Found problems: 2265
1978 Germany Team Selection Test, 5
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds:
(i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$
(ii) some plane contains exactly three points from $E.$
2014 Tournament of Towns., 6
A $3\times 3\times 3$ cube is made of $1\times 1\times 1$ cubes glued together. What is the maximal number of small cubes one can remove so the remaining solid has the following features:
1) Projection of this solid on each face of the original cube is a $3\times 3$ square,
2) The resulting solid remains face-connected (from each small cube one can reach any other small cube along a chain of consecutive cubes with common faces).
1978 IMO, 2
We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as edges$.$ Find the locus of the point $Q$ when $A, B, C$ take all the positions compatible with our problem.
2001 AMC 10, 21
A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter $ 10$ and altitude $ 12$, and the axes of the cylinder and cone coincide. Find the radius of the cylinder.
$ \textbf{(A)}\ \frac83 \qquad
\textbf{(B)}\ \frac{30}{11} \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ \frac{25}{8} \qquad
\textbf{(E)}\ \frac{7}{2}$
2004 Moldova Team Selection Test, 2
In the tetrahedron $ABCD$ the radius of its inscribed sphere is $r$ and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are $r_A, r_B, r_C, r_D.$ Prove the inequality $$\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.$$
2004 All-Russian Olympiad Regional Round, 11.8
Given a triangular pyramid $ABCD$. Sphere $S_1$ passing through points $A$, $B$, $C$, intersects edges $AD$, $BD$, $CD$ at points $K$, $L$, $M$, respectively; sphere $S_2$ passing through points $A$, $B$, $D$ intersects the edges $AC$, $BC$, $DC$ at points $P$, $Q$, $M$ respectively. It turned out that $KL \parallel PQ$. Prove that the bisectors of plane angles $KMQ$ and $LMP$ are the same.
1957 Kurschak Competition, 1
$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.
1959 AMC 12/AHSME, 1
Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is:
$ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750 $
2005 Purple Comet Problems, 14
Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$.
2002 AMC 10, 18
A $ 3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $ 3 \times 3 \times 3$ cube is
$ \text{(A)}\ 60 \qquad
\text{(B)}\ 72 \qquad
\text{(C)}\ 84 \qquad
\text{(D)}\ 90 \qquad
\text{(E)}\ 96$
1989 AMC 8, 23
An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint?
$\text{(A)}\ 21 \qquad \text{(B)}\ 24 \qquad \text{(C)}\ 33 \qquad \text{(D)}\ 37 \qquad \text{(E)}\ 42$
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draw((1.59,3.09)--(2.29,3.63)--(2.98,3.59));
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1976 IMO Longlists, 26
A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.
2002 Moldova National Olympiad, 4
The circumradius of a tetrahedron $ ABCD$ is $ R$, and the lenghts of the segments connecting the vertices $ A,B,C,D$ with the centroids of the opposite faces are equal to $ m_a,m_b,m_c$ and $ m_d$, respectively. Prove that:
$ m_a\plus{}m_b\plus{}m_c\plus{}m_d\leq \dfrac{16}{3}R$
2024 IFYM, Sozopol, 8
In space, there are \( 13 \) points, no four of which lie in the same plane. Three of the points are colored blue, and the triangle with these points as vertices will be called a [i]blue triangle[/i]. The remaining \( 10 \) points are colored red. We say that a triangle with three red vertices is [i]attached[/i] to the blue triangle if the boundary of the red triangle intersects the blue triangle (either in its interior or on its boundary) at exactly one point. Is it possible for the number of attached triangles to be \( 33 \)?
1977 All Soviet Union Mathematical Olympiad, 241
Every vertex of a convex polyhedron belongs to three edges. It is possible to circumscribe a circle around all its faces. Prove that the polyhedron can be inscribed in a sphere.
1980 Poland - Second Round, 3
There is a sphere $ K $ in space and points $ A, B $ outside the sphere such that the segment $ AB $ intersects the interior of the sphere. Prove that the set of points $ P $ for which the segments $ AP $ and $ BP $ are tangent to the sphere $ K $ is contained in a certain plane.
1989 Dutch Mathematical Olympiad, 4
Given is a regular $n$-sided pyramid with top $T$ and base $A_1A_2A_3... A_n$. The line perpendicular to the ground plane through a point $B$ of the ground plane within $A_1A_2A_3... A_n$ intersects the plane $TA_1A_2$ at $C_1$, the plane $TA_2A_3$ at $C_2$, and so on, and finally the plane $TA_nA_1$ at $C_n$. Prove that $BC_1 + BC_2 + ... + BC_n$ is independent of choice of $B$'s.
1978 IMO Shortlist, 14
Prove that it is possible to place $2n(2n + 1)$ parallelepipedic (rectangular) pieces of soap of dimensions $1 \times 2 \times (n + 1)$ in a cubic box with edge $2n + 1$ if and only if $n$ is even or $n = 1$.
[i]Remark[/i]. It is assumed that the edges of the pieces of soap are parallel to the edges of the box.
2010 All-Russian Olympiad, 1
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
1948 Moscow Mathematical Olympiad, 146
Consider two triangular pyramids $ABCD$ and $A'BCD$, with a common base $BCD$, and such that $A'$ is inside $ABCD$. Prove that the sum of planar angles at vertex $A'$ of pyramid $A'BCD$ is greater than the sum of planar angles at vertex $A$ of pyramid $ABCD$.
1988 IMO Longlists, 59
In $3$-dimensional space there is given a point $O$ and a finite set $A$ of segments with the sum of lengths equal to $1988$. Prove that there exists a plane disjoint from $A$ such that the distance from it to $O$ does not exceed $574$.
1946 Putnam, B3
In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$, where $k$ is a constant, find $\rho$ as a function of $r.$ Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center.
1974 Polish MO Finals, 1
In a tetrahedron $ABCD$ the edges $AB$ and $CD$ are perpendicular and $\angle ACB =\angle ADB$. Prove that the plane through $AB$ and the midpoint of the edge $CD$, is perpendicular to $CD$.
2014 Miklós Schweitzer, 10
To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.
1965 Vietnam National Olympiad, 2
$AB$ and $CD$ are two fixed parallel chords of the circle $S$. $M$ is a variable point on the circle. $Q$ is the intersection of the lines $MD$ and $AB$. $X$ is the circumcenter of the triangle $MCQ$.
Find the locus of $X$.
What happens to $X$ as $M$ tends to
(1) $D$,
(2) $C$?
Find a point $E$ outside the plane of $S$ such that the circumcenter of the tetrahedron $MCQE$ has the same locus as $X$.