $c=AB$) are given in space. Triangle $ABC$ is right with hypotenuse $AB$, $ABD$ is equilateral. Denote $\varphi$ the dihedral angle between planes $ABC,ABD$.
1) Determine the length of $CD$ in terms of $a=BC,b=CA,c$ and $\varphi$.
2) Determine all values of $\varphi$ such that the tetrahedron $ABCD$ has four sides of the same length.
Today's calculation of integrals, 768
Let $r$ be a real such that $0<r\leq 1$. Denote by $V(r)$ the volume of the solid formed by all points of $(x,\ y,\ z)$ satisfying
\[x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2\]
in $xyz$-space.
(1) Find $V(r)$.
(2) Find $\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.$
(3) Find $\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.$
2018 VJIMC, 3
In $\mathbb{R}^3$ some $n$ points are coloured. In every step, if four coloured points lie on the same line, Vojtěch can colour any other point on this line. He observes that he can colour any point $P \in \mathbb{R}^3$ in a finite number of steps (possibly depending on $P$). Find the minimal value of $n$ for which this could happen.