Found problems: 2265
1967 IMO Shortlist, 3
Which regular polygon can be obtained (and how) by cutting a cube with a plane ?
1991 AMC 8, 24
A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$
$\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$
2005 Abels Math Contest (Norwegian MO), 1b
In a pyramid, the base is a right-angled triangle with integer sides. The height of the pyramid is also integer. Show that the volume of the pyramid is even.
1999 Romania National Olympiad, 3
Let $ABCDA'B'C'D'$ be a right parallelepiped, $E$ and $F$ the projections of $A$ on the lines $A'D$, $A'C$, respectively, and $P, Q$ the projections of $B'$ on the lines $A'C'$ and $A'C$ Prove that
a) the planes $(AEF)$ and $(B'PQ)$ are parallel
b) the triangles $AEF$ and $B'PQ$ are similar.
1981 Polish MO Finals, 1
Two intersecting lines $a$ and $b$ are given in a plane. Consider all pairs of orthogonal planes $\alpha$, $\beta$ such that $a \subset \alpha$ and $b\subset \beta$. Prove that there is a circle such that every its point lies on the line $\alpha \cap \beta$ for some $\alpha$ and $\beta$.
2000 AIME Problems, 8
A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the liquid is $m-n\sqrt[3]{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the cube of any prime number. Find $m+n+p.$
1979 Kurschak Competition, 1
The base of a convex pyramid has an odd number of edges. The lateral edges of the pyramid are all equal, and the angles between neighbouring faces are all equal. Show that the base must be a regular polygon.
1970 Yugoslav Team Selection Test, Problem 3
If all edges of a non-planar quadrilateral tangent the faces of a sphere, prove that all of the points of tangency belong to a plane.
III Soros Olympiad 1996 - 97 (Russia), 10.2
Let $ABCD$ be a regular triangular pyramid with base $ABC$ (this means that $ABC$ is a regular triangle, and edges $AD$, $BD$ and $CD$ are equal) and plane angles at the opposite vertex equal to $a$. A plane parallel to $ABC$ intersects $AD$, $BD$ and $CD$, respectively, at points $A_1$, $B_1$ and $C_1$. The surface of the polyhedron $ABCA_1B_1C_1$ is cut along five edges: $A_1B_1$, $B_1C_1$, $C_1C$, $CA$ and $AB$, after which this surface is turned onto a plane. At what values of $a$ will the resulting scan necessarily cover itself?
2006 Kazakhstan National Olympiad, 6
In the tetrahedron $ ABCD $ from the vertex $ A $, the perpendiculars $ AB '$, $ AC' $ are drawn, $ AD '$ on planes dividing dihedral angles at edges $ CD $, $ BD $, $ BC $ in half. Prove that the plane $ (B'C'D ') $ is parallel to the plane $ (BCD) $.
2003 May Olympiad, 5
An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.
1984 Bulgaria National Olympiad, Problem 3
Points $P_1,P_2,\ldots,P_n,Q$ are given in space $(n\ge4)$, no four of which are in a plane. Prove that if for any three distinct points $P_\alpha,P_\beta,P_\gamma$ there is a point $P_\delta$ such that the tetrahedron $P_\alpha P_\beta P_\gamma P_\delta$ contains the point $Q$, then $n$ is an even number.
1973 Poland - Second Round, 5
Prove that if in the tetrahedron $ ABCD $ we have $ AB = CD $, $ AC = BD $, $ AD = BC $, then all faces of the tetrahedron are acute-angled triangles.
1989 Tournament Of Towns, (237) 1
Is it possible to choose a sphere, a triangular pyramid and a plane so that every plane, parallel to the chosen one, intersects the sphere and the pyramid in sections of equal area?
(Problem from Latvia)
2020 AMC 10, 19
As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
[asy]
import graph;
unitsize(4.5cm);
pair A = (0.082, 0.378);
pair B = (0.091, 0.649);
pair C = (0.249, 0.899);
pair D = (0.479, 0.939);
pair E = (0.758, 0.893);
pair F = (0.862, 0.658);
pair G = (0.924, 0.403);
pair H = (0.747, 0.194);
pair I = (0.526, 0.075);
pair J = (0.251, 0.170);
pair K = (0.568, 0.234);
pair L = (0.262, 0.449);
pair M = (0.373, 0.813);
pair N = (0.731, 0.813);
pair O = (0.851, 0.461);
path[] f;
f[0] = A--B--C--M--L--cycle;
f[1] = C--D--E--N--M--cycle;
f[2] = E--F--G--O--N--cycle;
f[3] = G--H--I--K--O--cycle;
f[4] = I--J--A--L--K--cycle;
f[5] = K--L--M--N--O--cycle;
draw(f[0]);
axialshade(f[1], white, M, gray(0.5), (C+2*D)/3);
draw(f[1]);
filldraw(f[2], gray);
filldraw(f[3], gray);
axialshade(f[4], white, L, gray(0.7), J);
draw(f[4]);
draw(f[5]);
[/asy]
$\textbf{(A) } 125 \qquad \textbf{(B) } 250 \qquad \textbf{(C) } 405 \qquad \textbf{(D) } 640 \qquad \textbf{(E) } 810$
2009 AMC 8, 22
How many whole numbers between 1 and 1000 do [b]not[/b] contain the digit 1?
$ \textbf{(A)}\ 512 \qquad \textbf{(B)}\ 648 \qquad \textbf{(C)}\ 720 \qquad \textbf{(D)}\ 728 \qquad \textbf{(E)}\ 800$
1996 Austrian-Polish Competition, 5
A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.
1987 IMO Longlists, 4
Let $a_1, a_2, a_3, b_1, b_2, b_3$ be positive real numbers. Prove that
\[(a_1b_2 + a_2b_1 + a_1b_3 + a_3b_1 + a_2b_3 + a_3b_2)^2 \geq 4(a_1a_2 + a_2a_3 + a_3a_1)(b_1b_2 + b_2b_3 + b_3b_1)\]
and show that the two sides of the inequality are equal if and only if $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3}.$
2002 Austrian-Polish Competition, 3
Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$. Prove that \[\frac{1}{3}\leq \frac{KS}{LS}\leq 3\]
1983 Federal Competition For Advanced Students, P2, 6
Planes $ \pi _1$ and $ \pi _2$ in Euclidean space $ \mathbb{R} ^3$ partition $ S\equal{}\mathbb{R} ^3 \setminus (\pi _1 \cup \pi _2)$ into several components. Show that for any cube in $ \mathbb{R} ^3$, at least one of the components of $ S$ meets at least three faces of the cube.
2010 AMC 12/AHSME, 9
Let $ n$ be the smallest positive integer such that $ n$ is divisible by $ 20$, $ n^2$ is a perfect cube, and $ n^3$ is a perfect square. What is the number of digits of $ n$?
$ \textbf{(A)}\ 3 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
Ukrainian TYM Qualifying - geometry, I.5
The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.
2006 Tournament of Towns, 7
An ant craws along a closed route along the edges of a dodecahedron, never going backwards.
Each edge of the route is passed exactly twice. Prove that one of the edges is passed both times in the same direction. (Dodecahedron has $12$ faces in the shape of pentagon, $30$ edges and $20$ vertices; each vertex emitting 3 edges). (8)
2013 Waseda University Entrance Examination, 5
Given a plane $P$ in space. For a figure $A$, call orthogonal projection the whole of points of intersection between the perpendicular drawn from each point in $A$ and $P$. Answer the following questions.
(1) Let a plane $Q$ intersects with the plane $P$ by angle $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$ between the planes, that is to say, the angles between two lines, is $\theta$, which can be generated by cuttng $P,\ Q$ by a plane which is perpendicular to the line of intersection of $P$ and $Q$. Find the maximum and minimum length of the orthogonal projection of the line segment in length 1 on $Q$ on to $P$..
(2) Consider $Q$ in (1). Find the area of the orthogonal projection of a equilateral triangle on $Q$ with side length 1 onto $P$.
(3) What's the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 on to $P'$, then find the max area of $T'$.
1949-56 Chisinau City MO, 61
Find the locus of the projections of a given point on all planes containing another point $B$.