Olympiad problems

2015 AMC 10, 21

Tags: geometry , 3D geometry , tetrahedron , analytic geometry , calculus , vector

Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

1993 Irish Math Olympiad, 5

Tags: geometry , rectangle , 3D geometry , geometry proposed

$ (a)$ The rectangle $ PQRS$ with $ PQ\equal{}l$ and $ QR\equal{}m$ $ (l,m \in \mathbb{N})$ is divided into $ lm$ unit squares. Prove that the diagonal $ PR$ intersects exactly $ l\plus{}m\minus{}d$ of these squares, where $ d\equal{}(l,m)$. $ (b)$ A box with edge lengths $ l,m,n \in \mathbb{N}$ is divided into $ lmn$ unit cubes. How many of the cubes does a main diagonal of the box intersect?

V Soros Olympiad 1998 - 99 (Russia), 11.2

Tags: geometry , 3D geometry , pyramid

Five edges of a triangular pyramid are equal to $1$. Find the sixth edge if it is known that the radius of the ball circumscribed about this pyramid is equal to $1$.

1970 IMO Longlists, 16

Tags: geometry , 3D geometry

Show that the equation $\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$ has no real roots.

2019 District Olympiad, 3

Tags: 3D geometry , geometry , angles , parallelepiped

Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.

1997 AMC 8, 22

Tags: geometry , 3D geometry , AMC

A two-inch cube $(2\times 2\times 2)$ of silver weighs 3 pounds and is worth \$200. How much is a three-inch cube of silver worth? $\textbf{(A)}\ 300\text{ dollars} \qquad \textbf{(B)}\ 375\text{ dollars} \qquad \textbf{(C)}\ 450\text{ dollars} \qquad \textbf{(D)}\ 560\text{ dollars} \qquad \textbf{(E)}\ 675\text{ dollars}$

2014 PUMaC Geometry B, 2

Tags: geometry , 3D geometry , pyramid

Consider the pyramid $OABC$. Let the equilateral triangle $ABC$ with side length $6$ be the base. Also $9=OA=OB=OC$. Let $M$ be the midpoint of $AB$. Find the square of the distance from $M$ to $OC$.

2006 Oral Moscow Geometry Olympiad, 2

Tags: geometry , 3D geometry , tetrahedron , triangle inequality

Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron? (S. Markelov)

1997 Polish MO Finals, 3

Tags: geometry , 3D geometry , tetrahedron , inequalities , trigonometry

In a tetrahedron $ABCD$, the medians of the faces $ABD$, $ACD$, $BCD$ from $D$ make equal angles with the corresponding edges $AB$, $AC$, $BC$. Prove that each of these faces has area less than or equal to the sum of the areas of the other two faces. [hide="Comment"][i]Equivalent version of the problem:[/i] $ABCD$ is a tetrahedron. $DE$, $DF$, $DG$ are medians of triangles $DBC$, $DCA$, $DAB$. The angles between $DE$ and $BC$, between $DF$ and $CA$, and between $DG$ and $AB$ are equal. Show that: area $DBC$ $\leq$ area $DCA$ + area $DAB$. [/hide]

2018 Yasinsky Geometry Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron , midpoint , 3-Dimensional Geometry

In the tetrahedron $SABC$, points $E, F, K, L$ are the midpoints of the sides $SA , BC, AC, SB$ respectively, . The lengths of the segments $EF$ and $KL$ are equal to $11 cm$ and $13 cm$ respectively, and the length of the segment $AB$ equals to $18 cm$. Find the length of the side $SC$ of the tetrahedron.

2008 AMC 10, 3

Tags: geometry , 3D geometry , AMC

Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$? $ \textbf{(A)}\ x^{1/6} \qquad \textbf{(B)}\ x^{1/4} \qquad \textbf{(C)}\ x^{3/8} \qquad \textbf{(D)}\ x^{1/2} \qquad \textbf{(E)}\ x$

1963 Vietnam National Olympiad, 4

Tags: geometry , 3D geometry , tetrahedron , geometry unsolved

The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\circ}$ and $ SB \equal{} SC \equal{} 1$. Find the volume of the tetrahedron.

2007 District Olympiad, 2

Tags: geometry , 3D geometry , rectangle , angles , rectnagle , perpendicular bisector , midpoint

Consider a rectangle $ABCD$ with $AB = 2$ and $BC = \sqrt3$. The point $M$ lies on the side $AD$ so that $MD = 2 AM$ and the point $N$ is the midpoint of the segment $AB$. On the plane of the rectangle rises the perpendicular MP and we choose the point $Q$ on the segment $MP$ such that the measure of the angle between the planes $(MPC)$ and $(NPC)$ shall be $45^o$, and the measure of the angle between the planes $(MPC)$ and $(QNC)$ shall be $60^o$. a) Show that the lines $DN$ and $CM$ are perpendicular. b) Show that the point $Q$ is the midpoint of the segment $MP$.

1992 ITAMO, 1

Tags: combinatorial geometry , geometry , 3D geometry , cube , Plane

A cube is divided into $27$ equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.

2017 Iranian Geometry Olympiad, 5

Tags: IGO , Iran , geometry , 3D geometry , sphere , tetrahedron

Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$. [i]Proposed by Alexey Zaslavsky (Russia)[/i]

1951 Moscow Mathematical Olympiad, 197

Tags: perfect cube , number theory , geometry , 3D geometry

Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.

1996 AMC 12/AHSME, 28

Tags: geometry , 3D geometry , pyramid , vector , Pythagorean Theorem , area of a triangle , Heron's formula

On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing $A$, $B$, and $C$ is closest to $\text{(A)}\ 1.6 \qquad \text{(B)}\ 1.9 \qquad \text{(C)}\ 2.1 \qquad \text{(D)}\ 2.7 \qquad \text{(E)}\ 2.9$

2000 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , 3D geometry

Prove that every multiple of $3$ can be written as a sum of four cubes (positive or negatives).

1978 Polish MO Finals, 6

Tags: geometry , 3D geometry , tetrahedron

Prove that if $h_1,h_2,h_3,h_4$ are the altitudes of a tetrahedron and $d_1,d_2,d_3$ the distances between the pairs of opposite edges of the tetrahedron, then $$\frac{1}{h_1^2} +\frac{1}{h_2^2} +\frac{1}{h_3^2} +\frac{1}{h_4^2} =\frac{1}{d_1^2} +\frac{1}{d_2^2} +\frac{1}{d_3^2}.$$

1963 IMO, 2

Tags: geometry , 3D geometry , sphere , Locus , Locus problems , IMO , IMO 1963

Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

2016 Bundeswettbewerb Mathematik, 4

Tags: geometry , 3D geometry , dodecahedron , combinatorics , combinatorial geometry

Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint [i]regions[/i]. Find the number of such regions.

1976 IMO Longlists, 5

Tags: geometry , 3D geometry , pyramid , parallelogram , geometry unsolved

Let $ABCDS$ be a pyramid with four faces and with $ABCD$ as a base, and let a plane $\alpha$ through the vertex $A$ meet its edges $SB$ and $SD$ at points $M$ and $N$, respectively. Prove that if the intersection of the plane $\alpha$ with the pyramid $ABCDS$ is a parallelogram, then $SM \cdot SN > BM \cdot DN$.

1999 French Mathematical Olympiad, Problem 1

Tags: geometry , 3D geometry

What is the maximum possible volume of a cylinder inscribed in a cone and having the same axis of symmetry as the cone? What is the maximum possible volume of a ball inscribed in the cone with center on the axis of symmetry of the cone? Compare these three volumes.

2020 BMT Fall, 12

Tags: geometry , 3D geometry , prism , geometric inequality

A hollow box (with negligible thickness) shaped like a rectangular prism has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?

1990 AMC 12/AHSME, 10

Tags: geometry , 3D geometry , AMC

An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? $\textbf{(A) }328\qquad \textbf{(B) }329\qquad \textbf{(C) }330\qquad \textbf{(D) }331\qquad \textbf{(E) }332\qquad$