Found problems: 2265
1978 USAMO, 4
(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
2010 Oral Moscow Geometry Olympiad, 5
All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.
1947 Moscow Mathematical Olympiad, 140
Prove that if the four faces of a tetrahedron are of the same area they are equal.
1955 Polish MO Finals, 6
Through points $ A $ and $ B $ two oblique lines $ m $ and $ n $ are drawn perpendicular to the line $ AB $. On line $ m $ the point $ C $ (different from $ A $) is taken, and on line $ n $ the point $ D $ (different from $ B $) is taken. Given the lengths of segments $ AB = d $ and $ CD = l $ and the angle $ \varphi $ formed by the oblique lines $ m $ and $ n $, calculate the radius of the surface of the sphere passing through the points $ A $, $ B $, $ C $, $ D $.
1962 Czech and Slovak Olympiad III A, 3
Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.
2008 Putnam, B3
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1?
2016 Nigerian Senior MO Round 2, Problem 5
A solid pyramid $TABCD$, with a quadrilateral base $ABCD$ is to be coloured on each of the five faces such that no two faces with a common edge will have the same colour. If five different colours are available, what is the number of ways to colour the pyramid?
2006 All-Russian Olympiad, 6
Consider a tetrahedron $SABC$. The incircle of the triangle $ABC$ has the center $I$ and touches its sides $BC$, $CA$, $AB$ at the points $E$, $F$, $D$, respectively. Let $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the points on the segments $SA$, $SB$, $SC$ such that $AA^{\prime}=AD$, $BB^{\prime}=BE$, $CC^{\prime}=CF$, and let $S^{\prime}$ be the point diametrically opposite to the point $S$ on the circumsphere of the tetrahedron $SABC$. Assume that the line $SI$ is an altitude of the tetrahedron $SABC$. Show that $S^{\prime}A^{\prime}=S^{\prime}B^{\prime}=S^{\prime}C^{\prime}$.
2017 AMC 12/AHSME, 14
An ice-cream novelty item consists of a cup in the shape of a $4$-inch-tall frustum of a right circular cone, with a $2$-inch-diameter base at the bottom and a $4$-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height $4$ inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
$\textbf{(A)}\ 8\pi\qquad\textbf{(B)}\ \frac{28\pi}{3}\qquad\textbf{(C)}\ 12\pi\qquad\textbf{(D)}\ 14\pi\qquad\textbf{(E)}\ \frac{44\pi}{3}$
1979 IMO Shortlist, 25
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
1998 Harvard-MIT Mathematics Tournament, 2
A cube with sides 1m in length is filled with water, and has a tiny hole through which the water drains into a cylinder of radius $1\text{ m}$. If the water level in the cube is falling at a rate of $1 \text{ cm/s}$, at what rate is the water level in the cylinder rising?
2021 Bundeswettbewerb Mathematik, 1
A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut.
What is the smallest possible volume of the largest of the three cuboids?
1990 IMO Longlists, 63
Let $ P$ be a point inside a regular tetrahedron $ T$ of unit volume. The four planes passing through $ P$ and parallel to the faces of $ T$ partition $ T$ into 14 pieces. Let $ f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $ f(P)$ as $ P$ varies over $ T.$
1983 IMO Longlists, 71
Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$
1994 Dutch Mathematical Olympiad, 3
$ (a)$ Prove that every multiple of $ 6$ can be written as a sum of four cubes.
$ (b)$ Prove that every integer can be written as a sum of five cubes.
1964 Putnam, B4
Into how many regions do $n$ great circles, no three of which meet at a point, divide a sphere?
2020 Adygea Teachers' Geometry Olympiad, 3
Is it true that of the four heights of an arbitrary tetrahedron, three can be selected from which a triangle can be made?
2014 Moldova Team Selection Test, 1
Prove that there do not exist $4$ points in the plane such that the distances between any pair of them is an odd integer.
2016 CHMMC (Fall), 5
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that the entire cube is one solid color can be written as $\frac{1}{2^n}$ for some positive integer $n$. Find $n$.
1990 Iran MO (2nd round), 1
[b](a)[/b] Consider the set of all triangles $ABC$ which are inscribed in a circle with radius $R.$ When is $AB^2+BC^2+CA^2$ maximum? Find this maximum.
[b](b)[/b] Consider the set of all tetragonals $ABCD$ which are inscribed in a sphere with radius $R.$ When is the sum of squares of the six edges of $ABCD$ maximum? Find this maximum, and in this case prove that all of the edges are equal.
PEN H Problems, 13
Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]
2017 Yasinsky Geometry Olympiad, 2
In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.
2021 AIME Problems, 6
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$
1979 IMO Longlists, 68
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
2013-2014 SDML (High School), 2
A semicircle is joined to the side of a triangle, with the common edge removed. Sixteen points are arranged on the figure, as shown below. How many non-degenerate triangles can be drawn from the given points?
[asy]
draw((0,-2)--arc((0,0),1,0,180)--cycle);
dot((-0.8775,-0.245));
dot((-0.735,-0.53));
dot((-0.5305,-0.939));
dot((-0.3875,-1.225));
dot((-0.2365,-1.527));
dot((0.155,-1.69));
dot((0.306,-1.388));
dot((0.4,-1.2));
dot((0.551,-0.898));
dot((0.837,-0.326));
dot(dir(25));
dot(dir(50));
dot(dir(65));
dot(dir(100));
dot(dir(115));
dot(dir(140));
[/asy]