Olympiad problems

2004 All-Russian Olympiad, 1

Are there such pairwise distinct natural numbers $ m, n, p, q$ satisfying $ m \plus{} n \equal{} p \plus{} q$ and $ \sqrt{m} \plus{} \sqrt[3]{n} \equal{} \sqrt{p} \plus{} \sqrt[3]{q} > 2004$ ?

2006 Estonia Math Open Junior Contests, 6

Tags: geometry , 3d geometry , quadratic , algebra

Find all real numbers with the following property: the difference of its cube and its square is equal to the square of the difference of its square and the number itself.

2012 Stanford Mathematics Tournament, 3

Tags: geometry , 3d geometry

Express $\frac{2^3-1}{2^3+1}\times\frac{3^3-1}{3^3+1}\times\frac{4^3-1}{4^3+1}\times\dots\times\frac{16^3-1}{16^3+1}$ as a fraction in lowest terms.

2014 Middle European Mathematical Olympiad, 8

Tags: modular arithmetic , quadratic , inequalities , geometry , 3d geometry , number theory

Determine all quadruples $(x,y,z,t)$ of positive integers such that \[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]

1998 Romania National Olympiad, 3

Tags: geometry , trapezoid , 3d geometry

In the right-angled trapezoid $AB CD$, $AB \parallel CD$, $m( \angle A) = 90°$, $AD = DC = a$ and $AB =2a$. On the perpendiculars raised in $C$ and $D$ on the plane containing the trapezoid one considers points $E$ and $F$ (on the same side of the plane) such that $CE = 2a$ and $DF = a$. Find the distance from the point $B$ to the plane $(AEF)$ and the measure of the angle between the lines $AF$ and $BE$.

1997 All-Russian Olympiad Regional Round, 10.4

Tags: combinatorics , combinatorial geometry , geometry , 3d geometry

Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?

1999 All-Russian Olympiad Regional Round, 8.8

Tags: combinatorics , combinatorial geometry , 3d geometry , geometry

An open chain was made from $54$ identical single cardboard squares, connecting them hingedly at the vertices. Any square (except for the extreme ones) is connected to its neighbors by two opposite vertices. Is it possible to completely cover a $3\times 3 \times3$ surface with this chain of squares?

1993 Poland - First Round, 12

Tags: geometry , 3d geometry , tetrahedron

Prove that the sums of the opposite dihedral angles of a tetrahedron are equal if and only if the sums of the opposite edges of this tetrahedron are equal.

2016 PUMaC Combinatorics B, 3

Tags: geometry , 3d geometry , rotation

Chitoge is painting a cube; she can paint each face either black or white, but she wants no vertex of the cube to be touching three faces of the same color. In how many ways can Chitoge paint the cube? Two paintings of a cube are considered to be the same if you can rotate one cube so that it looks like the other cube.

2019 PUMaC Geometry A, 1

Tags: geometry , 3d geometry

A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?

LMT Speed Rounds, 7

Tags: geometry , 3d geometry , prism

Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$. [i]Proposed by Isabella Li[/i]

2021 AIME Problems, 10

Tags: geometry , 3d geometry , sphere

Two spheres with radii $36$ and one sphere with radius $13$ are each externally tangent to the other two spheres and to two different planes $\mathcal{P}$ and $\mathcal{Q}$. The intersection of planes $\mathcal{P}$ and $\mathcal{Q}$ is the line $\ell$. The distance from line $\ell$ to the point where the sphere with radius $13$ is tangent to plane $\mathcal{P}$ is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://imgur.com/1mfBNNL.png[/img]

2005 Romania National Olympiad, 2

Tags: geometry , 3d geometry , pyramid , trigonometry , geometric transformation , rotation , conic

The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if \[\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},\] then the pyramid is regular.

2007 Harvard-MIT Mathematics Tournament, 1

Tags: 3d geometry , geometry

A cube of edge length $s>0$ has the property that its surface area is equal to the sum of its volume and five times its edge length. Compute all possible values of $s$.

Kyiv City MO 1984-93 - geometry, 1986.9.2

Tags: geometry , 3d geometry , polyhedron , rhombus

The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.

2008 Sharygin Geometry Olympiad, 5

Tags: 3d geometry , tetrahedron , geometry

(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?

2023 LMT Fall, 7

Tags: geometry , 3d geometry , prism

Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$. [i]Proposed by Isabella Li[/i]

2014 Contests, 2

Tags: geometry , 3d geometry , prism , algorithm , geometric transformation , reflection , induction

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

1993 All-Russian Olympiad Regional Round, 11.3

Tags: 3d geometry , sphere , pyramid , geometry

Point $O$ is the foot of the altitude of a quadrilateral pyramid. A sphere with center $O$ is tangent to all lateral faces of the pyramid. Points $A,B,C,D$ are taken on successive lateral edges so that segments $AB$, $BC$, and $CD$ pass through the three corresponding tangency points of the sphere with the faces. Prove that the segment $AD$ passes through the fourth tangency point

2021 AMC 10 Spring, 13

Tags: geometry , 3d geometry , tetrahedron , prob

What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\sqrt{13}, BD=2\sqrt{5},$ and $CD=5$? $\textbf{(A) }3 \qquad \textbf{(B) }2\sqrt{3} \qquad \textbf{(C) }4 \qquad \textbf{(D) }3\sqrt{3} \qquad \textbf{(E) }6$

2009 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , 3d geometry

The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?

2008 District Olympiad, 3

Tags: geometry , 3d geometry , collinear , perpendicular , symmetric

Let $ABCDA' B' C' D '$ be a cube , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.

2010 Middle European Mathematical Olympiad, 11

Tags: geometry , 3d geometry , number theory

For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation \[1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}\] Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 7)[/i]

1985 IMO Longlists, 87

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

Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$

2011 Iran MO (3rd Round), 1

Tags: 3d geometry , dodecahedron , rotation , geometric transformation , group theory , geometry

A regular dodecahedron is a convex polyhedra that its faces are regular pentagons. The regular dodecahedron has twenty vertices and there are three edges connected to each vertex. Suppose that we have marked ten vertices of the regular dodecahedron. [b]a)[/b] prove that we can rotate the dodecahedron in such a way that at most four marked vertices go to a place that there was a marked vertex before. [b]b)[/b] prove that the number four in previous part can't be replaced with three. [i]proposed by Kasra Alishahi[/i]