Found problems: 2265
1955 AMC 12/AHSME, 43
The pairs of values of $ x$ and $ y$ that are the common solutions of the equations $ y\equal{}(x\plus{}1)^2$ and $ xy\plus{}y\equal{}1$ are:
$ \textbf{(A)}\ \text{3 real pairs} \qquad
\textbf{(B)}\ \text{4 real pairs} \qquad
\textbf{(C)}\ \text{4 imaginary pairs} \\
\textbf{(D)}\ \text{2 real and 2 imaginary pairs} \qquad
\textbf{(E)}\ \text{1 real and 2 imaginary pairs}$
1986 Polish MO Finals, 2
Find the maximum possible volume of a tetrahedron which has three faces with area $1$.
1995 Brazil National Olympiad, 4
A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?
2002 Croatia National Olympiad, Problem 3
Points $E$ and $F$ are taken on the diagonals $AB_1$ and $CA_1$ of the lateral faces $ABB_1A_1$ and $CAA_1C_1$ of a triangular prism $ABCA_1B_1C_1$ so that $EF\parallel BC_1$. Find the ratio of the lengths of $EF$ and $BC_1$.
2012 Switzerland - Final Round, 8
Consider a cube and two of its vertices $A$ and $B$, which are the endpoints of a face diagonal. A [i]path [/i] is a sequence of cube angles, each step of one angle along a cube edge is walked to one of the three adjacent angles. Let $a$ be the number of paths of length $2012$ that starts at point $A$ and ends at $A$ and let b be the number of ways of length $2012$ that starts in $A$ and ends in $B$. Decide which of the two numbers $a$ and $b$ is the larger.
2022 Sharygin Geometry Olympiad, 10.8
Let $ABCA'B'C'$ be a centrosymmetric octahedron (vertices $A$ and $A'$, $B$ and $B'$, $C$ and $C'$ are opposite) such that the sums of four planar angles equal $240^o$ for each vertex. The Torricelli points $T_1$ and $T_2$ of triangles $ABC$ and $A'BC$ are marked. Prove that the distances from $T_1$ and $T_2$ to $BC$ are equal.
1948 Putnam, B6
Answer wither (i) or (ii):
(i) Let $V, V_1 , V_2$ and $V_3$ denote four vertices of a cube such that $V_1 , V_2 , V_3 $ are adjacent to $V.$ Project the cube orthogonally on a plane of which the points are marked with complex numbers. Let the projection of $V$ fall in the origin and the projections of $V_1 , V_2 , V_3 $ in points marked with the complex numbers $z_1 , z_2 , z_3$, respectively. Show that $z_{1}^{2} +z_{2}^{2} +z_{3}^{2}=0.$
(ii) Let $(a_{ij})$ be a matrix such that
$$|a_{ii}| > |a_{i1}| + |a_{i2}|+\ldots +|a_{i i-1}|+ |a_{i i+1}| +\ldots +|a_{in}|$$
for all $i.$ Show that the determinant is not equal to $0.$
1979 IMO Shortlist, 4
We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.
2019 AMC 12/AHSME, 18
Square pyramid $ABCDE$ has base $ABCD,$ which measures $3$ cm on a side, and altitude $\overline{AE}$ perpendicular to the base$,$ which measures $6$ cm. Point $P$ lies on $\overline{BE},$ one third of the way from $B$ to $E;$ point $Q$ lies on $\overline{DE},$ one third of the way from $D$ to $E;$ and point $R$ lies on $\overline{CE},$ two thirds of the way from $C$ to $E.$ What is the area, in square centimeters, of $\triangle PQR?$
$\textbf{(A) } \frac{3\sqrt2}{2} \qquad\textbf{(B) } \frac{3\sqrt3}{2} \qquad\textbf{(C) } 2\sqrt2 \qquad\textbf{(D) } 2\sqrt3 \qquad\textbf{(E) } 3\sqrt2$
2001 AIME Problems, 12
A sphere is inscribed in the tetrahedron whose vertices are $A=(6,0,0), B=(0,4,0), C=(0,0,2),$ and $D=(0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
1976 Poland - Second Round, 3
We consider a spherical bowl without any great circle. The distance between points $A$ and $B$ on such a bowl is defined as the length of the arc of the great circle of the sphere with ends at points $A$ and $B$, which is contained in the bowl. Prove that there is no isometry mapping this bowl to a subset of the plane.
Attention. A spherical bowl is each of the two parts into which the surface of the sphere is divided by a plane intersecting the sphere.
PEN P Problems, 2
Show that each integer $n$ can be written as the sum of five perfect cubes (not necessarily positive).
1977 IMO Longlists, 44
Let $E$ be a finite set of points in space such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that $E$ contains the vertices of a tetrahedron $T = ABCD$ such that $T \cap E = \{A,B,C,D\}$ (including interior points of $T$ ) and such that the projection of $A$ onto the plane $BCD$ is inside a triangle that is similar to the triangle $BCD$ and whose sides have midpoints $B,C,D.$
1972 AMC 12/AHSME, 5
From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are
\[ \begin{array}{rlrlrlrl} \hbox {(A)}& 3^{1/3},\ 2^{1/2} \quad & \hbox {(B)}& 3^{1/3},\ 8^{1/8} \quad & \hbox {(C)}& 3^{1/3},\ 9^{1/9} \quad & \hbox {(D)}& 8^{1/8},\ 9^{1/9} \\ \hbox {(E)}& \multicolumn{3}{l}{\hbox{None of these}} \end{array} \]
2020 AMC 8 -, 9
Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?
[asy]
/*
Created by SirCalcsALot and sonone
Code modfied from https://artofproblemsolving.com/community/c3114h2152994_the_old__aops_logo_with_asymptote
*/
import three;
currentprojection=orthographic(1.75,7,2);
//++++ edit colors, names are self-explainatory ++++
//pen top=rgb(27/255, 135/255, 212/255);
//pen right=rgb(254/255,245/255,182/255);
//pen left=rgb(153/255,200/255,99/255);
pen top = rgb(170/255, 170/255, 170/255);
pen left = rgb(81/255, 81/255, 81/255);
pen right = rgb(165/255, 165/255, 165/255);
pen edges=black;
int max_side = 4;
//+++++++++++++++++++++++++++++++++++++++
path3 leftface=(1,0,0)--(1,1,0)--(1,1,1)--(1,0,1)--cycle;
path3 rightface=(0,1,0)--(1,1,0)--(1,1,1)--(0,1,1)--cycle;
path3 topface=(0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle;
for(int i=0; i<max_side; ++i){
for(int j=0; j<max_side; ++j){
draw(shift(i,j,-1)*surface(topface),top);
draw(shift(i,j,-1)*topface,edges);
draw(shift(i,-1,j)*surface(rightface),right);
draw(shift(i,-1,j)*rightface,edges);
draw(shift(-1,j,i)*surface(leftface),left);
draw(shift(-1,j,i)*leftface,edges);
}
}
picture CUBE;
draw(CUBE,surface(leftface),left,nolight);
draw(CUBE,surface(rightface),right,nolight);
draw(CUBE,surface(topface),top,nolight);
draw(CUBE,topface,edges);
draw(CUBE,leftface,edges);
draw(CUBE,rightface,edges);
// begin made by SirCalcsALot
int[][] heights = {{4,4,4,4},{4,4,4,4},{4,4,4,4},{4,4,4,4}};
for (int i = 0; i < max_side; ++i) {
for (int j = 0; j < max_side; ++j) {
for (int k = 0; k < min(heights[i][j], max_side); ++k) {
add(shift(i,j,k)*CUBE);
}
}
}
[/asy]
$\textbf{(A)}\ 12\qquad~~\textbf{(B)}\ 16\qquad~~\textbf{(C)}\ 18\qquad~~\textbf{(D)}\ 20\qquad~~\textbf{(E)}\ 24$
2006 AIME Problems, 14
A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)
2007 AMC 10, 21
A sphere is inscribed in a cube that has a surface area of $ 24$ square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$
1972 IMO Shortlist, 5
Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if
\[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\]
2011 USAMTS Problems, 4
Renata the robot packs boxes in a warehouse. Each box is a cube of side length $1$ foot. The warehouse floor is a square, $12$ feet on each side, and is divided into a $12$-by-$12$ grid of square tiles $1$ foot on a side. Each tile can either support one box or be empty. The warehouse has exactly one door, which opens onto one of the corner tiles.
Renata fits on a tile and can roll between tiles that share a side. To access a box, Renata must be able to roll along a path of empty tiles starting at the door and ending at a tile sharing a side with that box.
[list=a]
[*]Show how Renata can pack $91$ boxes into the warehouse and still be able to access any box.
[*]Show that Renata [b]cannot[/b] pack $95$ boxes into the warehouse and still be able to access any box.[/list]
1986 IMO Shortlist, 19
A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$
1976 Bundeswettbewerb Mathematik, 4
Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !
2022 Purple Comet Problems, 20
Let $\mathcal{S}$ be a sphere with radius $2.$ There are $8$ congruent spheres whose centers are at the vertices of a cube, each has radius $x,$ each is externally tangent to $3$ of the other $7$ spheres with radius $x,$ and each is internally tangent to $\mathcal{S}.$ There is a sphere with radius $y$ that is the smallest sphere internally tangent to $\mathcal{S}$ and externally tangent to $4$ spheres with radius $x.$ There is a sphere with radius $z$ centered at the center of $\mathcal{S}$ that is externally tangent to all $8$ of the spheres with radius $x.$ Find $18x + 5y + 4z.$
1965 IMO Shortlist, 3
Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.
2005 Harvard-MIT Mathematics Tournament, 7
Let $ABCD$ be a tetrahedron such that edges $AB$, $AC$, and $AD$ are mutually perpendicular. Let the areas of triangles $ABC$, $ACD$, and $ADB$ be denoted by $x$, $y$, and $z$, respectively. In terms of $x$, $y$, and $z$, find the area of triangle $BCD$.
1996 Tournament Of Towns, (514) 1
Consider three edges $a, b, c$ of a cube such that no two of these edges lie in one plane. Find the locus of points inside the cube which are equidistant from $a$, $b$ and $c$.
(V Proizvolov,)