Found problems: 242
1999 Harvard-MIT Mathematics Tournament, 10
Pyramid $EARLY$ is placed in $(x,y,z)$ coordinates so that $E=(10,10,0),A=(10,-10,0)$, $R=(-10,-10,0)$, $L=(-10,10,0)$, and $Y=(0,0,10)$. Tunnels are drilled through the pyramid in such a way that one can move from $(x,y,z)$ to any of the $9$ points $(x,y,z-1)$, $(x\pm 1,y,z-1)$, $(x,y\pm 1, z-1)$, $(x\pm 1, y\pm 1, z-1)$. Sean starts at $Y$ and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability $\dfrac{1}{9}$. What is the probability that he ends up at the point $(8,9,0)$?
1999 National High School Mathematics League, 12
The bottom surface of triangular pyramid $S-ABC$ is a regular triangle. Projection of $A$ on plane $SBC$ is $H$, which is the orthocenter of $\triangle SBC$. If $H-AB-C=30^{\circ},SA=2\sqrt3$, then the volume of $S-ABC$ is________.
2004 Tournament Of Towns, 3
The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.
2004 Romania National Olympiad, 3
Let $ABCD A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ be a truncated regular pyramid in which $BC^{\prime}$ and $DA^{\prime}$ are perpendicular.
(a) Prove that $\measuredangle \left( AB^{\prime},DA^{\prime} \right) = 60^{\circ}$;
(b) If the projection of $B^{\prime}$ on $(ABC)$ is the center of the incircle of $ABC$, then prove that $d \left( CB^{\prime},AD^{\prime} \right) = \frac12 BC^{\prime}$.
[i]Mircea Fianu[/i]
1978 Bulgaria National Olympiad, Problem 6
The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$.
[i]Jordan Tabov[/i]
2016 Tournament Of Towns, 5
In convex hexagonal pyramid 11 edges are equal to 1.Find all possible values of 12th edge.
2021 Saint Petersburg Mathematical Olympiad, 3
In the pyramid $SA_1A_2 \cdots A_n$, all sides are equal. Let point $X_i$ be the midpoint of arc $A_iA_{i+1}$ in the circumcircle of $\triangle SA_iA_{i+1}$ for $1 \le i \le n$ with indices taken mod $n$. Prove that the circumcircles of $X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1$ have a common point.
2014-2015 SDML (Middle School), 8
Two regular square pyramids have all edges $12$ cm in length. The pyramids have parallel bases and those bases have parallel edges, and each pyramid has its apex at the center of the other pyramid's base. What is the total number of cubic centimeters in the volume of the solid of intersection of the two pyramids?
2007 Princeton University Math Competition, 9
There are four spheres each of radius $1$ whose centers form a triangular pyramid where each side has length $2$. There is a 5th sphere which touches all four other spheres and has radius less than $1$. What is its radius?
1993 Bulgaria National Olympiad, 3
it is given a polyhedral constructed from two regular pyramids with bases heptagons (a polygon with $7$ vertices) with common base $A_1A_2A_3A_4A_5A_6A_7$ and vertices respectively the points $B$ and $C$. The edges $BA_i , CA_i$ $(i = 1,...,7$), diagonals of the common base are painted in blue or red. Prove that there exists three vertices of the polyhedral given which forms a triangle with all sizes in the same color.
Champions Tournament Seniors - geometry, 2013.3
On the base of the $ABC$ of the triangular pyramid $SABC$ mark the point $M$ and through it were drawn lines parallel to the edges $SA, SB$ and $SC$, which intersect the side faces at the points $A1_, B_1$ and $C_1$, respectively. Prove that $\sqrt{MA_1}+ \sqrt{MB_1}+ \sqrt{MC_1}\le \sqrt{SA+SB+SC}$
2022 Sharygin Geometry Olympiad, 24
Let $OABCDEF$ be a hexagonal pyramid with base $ABCDEF$ circumscribed around a sphere $\omega$. The plane passing through the touching points of $\omega$ with faces $OFA$, $OAB$ and $ABCDEF$ meets $OA$ at point $A_1$, points $B_1$, $C_1$, $D_1$, $E_1$ and $F_1$ are defined similarly. Let $\ell$, $m$ and $n$ be the lines $A_1D_1$, $B_1E_1$ and $C_1F_1$ respectively. It is known that $\ell$ and $m$ are coplanar, also $m$ and $n$ are coplanar. Prove that $\ell$ and $n$ are coplanar.
2008 AMC 12/AHSME, 18
A pyramid has a square base $ ABCD$ and vertex $ E$. The area of square $ ABCD$ is $ 196$, and the areas of $ \triangle{ABE}$ and $ \triangle{CDE}$ are $ 105$ and $ 91$, respectively. What is the volume of the pyramid?
$ \textbf{(A)}\ 392 \qquad
\textbf{(B)}\ 196\sqrt{6} \qquad
\textbf{(C)}\ 392\sqrt2 \qquad
\textbf{(D)}\ 392\sqrt3 \qquad
\textbf{(E)}\ 784$
1986 Vietnam National Olympiad, 2
Let $ R$, $ r$ be respectively the circumradius and inradius of a regular $ 1986$-gonal pyramid. Prove that \[ \frac{R}{r}\ge 1\plus{}\frac{1}{\cos\frac{\pi}{1986}}\] and find the total area of the surface of the pyramid when the equality occurs.
1998 Hong kong National Olympiad, 2
The underside of a pyramid is a convex nonagon , paint all the diagonals of the nonagon and all the ridges of the pyramid into white and black , prove : there exists a triangle ,the colour of its three sides are the same . ( PS:the sides of the nonagon is not painted. )
2012 NIMO Problems, 6
A square is called [i]proper[/i] if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$.
[i]Proposed by Lewis Chen[/i]
1999 Romania National Olympiad, 4
Let $SABC$ be a regular pyramid, $O$ the center of basis $ABC$, and $M$ the midpoint of $[BC]$. If $N \in [SA]$ such that $SA = 25 \cdot NS$ and $SO \cap MN=\{P\}$, $AM=2\cdot SO$, prove that the planes $(ABP)$ and $(SBC)$ are perpendicular.
1998 Polish MO Finals, 3
$PABCDE$ is a pyramid with $ABCDE$ a convex pentagon. A plane meets the edges $PA, PB, PC, PD, PE$ in points $A', B', C', D', E'$ distinct from $A, B, C, D, E$ and $P$. For each of the quadrilaterals $ABB'A', BCC'B, CDD'C', DEE'D', EAA'E'$ take the intersection of the diagonals. Show that the five intersections are coplanar.
1998 Brazil Team Selection Test, Problem 1
Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.
Kyiv City MO 1984-93 - geometry, 1991.11.5
Lines that are drawn perpendicular to the faces of a triangular pyramid through the centers of the inscribed circles intersect at one point. Prove that the sums of the opposite edges of such a pyramid are equal to each other.
2001 Federal Math Competition of S&M, Problem 4
Parallelogram $ABCD$ is the base of a pyramid $SABCD$. Planes determined by triangles $ASC$ and $BSD$ are mutually perpendicular. Find the area of the side $ASD$, if areas of sides $ASB,BSC$ and $CSD$ are equal to $x,y$ and $z$, respectively.
1935 Moscow Mathematical Olympiad, 012
The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.
1994 All-Russian Olympiad Regional Round, 11.7
Points $A_1$, $B_1$ and $C_1$ are taken on the respective edges $SA$, $SB$, $SC$ of a regular triangular pyramid $SABC$ so that the planes $A_1B_1C_1$ and $ABC$ are parallel. Let $O$ be the center of the sphere passing through $A$, $B$, $C_1$ and $S$. Prove that the line $SO$ is perpendicular to the plane $A_1B_1C$.
1992 Polish MO Finals, 2
The base of a regular pyramid is a regular $2n$-gon $A_1A_2...A_{2n}$. A sphere passing through the top vertex $S$ of the pyramid cuts the edge $SA_i$ at $B_i$ (for $i = 1, 2, ... , 2n$). Show that $\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}$.
1998 Brazil Team Selection Test, Problem 1
Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.