Found problems: 242
The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.
The inscribed and exscribed sphere of a triangular pyramid $ABCD$ touch her face $BCD$ at different points $X$ and $Y$. Prove that the triangle $AXY$ is obtuse triangle.
In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid.
[asy]
draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0));
draw((2.5,2.5)--(4,0)--(5.5,2.5)--(8,4)--(5.5,5.5)--(4,8)--(2.5,5.5)--(0,4)--(2.5,2.5)--(5.5,2.5)--(5.5,5.5)--(2.5,5.5)--(2.5,2.5));
[/asy]
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.
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?
A given truncated pyramid has triangular bases. The areas of the bases are $B_1$ and $B_2$ and the area of the surface is $S$. Prove that if there exists a plane parallel to the bases whose intersection divides the pyramid to two truncated pyramids in which may be inscribed by spheres then
$$S=(\sqrt{B_1}+\sqrt{B_2})(\sqrt[4]{B_1}+\sqrt[4]{B_2})^2$$
[i]G. Gantchev[/i]
The height $SO$ of a regular quadrangular pyramid $SABCD$ forms an angle $60^o$ with a side edge , the volume of this pyramid is equal to $18$ cm$^3$ . The vertex of the second regular quadrangular pyramid is at point $S$, the center of the base is at point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids. (The common part of the pyramids is the set of all such points in space that lie inside or on the surface of both pyramids).
A pyramid with a square base, and all its edges of length $2$, is joined to a regular tetrahedron, whose edges are also of length $2$, by gluing together two of the triangular faces. Find the sum of the lengths of the edges of the resulting solid.
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.
Consider the regular pyramid $VABCD$ with the vertex in $V$ which measures the angle formed by two opposite lateral edges is $45^o$. The points $M,N,P$ are respectively, the projections of the point $A$ on the line $VC$, the symmetric of the point $M$ with respect to the plane $(VBD)$ and the symmetric of the point $N$ with respect to $O$. ($O$ is the center of the base of the pyramid.)
a) Show that the polyhedron $MDNBP$ is a regular pyramid.
b) Determine the measure of the angle between the line $ND$ and the plane $(ABC) $
Let there be given a pyramid $SABCD$ whose base $ABCD$ is a parallelogram. Let $N$ be the midpoint of $BC$. A plane $\lambda$ intersects the lines $SC,SA,AB$ at points $P,Q,R$ respectively such that $\overline{CP}/\overline{CS}=\overline{SQ}/\overline{SA}=\overline{AR}/\overline{AB}$. A point $M$ on the line $SD$ is such that the line $MN$ is parallel to $\lambda$. Show that the locus of points $M$, when $\lambda$ takes all possible positions, is a segment of the length $\frac{\sqrt5}2SD$.
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.
In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true
$$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$
where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$.
[i]V. Petkov[/i]
Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$.
Find max value capacity(volume) of the pyramid $SABC$.
For a given integer $n\geq2$, a pyramid of height $n$ if defined as a collection of $1^2+2^2+\dots+n^2$ stone cubes of equal size stacked in $n$ layers such that the cubes in the $k$-th layer form a square with sidelength $n+1-k$ and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?
A square piece of paper has sides of length $ 100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance $ \sqrt {17}$ from the corner, and they meet on the diagonal at an angle of $ 60^\circ$ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form $ \sqrt [n]{m}$, where $ m$ and $ n$ are positive integers, $ m < 1000$, and $ m$ is not divisible by the $ n$th power of any prime. Find $ m \plus{} n$.
[asy]import math;
unitsize(5mm);
defaultpen(fontsize(9pt)+Helvetica()+linewidth(0.7));
pair O=(0,0);
pair A=(0,sqrt(17));
pair B=(sqrt(17),0);
pair C=shift(sqrt(17),0)*(sqrt(34)*dir(75));
pair D=(xpart(C),8);
pair E=(8,ypart(C));
draw(O--(0,8));
draw(O--(8,0));
draw(O--C);
draw(A--C--B);
draw(D--C--E);
label("$\sqrt{17}$",(0,2),W);
label("$\sqrt{17}$",(2,0),S);
label("cut",midpoint(A--C),NNW);
label("cut",midpoint(B--C),ESE);
label("fold",midpoint(C--D),W);
label("fold",midpoint(C--E),S);
label("$30^\circ$",shift(-0.6,-0.6)*C,WSW);
label("$30^\circ$",shift(-1.2,-1.2)*C,SSE);[/asy]
Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$.
[i]Author: Alex Zhu[/i]
Let $ABCDS$ be a (not neccessarily straight) pyramid with a rectangular base $ABCD$ and acute triangular faces $ABS,BCS,CDS,DAS$. We consider all cuboids which are inscribed inside the pyramid with its base being in the plane $ABCD$ and its upper vertexes are in the triangular faces (one in each).
Find the locus of the midpoints of these cuboids.
An arbitrary point $M$ is taken on the basis of a regular triangular pyramid. Let $K$, $L$, $N$ be the projections of $M$ onto the lateral faces of this pyramid, and $P$ be the intersection point of the medians of the triangle $KLN$. Prove that the straight line passing through the points $M$ and$ P$ intersects the height of the pyramid (or its extension). Let us denote this intersection point by $E$. Find $MP: PE$ if the dihedral angles at the base of the pyramid are equal to $a$.
A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal.
(M. Volchkevich)
P6. It is given the integer $Y$ with
$Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.$
Determine the sum of all the digits of such $Y$. (It is implied that $Y$ is written with a decimal representation.)
P7. Three groups of lines divides a plane into $D$ regions. Every pair of lines in the same group are parallel. Let $x, y$ and $z$ respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is ....
P8. It is known a frustum $ABCD.EFGH$ where $ABCD$ and $EFGH$ are squares with both planes being parallel. The length of the sides of $ABCD$ and $EFGH$ respectively are $6a$ and $3a$, and the height of the frustum is $3t$. Points $M$ and $N$ respectively are intersections of the diagonals of $ABCD$ and $EFGH$ and the line $MN$ is perpendicular to the plane $EFGH$. Construct the pyramids $M.EFGH$ and $N.ABCD$ and calculate the volume of the 3D figure which is the intersection of pyramids $N.ABCD$ and $M.EFGH$.
P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number $19$ is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number $26$ is on the 3rd row, 5th column, and 7th diagonal.
(Image should be placed here, look at attachment.)
a) Determine the position of the number $2018$ based on its row, column, and diagonal.
b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to $2018$.
P10. It is known that $A$ is the set of 3-digit integers not containing the digit $0$. Define a [i]gadang[/i] number to be the element of $A$ whose digits are all distinct and the digits contained in such number are not prime, and (a [i]gadang[/i] number leaves a remainder of 5 when divided by 7. If we pick an element of $A$ at random, what is the probability that the number we picked is a [i]gadang[/i] number?
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?
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.
Two triangular pyramids have common base. One pyramid contains the other. Can the sum of the lengths of the edges of the inner pyramid be longer than that of the outer one?
A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
$ \textbf{(A)}\ 5\sqrt{2}-7 \qquad
\textbf{(B)}\ 7-4\sqrt{3} \qquad
\textbf{(C)}\ \frac{2\sqrt{2}}{27} \qquad
\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad
\textbf{(E)}\ \frac{\sqrt{3}}{9} $