Olympiad problems

2018 Adygea Teachers' Geometry Olympiad, 4

Tags: geometry , 3d geometry , cube , angle , plane

Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$. a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$. b) Find the angle between the plane $a$ and the plane $BB_1C_1$.

2005 AIME Problems, 9

Tags: geometry , 3d geometry , probability , rotation

Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The $27$ cubes are randomly arranged to form a $3\times 3 \times 3$ cube. Given the probability of the entire surface area of the larger cube is orange is $\frac{p^a}{q^br^c},$ where $p$,$q$, and $r$ are distinct primes and $a$,$b$, and $c$ are positive integers, find $a+b+c+p+q+r$.

2004 Postal Coaching, 17

Tags: 3d geometry , number theory , geometry

In a system of numeration with base $B$ , there are $n$ one-digit numbers less than $B$ whose cubes have $B-1$ in the units-digits place. Determine the relation between $n$ and $B$

2000 ITAMO, 3

Tags: ratio , geometry , 3d geometry , pyramid , volume , sphere

A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.

1992 Baltic Way, 3

Tags: geometry , 3d geometry , arithmetic sequence , algebra , factorization , sum of cubes , number theory

Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).

1975 Poland - Second Round, 5

Tags: geometry , 3d geometry , coloring , combinatorial geometry

Prove that if a sphere can be inscribed in a convex polyhedron and each face of this polyhedron can be painted in one of two colors such that any two faces sharing a common edge are of different colors, then the sum of the areas of the faces of one color is equal to the sum of the areas of the faces of the other color.

2008 Sharygin Geometry Olympiad, 24

Tags: inequalities , 3d geometry , tetrahedron , geometry

(I.Bogdanov, 11) Let $ h$ be the least altitude of a tetrahedron, and $ d$ the least distance between its opposite edges. For what values of $ t$ the inequality $ d>th$ is possible?

2013 IPhOO, 9

Tags: limit , geometry , 3d geometry , sphere , buoyancy

Bob, a spherical person, is floating around peacefully when Dave the giant orange fish launches him straight up 23 m/s with his tail. If Bob has density 100 $\text{kg/m}^3$, let $f(r)$ denote how far underwater his centre of mass plunges underwater once he lands, assuming his centre of mass was at water level when he's launched up. Find $\lim_{r\to0} \left(f(r)\right) $. Express your answer is meters and round to the nearest integer. Assume the density of water is 1000 $\text{kg/m}^3$. [i](B. Dejean, 6 points)[/i]

2024 All-Russian Olympiad, 1

Tags: 3d geometry , tetrahedron , geometry

We are given an infinite cylinder in space (i.e. the locus of points of a given distance $R>0$ from a given straight line). Can six straight lines containing the edges of a tetrahedron all have exactly one common point with this cylinder? [i]Proposed by A. Kuznetsov[/i]

1971 IMO Longlists, 49

Tags: polyhedron , 3d geometry , combinatorial geometry , volume , geometry

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

2013 Sharygin Geometry Olympiad, 3

Tags: incenter , 3d geometry , tetrahedron , sphere , geometry

Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent. [hide]Of course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))[/hide]

1997 French Mathematical Olympiad, Problem 3

Tags: 3d geometry , geometry

Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$.

2000 Croatia National Olympiad, Problem 3

Tags: geometry , 3d geometry

A plane intersects a rectangular parallelepiped in a regular hexagon. Prove that the rectangular parallelepiped is a cube.

1983 Polish MO Finals, 6

Tags: tetrahedron , dihedral angle , faces , 3d geometry , geometry

Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.

2003 Iran MO (3rd Round), 12

Tags: geometry , 3d geometry , sphere , prism , combinatorics

There is a lamp in space.(Consider lamp a point) Do there exist finite number of equal sphers in space that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)

2014 NIMO Problems, 8

Tags: calculus , geometry , 3d geometry

Let $x$ be a positive real number. Define \[ A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}. \] Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$. [i]Proposed by Evan Chen[/i]

1991 Arnold's Trivium, 100

Tags: geometry , 3d geometry , sphere , symmetry

Find the mathematical expectation of the area of the projection of a cube with edge of length $1$ onto a plane with an isotropically distributed random direction of projection.

1997 Vietnam Team Selection Test, 1

Tags: 3d geometry , tetrahedron , circumcircle , inequalities , analytic geometry , geometry

Let $ ABCD$ be a given tetrahedron, with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$, $ DA \equal{} a_1$, $ DB \equal{} b_1$, $ DC \equal{} c_1$. Prove that there is a unique point $ P$ satisfying \[ PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2 \] and for this point $ P$ we have $ PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2$, where $ R$ is the circumradius of the tetrahedron $ ABCD$. Find the necessary and sufficient condition so that this inequality is an equality.

1977 IMO Longlists, 48

Tags: 3d geometry , tetrahedron , perimeter , geometry

The intersection of a plane with a regular tetrahedron with edge $a$ is a quadrilateral with perimeter $P.$ Prove that $2a \leq P \leq 3a.$

1964 IMO, 6

Tags: geometry , 3d geometry , tetrahedron , analytic geometry

In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

1967 IMO Shortlist, 5

Tags: geometry , 3d geometry , sphere , polyhedron

Faces of a convex polyhedron are six squares and 8 equilateral triangles and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge, are equal. Prove that it is possible to circumscribe a sphere around the polyhedron, and compute the ratio of the squares of volumes of that polyhedron and of the ball whose boundary is the circumscribed sphere.

2012 NIMO Problems, 6

Tags: analytic geometry , geometry , perimeter , floor function , 3d geometry , pyramid , calculus

A square is called [i]proper[/i] if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$. [i]Proposed by Lewis Chen[/i]

1996 Romania National Olympiad, 4

Tags: inequalities , geometric inequality , geometric inequalities , 3d geometry , geometry , algebra

a) Let $AB CD$ be a regular tetrahedron. On the sides $AB$, $AC$ and $AD$, the points $M$, $N$ and $P$, are considered. Determine the volume of the tetrahedron $AMNP$ in terms of $x, y, z$, where $x=AM$, $y=AN$, $z=AP$. b) Show that for any real numbers $x, y, z, t, u, v \in (0, 1)$ : $$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$

2017 Sharygin Geometry Olympiad, P24

Tags: geometry , tetrahedron , similarity , 3d geometry

Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?

I Soros Olympiad 1994-95 (Rus + Ukr), 11.4

Tags: geometry , 3d geometry

The wire is bent in the form of a square with side $2$. Find the volume of the body consisting of all points in space located at a distance not exceeding $1$ from at least one point of the wire.