Found problems: 2265
1977 Dutch Mathematical Olympiad, 2
Four masts stand on a flat horizontal piece of land at the vertices of a square $ABCD$. The height of the mast on $A$ is $7$ meters, of the mast on $B$ $13$ meters, and of the mast on $C$ $15$ meters. Within the square there is a point $P$ on the ground equidistant from each of the tops of these three masts.
(a) What length must the sides of the square be at least for this to be possible?
(b) The distance from $P$ to the top of the mast on $D$ is equal to the distance from$ P$ to each of the tops of the three other masts. Calculate the height of the mast at $D$.
1979 Romania Team Selection Tests, 2.
Let $VA_1A_2A_3A_4$ be a pyramid with the vertex at $V$. Let $M,\, N,\, P$ be the midpoints of the segments $VA_1$, $VA_3$, and $A_2A_4$. Show that the plane $(MNP)$ cuts the pyramid into two parts with the same volume.
[i]Radu Gologan[/i]
2008 ISI B.Stat Entrance Exam, 3
Study the derivatives of the function
\[y=\sqrt{x^3-4x}\]
and sketch its graph on the real line.
2008 Harvard-MIT Mathematics Tournament, 9
Consider a circular cone with vertex $ V$, and let $ ABC$ be a triangle inscribed in the base of the cone, such that $ AB$ is a diameter and $ AC \equal{} BC$. Let $ L$ be a point on $ BV$ such that the volume of the cone is 4 times the volume of the tetrahedron $ ABCL$. Find the value of $ BL/LV$.
2014 Oral Moscow Geometry Olympiad, 2
Is it possible to cut a regular triangular prism into two equal pyramids?
2007 ISI B.Stat Entrance Exam, 7
Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.
2018 AMC 10, 4
4. A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X + Y + Z$?
$\textbf{(A)} \text{ 18} \qquad \textbf{(B)} \text{ 22} \qquad \textbf{(C)} \text{ 24} \qquad \textbf{(D)} \text{ 30} \qquad \textbf{(E)} \text{ 36}$
1988 Bulgaria National Olympiad, Problem 5
The points of space are painted in two colors. Prove that there is a tetrahedron such that all its vertices and its centroid are of the same color.
2022 AMC 10, 21
A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?
[asy]
size(200);
defaultpen(linewidth(0.8));
draw((342,-662) -- (600, -727) -- (757,-619) -- (967,-400) -- (1016,-300) -- (912,-116) -- (651,-46) -- (238,-90) -- (82,-204) -- (184, -388) -- (447,-458) -- (859,-410) -- (1016,-300));
draw((82,-204) -- (133,-490) -- (342, -662));
draw((652,-626) -- (600,-727));
draw((447,-458) -- (652,-626) -- (859,-410));
draw((133,-490) -- (184, -388));
draw((967,-400) -- (912,-116)^^(342,-662) -- (496, -545) -- (757,-619)^^(496, -545) -- (446, -262) -- (238, -90)^^(446, -262) -- (651, -46),linewidth(0.6)+linetype("5 5")+gray(0.4));
[/asy]
$\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }5+2\sqrt{2}\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2021 Math Prize for Girls Problems, 19
Let $T$ be a regular tetrahedron. Let $t$ be the regular tetrahedron whose vertices are the centers of the faces of $T$. Let $O$ be the circumcenter of either tetrahedron. Given a point $P$ different from $O$, let $m(P)$ be the midpoint of the points of intersection of the ray $\overrightarrow{OP}$ with $t$ and $T$. Let $S$ be the set of eight points $m(P)$ where $P$ is a vertex of either $t$ or $T$. What is the volume of the convex hull of $S$ divided by the volume of $t$?
1952 Moscow Mathematical Olympiad, 220
A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.
1960 Poland - Second Round, 6
Calculate the volume of the tetrahedron $ ABCD $ given the edges $ AB = b $, $ AC = c $, $ AD = d $ and the angles $ \measuredangle CAD = \beta $, $ \measuredangle DAB = \gamma $ and $ \measuredangle BAC = \delta$.
1997 Yugoslav Team Selection Test, Problem 1
Consider a regular $n$-gon $A_1A_2\ldots A_n$ with area $S$. Let us draw the lines $l_1,l_2,\ldots,l_n$ perpendicular to the plane of the $n$-gon at $A_1,A_2,\ldots,A_n$ respectively. Points $B_1,B_2,\ldots,B_n$ are selected on lines $l_1,l_2,\ldots,l_n$ respectively so that:
(i) $B_1,B_2,\ldots,B_n$ are all on the same side of the plane of the $n$-gon;
(ii) Points $B_1,B_2,\ldots,B_n$ lie on a single plane;
(iii) $A_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n$.
Express the volume of polyhedron $A_1A_2\ldots A_nB_1B_2\ldots B_n$ as a function in $S,h_1,\ldots,h_n$.
2000 AMC 12/AHSME, 25
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
[asy]import three;
import math;
size(180);
defaultpen(linewidth(.8pt));
currentprojection=orthographic(2,0.2,1);
triple A=(0,0,1);
triple B=(sqrt(2)/2,sqrt(2)/2,0);
triple C=(sqrt(2)/2,-sqrt(2)/2,0);
triple D=(-sqrt(2)/2,-sqrt(2)/2,0);
triple E=(-sqrt(2)/2,sqrt(2)/2,0);
triple F=(0,0,-1);
draw(A--B--E--cycle);
draw(A--C--D--cycle);
draw(F--C--B--cycle);
draw(F--D--E--cycle,dotted+linewidth(0.7));[/asy]$ \textbf{(A)}\ 210 \qquad \textbf{(B)}\ 560 \qquad \textbf{(C)}\ 840 \qquad \textbf{(D)}\ 1260 \qquad \textbf{(E)}\ 1680$
2016 Czech-Polish-Slovak Junior Match, 3
Find all integers $n \ge 3$ with the following property:
it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$.
Poland
2021/2022 Tournament of Towns, P4
Given is a segment $AB$. Three points $X, Y, Z$ are picked in the space so that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocenters of all triangles $XYZ$ obtained in this way belong to a fixed circle.
[i]Alexandr Matveev[/i]
2006 All-Russian Olympiad, 2
If an integer $a > 1$ is given such that $\left(a-1\right)^3+a^3+\left(a+1\right)^3$ is the cube of an integer, then show that $4\mid a$.
1966 Polish MO Finals, 3
Prove that the sum of the squares of the areas of the projections of the faces of a rectangular parallelepiped on a plane is the same for all positions of the plane if and only if the parallelepiped is a cube.
1982 Poland - Second Round, 4
Let $ A $ be a finite set of points in space having the property that for any of its points $ P, Q $ there is an isometry of space that transforms the set $ A $ into the set $ A $ and the point $ P $ into the point $ Q $. Prove that there is a sphere passing through all points of the set $ A $.
2002 All-Russian Olympiad Regional Round, 11.2
The altitude of a quadrangular pyramid $SABCD$ passes through the intersection point of the diagonals of its base $ABCD$. From the tops of the base perpendiculars $AA_1$, $BB_1$, $CC_1$, $DD_1$ are dropped onto lines $SC$, $SD,$ $SA$ and $SB$ respectively. It turned out that the points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are different and lie on the same sphere. Prove that lines $AA_1$, $ BB_1$, $CC_1$, $DD_1$ pass through one point.
2022 Israel National Olympiad, P5
A paper convex quadrilateral will be called [b]folding[/b] if there are points $P,Q,R,S$ on the interiors of segments $AB,BC,CD,DA$ respectively so that if we fold in the triangles $SAP, PBQ, QCR, RDS$, they will exactly cover the quadrilateral $PQRS$. In other words, if the folded triangles will cover the quadrilateral $PQRS$ but won't cover each other.
Prove that if quadrilateral $ABCD$ is folding, then $AC\perp BD$ or $ABCD$ is a trapezoid.
1994 Poland - Second Round, 3
A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.
1978 Romania Team Selection Test, 6
Show that there is no polyhedron whose projection on the plane is a nondegenerate triangle.
1995 Poland - First Round, 12
Find out whether there exist two congruent cubes with a common center such that each face of one cube has a common point with each face of the other.
2007 QEDMO 4th, 5
Let $ ABC$ be a triangle, and let $ X$, $ Y$, $ Z$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Denote by $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ the reflections of these points $ X$, $ Y$, $ Z$ in the midpoints of the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \left\vert XYZ\right\vert \equal{}\left\vert X^{\prime}Y^{\prime}Z^{\prime}\right\vert$.