Found problems: 2265
2011 USAMTS Problems, 4
Let $ABCDEF$ and $ABC'D'E'F'$ be regular planar hexagons in three-dimensional space with side length $1$, such that $\angle EAE'=60^{\circ}$. Let $P$ be the convex polyhedron whose vertices are $A$, $B$, $C$, $C'$, $D$, $D'$, $E$, $E'$, $F$, and $F'$.
(a) Find the radius $r$ of the largest sphere that can be enclosed in polyhedron $P$.
(b) Let $S$ be a sphere enclosed in polyhedron $P$ with radius $r$ (as derived in part (a)). The set of possible centers of $S$ is a line segment $\overline{XY}$. Find the length $XY$.
1991 IMO Shortlist, 7
$ ABCD$ is a terahedron: $ AD\plus{}BD\equal{}AC\plus{}BC,$ $ BD\plus{}CD\equal{}BA\plus{}CA,$ $ CD\plus{}AD\equal{}CB\plus{}AB,$ $ M,N,P$ are the mid points of $ BC,CA,AB.$ $ OA\equal{}OB\equal{}OC\equal{}OD.$ Prove that $ \angle MOP \equal{} \angle NOP \equal{}\angle NOM.$
2004 Croatia National Olympiad, Problem 3
The altitudes of a tetrahedron meet at a single point. Prove that this point, the centroid of one face of the tetrahedron, the foot of the altitude on that face, and the three points dividing the other three altitudes in ratio $2:1$ (closer to the feet) all lie on a sphere.
1990 IMO Longlists, 60
Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let $ A$ be the beginning vertex and $ B$ be the end vertex. Let there be $ p \times q \times r$ cubes on the string $ (p, q, r \geq 1).$
[i](a)[/i] Determine for which values of $ p, q,$ and $ r$ it is possible to build a block with dimensions $ p, q,$ and $ r.$ Give reasons for your answers.
[i](b)[/i] The same question as (a) with the extra condition that $ A \equal{} B.$
2016 CHMMC (Fall), 12
For a positive real number $a$, let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$. If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$, find $k$.
2021 Yasinsky Geometry Olympiad, 3
Given a rectangular parallelepiped $ABCDA_1B_1C_1D_1$, which has $AD= DC = 3\sqrt2$ cm, and $DD_1 = 8$ cm. Through the diagonal $B_1D$ of the parallelepiped $m$ parallel to line $A_1C_1$ is drawn on the plane $\gamma$.
a) Draw a section of a parallelepiped with plane $\gamma$.
b) Justify what geometric figure is this section, and find its area.
(Alexander Shkolny)
1997 French Mathematical Olympiad, Problem 3
Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$.
2016 Tournament Of Towns, 4
A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.)
[i](8 points)[/i]
[i]Mikhail Evdokimov[/i]
1967 IMO Shortlist, 2
In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.
2008 China Team Selection Test, 3
Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.
1992 National High School Mathematics League, 9
From eight edges and eight diagonal of surfaces of a cube, choose $k$ lines. If any two lines of them are skew lines, then the maximum value of $k$ is________.
1980 AMC 12/AHSME, 16
Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron.
$\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ \sqrt 3 \qquad \text{(C)} \ \sqrt{\frac{3}{2}} \qquad \text{(D)} \ \frac{2}{\sqrt{3}} \qquad \text{(E)} \ 2$
2021 Brazil EGMO TST, 5
Let $S$ be a set, such that for every positive integer $n$, we have $|S\cap T|=1$, where $T=\{n,2n,3n\}$. Prove that if $2\in S$, then $13824\notin S$.
2017 Princeton University Math Competition, A3/B5
A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
1989 All Soviet Union Mathematical Olympiad, 505
$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$.
1986 Balkan MO, 2
Let $ABCD$ be a tetrahedron and let $E,F,G,H,K,L$ be points lying on the edges $AB,BC,CD$ $,DA,DB,DC$ respectively, in such a way that
\[AE \cdot BE = BF \cdot CF = CG \cdot AG= DH \cdot AH=DK \cdot BK=DL \cdot CL.\]
Prove that the points $E,F,G,H,K,L$ all lie on a sphere.
2008 iTest Tournament of Champions, 5
Two squares of side length $2$ are glued together along their boundary so that the four vertices of the first square are glued to the midpoints of the four sides of the other square, and vice versa. This gluing results in a convex polyhedron. If the square of the volume of this polyhedron is written in simplest form as $\tfrac{a+b\sqrt c}d$, what is the value of $a+b+c+d$?
1997 Croatia National Olympiad, Problem 3
The areas of the faces $ABD,ACD,BCD,BCA$ of a tetrahedron $ABCD$ are $S_1,S_2,Q_1,Q_2$, respectively. The angle between the faces $ABD$ and $ACD$ equals $\alpha$, and the angle between $BCD$ and $BCA$ is $\beta$. Prove that
$$S_1^2+S_2^2-2S_1S_2\cos\alpha=Q_1^2+Q_2^2-2Q_1Q_2\cos\beta.$$
2006 Moldova MO 11-12, 4
Let $ABCDE$ be a right quadrangular pyramid with vertex $E$ and height $EO$. Point $S$ divides this height in the ratio $ES: SO=m$. In which ratio does the plane $(ABC)$ divide the lateral area of the pyramid.
1968 IMO Shortlist, 3
Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.
MathLinks Contest 3rd, 3
We say that a tetrahedron is [i]median [/i] if and only if for each vertex the plane that passes through the midpoints of the edges emerging from the vertex is tangent to the inscribed sphere. Also a tetrahedron is called [i]regular [/i] if all its faces are congruent. Prove that a tetrahedron is regular if and only if it is median.
1987 Bulgaria National Olympiad, Problem 3
Let $MABCD$ be a pyramid with the square $ABCD$ as the base, in which $MA=MD$, $MA^2+AB^2=MB^2$ and the area of $\triangle ADM$ is equal to $1$. Determine the radius of the largest ball that is contained in the given pyramid.
Kyiv City MO Seniors 2003+ geometry, 2008.11.4
In the tetrahedron $SABC $ at the height $SH$ the following point $O$ is chosen, such that: $$\angle AOS + \alpha = \angle BOS + \beta = \angle COS + \gamma = 180^o, $$ where $\alpha, \beta, \gamma$ are dihedral angles at the edges $BC, AC, AB $, respectively, at this point $H$ lies inside the base $ABC$. Let ${{A} _ {1}}, \, {{B} _ {1}}, \, {{C} _ {1}} $be the points of intersection of lines and planes: ${{A} _ {1}} = AO \cap SBC $, ${{B} _ {1}} = BO \cap SAC $, ${{C} _ {1}} = CO \cap SBA$ . Prove that if the planes $ABC $ and ${{A} _ {1}} {{B} _ {1}} {{C} _ {1}} $ are parallel, then $SA = SB = SC $.
(Alexey Klurman)
1969 IMO Longlists, 12
$(CZS 1)$ Given a unit cube, find the locus of the centroids of all tetrahedra whose vertices lie on the sides of the cube.
2023 CMWMC, R6
[b]p16.[/b] Let $P(x)$ be a quadratic such that $P(-2) = 10$, $P(0) = 5$, $P(3) = 0$. Then, find the sum of the coefficients of the polynomial equal to $P(x)P(-x)$.
[b]p17.[/b] Suppose that $a < b < c < d$ are positive integers such that the pairwise differences of $a, b, c, d$ are all distinct, and $a + b + c + d$ is divisible by $2023$. Find the least possible value of $d$.
[b]p18.[/b] Consider a right rectangular prism with bases $ABCD$ and $A'B'C'D'$ and other edges $AA'$, $BB'$, $CC'$ and $DD'$. Suppose $AB = 1$, $AD = 2$, and $AA' = 1$.
$\bullet$ Let $X$ be the plane passing through $A$, $C'$, and the midpoint of $BB'$.
$\bullet$ Let $Y$ be the plane passing through $D$, $B'$, and the midpoint of $CC'$.
Then the intersection of $X$, $Y$ , and the prism is a line segment of length $\ell$. Find $\ell$.
PS. You should use hide for answers.