Found problems: 2265
1971 IMO Longlists, 14
Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.
1997 AMC 8, 17
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have?
[asy]draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));
draw((3,0)--(3,3));
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);
draw((2.5,4)--(2.5,1),dashed);
label("$x$",(2.75,3.5),NNE);
label("$y$",(4.125,1.5),NNE);
[/asy]
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$
2011 Polish MO Finals, 2
In a tetrahedron $ABCD$, the four altitudes are concurrent at $H$. The line $DH$ intersects the plane $ABC$ at $P$ and the circumsphere of $ABCD$ at $Q\neq D$. Prove that $PQ=2HP$.
2022 Chile National Olympiad, 4
In a right circular cone of wood, the radius of the circumference $T$ of the base circle measures $10$ cm, while every point on said circumference is $20$ cm away. from the apex of the cone. A red ant and a termite are located at antipodal points of $T$. A black ant is located at the midpoint of the segment that joins the vertex with the position of the termite. If the red ant moves to the black ant's position by the shortest possible path, how far does it travel?
1953 Moscow Mathematical Olympiad, 243
Given a right circular cone and a point $A$. Find the set of vertices of cones equal to the given one, with axes parallel to that of the given one, and with $A$ inside them. We shall assume that the cone is infinite in one side.
2018 ITAMO, 1
$1.$A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.
Find the height of the bottle.
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$
2014 NIMO Problems, 1
Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.
[i]Proposed by Kevin Sun[/i]
1981 IMO Shortlist, 2
A sphere $S$ is tangent to the edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at the points $E,F,G,H$ respectively. The points $E,F,G,H$ are the vertices of a square. Prove that if the sphere is tangent to the edge $AC$, then it is also tangent to the edge $BD.$
2008 Purple Comet Problems, 9
Find the sum of all the integers $N > 1$ with the properties that the each prime factor of $N $ is either $2, 3,$ or $5,$ and $N$ is not divisible by any perfect cube greater than $1.$
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?
1968 IMO Shortlist, 3
Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.
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.
2008 iTest Tournament of Champions, 5
Two squares of side length $2$ are glued together along their boundary so that the four vertices of the first square are glued to the midpoints of the four sides of the other square, and vice versa. This gluing results in a convex polyhedron. If the square of the volume of this polyhedron is written in simplest form as $\tfrac{a+b\sqrt c}d$, what is the value of $a+b+c+d$?
ICMC 7, 4
Points $A, B, C,$ and $D{}$ lie on the surface of a sphere with diameter 1. Determine the maximum possible volume of tetrahedron $ABCD.$
[i]Proposed by Fredy Yip[/i]
Ukrainian TYM Qualifying - geometry, 2010.16
Points $A, B, C, D$ lie on the sphere of radius $1$. It is known that $AB\cdot AC\cdot AD\cdot BC\cdot BD\cdot CD=\frac{512}{27}$. Prove that $ABCD$ is a regular tetrahedron.
2016 Chile National Olympiad, 6
Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$, let $X_1$ denote its reflection with respect to $P_1$, and $X_2$ its reflection with respect to $P_2$. Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.
1977 Poland - Second Round, 4
A pyramid with a quadrangular base is given such that each pair of circles inscribed in adjacent faces has a common point. Prove that the touchpoints of these circles with the base of the pyramid lie on one circle.
2009 Sharygin Geometry Olympiad, 24
A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected to the edges of the base. Prove that these projections are concyclic.
1983 Bundeswettbewerb Mathematik, 1
The surface of a soccer ball is made up of black pentagons and white hexagons together. On the sides of each pentagon are nothing but hexagons, while on the sides of each border of hexagons alternately pentagons and hexagons. Determine from this information about the soccer ball , the number of its pentagons and its hexagons.
2009 IMC, 5
Let $n$ be a positive integer. An $n-\emph{simplex}$ in $\mathbb{R}^n$ is given by $n+1$ points $P_0, P_1,\cdots , P_n$, called its vertices, which do not all belong to the same hyperplane. For every $n$-simplex $\mathcal{S}$ we denote by $v(\mathcal{S})$ the volume of $\mathcal{S}$, and we write $C(\mathcal{S})$ for the center of the unique sphere containing all the vertices of $\mathcal{S}$.
Suppose that $P$ is a point inside an $n$-simplex $\mathcal{S}$. Let $\mathcal{S}_i$ be the $n$-simplex obtained from $\mathcal{S}$ by replacing its $i^{\text{th}}$ vertex by $P$. Prove that :
\[ \sum_{j=0}^{n}v(\mathcal{S}_j)C(\mathcal{S}_j)=v(\mathcal{S})C(\mathcal{S}) \]
1993 Irish Math Olympiad, 5
$ (a)$ The rectangle $ PQRS$ with $ PQ\equal{}l$ and $ QR\equal{}m$ $ (l,m \in \mathbb{N})$ is divided into $ lm$ unit squares. Prove that the diagonal $ PR$ intersects exactly $ l\plus{}m\minus{}d$ of these squares, where $ d\equal{}(l,m)$.
$ (b)$ A box with edge lengths $ l,m,n \in \mathbb{N}$ is divided into $ lmn$ unit cubes. How many of the cubes does a main diagonal of the box intersect?
2011 Tokyo Instutute Of Technology Entrance Examination, 2
For a positive real number $t$, in the coordiante space, consider 4 points $O(0,\ 0,\ 0),\ A(t,\ 0,\ 0),\ B(0,\ 1,\ 0),\ C(0,\ 0,\ 1)$.
Let $r$ be the radius of the sphere $P$ which is inscribed to all faces of the tetrahedron $OABC$.
When $t$ moves, find the maximum value of $\frac{\text{vol[P]}}{\text{vol[OABC]}}.$
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 $\phi$.
2010 Paenza, 5
In $4$-dimensional space, a set of $1 \times 2 \times 3 \times 4$ bricks is given. Decide whether it is possible to build boxes of the following sizes using these bricks:
[list]i) $2 \times 5 \times 7 \times 12$
ii) $5 \times 5 \times 10 \times 12$
iii) $6 \times 6 \times 6 \times 6$.[/list]