Olympiad problems

2019 Jozsef Wildt International Math Competition, W. 68

Tags: tetrahedron , 3D geometry , inequalities , exradius , inradius

In all tetrahedron $ABCD$ holds [list=1] [*] $\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}$ [*] $\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}$ [/list] for all $t\in [0,1]$

1972 AMC 12/AHSME, 34

Tags: geometry , 3D geometry , number theory , relatively prime , AMC

Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is $\textbf{(A) }42\qquad\textbf{(B) }46\qquad\textbf{(C) }122\qquad\textbf{(D) }290\qquad \textbf{(E) }326$

2022 Princeton University Math Competition, B2

Tags: geometry , 3D geometry

Three spheres are all externally tangent to a plane and to each other. Suppose that the radii of these spheres are $6$, $8$, and, $10$. The tangency points of these spheres with the plane form the vertices of a triangle. Determine the largest integer that is smaller than the perimeter of this triangle.

1961 Polish MO Finals, 3

Tags: geometry , parallelogram , tetrahedron , 3D geometry , Geometric Inequalities

Prove that if a plane section of a tetrahedron is a parallelogram, then half of its perimeter is contained between the length of the smallest and the length of the largest edge of the tetrahedron.

2015 Israel National Olympiad, 5

Tags: geometry , 3D geometry , tetrahedron , sphere

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$, tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$. Let $K$ be the tangency point of $S_1$ to the face $ABC$, and let $L$ be the tangency point of $S_2$ to the face $ABC$. Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$. Prove that $L,T,K$ lie on one line.

2023 Portugal MO, 3

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

A crate with a base of $4 \times 2$ and a height of $2$ is open at the top. Tomas wants to completely fill the crate with some of his cubes. It has $16$ equal cubes of volume $1$ and two equal cubes of volume $8$. A cube of volume $1$ can only be placed on the top layer if the cube on the bottom layer has already been placed. In how many ways can Tom'as fill the box with cubes, placing them one by one?

2004 India Regional Mathematical Olympiad, 2

Tags: geometry , 3D geometry

Positive integers are written on all the faces of a cube, one on each. At each corner of the cube, the product of the numbers on the faces that meet at the vertex is written. The sum of the numbers written on the corners is 2004. If T denotes the sum of the numbers on all the faces, find the possible values of T.

1982 Polish MO Finals, 6

Tags: geometry , 3D geometry , tetrahedron , dihedral angle

Prove that the sum of dihedral angles in an arbitrary tetrahedron is greater than $2\pi$

2007 Oral Moscow Geometry Olympiad, 1

Tags: geometry , 3D geometry , cube

Given a rectangular strip of measure $12 \times 1$. Paste this strip in two layers over the cube with edge $1$ (the strip can be bent, but cannot be cut). (V. Shevyakov)

1978 Romania Team Selection Test, 2

Tags: geometry , 3D geometry , tetrahedron , rotation , Locus , Pure geometry , geometric transformation

Points $ A’,B,C’ $ are arbitrarily taken on edges $ SA,SB, $ respectively, $ SC $ of a tetrahedron $ SABC. $ Plane forrmed by $ ABC $ intersects the plane $ \rho , $ formed by $ A’B’C’, $ in a line $ d. $ Prove that, meanwhile the plane $ \rho $ rotates around $ d, $ the lines $ AA’,BB’ $ and $ CC’ $ are, and remain concurrent. Find de locus of the respective intersections.

2004 Romania National Olympiad, 2

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

Let $ABCD$ be a tetrahedron in which the opposite sides are equal and form equal angles. Prove that it is regular.

2002 AMC 12/AHSME, 24

Tags: geometry , 3D geometry , tetrahedron , prism

Let $ABCD$ be a regular tetrahedron and let $E$ be a point inside the face $ABC$. Denote by $s$ the sum of the distances from $E$ to the faces $DAB$, $DBC$, $DCA$, and by $S$ the sum of the distances from $E$ to the edges $AB$, $BC$, $CA$. Then $\dfrac sS$ equals $\textbf{(A) }\sqrt2\qquad\textbf{(B) }\dfrac{2\sqrt2}3\qquad\textbf{(C) }\dfrac{\sqrt6}2\qquad\textbf{(D) }2\qquad\textbf{(E) }3$

2002 Tournament Of Towns, 5

Tags: geometry , 3D geometry , prism , geometry proposed

Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)

2002 AMC 10, 16

Tags: geometry , 3D geometry , prism

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

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 $VABCD$.

1962 German National Olympiad, 5

Tags: geometry , Locus , 3D geometry

Given a plane $P$ and two fixed points $A$ and $B$ that do not lie in this plane. Denote two points $A'$ and $B'$ on plane $P$ and $M ,N$ the midpoints of the segments $AA'$, $BB'$. a) Determine the locus of the midpoint of the segment MN if the points are $A'$ and $B'$ move arbitrarily in plane $P$. b) A circle $O$ is considered in the plane $P$. Determine the locus $L$ of the midpoint of the segment $MN$ if the points $A'$ and $B'$ lie on the circle $O$ or inside it . c) $A'$ is assumed to be fixed on the circle $O$ or inside it and $B'$ is assumed to be movable inside it , except for $O$. Determine the locus of the point $B'$ such the above certain locus $L$ remains the same . Note: For b) and c) the following cases should be considered: 1. $A'$ and $B'$ are different, 2. $A'$ and $B'$ coincide.

2000 Austria Beginners' Competition, 4

Tags: geometry , circumcircle , 3D geometry , dodecahedron

Let $ABCDEFG$ be half of a regular dodecagon . Let $P$ be the intersection of the lines $AB$ and $GF$, and let $Q$ be the intersection of the lines $AC$ and $GE$. Prove that $Q$ is the circumcenter of the triangle $AGP$.

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.

2017 HMNT, 8

Tags: 3D geometry , geometry

[b]U[/b]ndecillion years ago in a galaxy far, far away, there were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Admiral Ackbar wanted to establish a base somewhere in space such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square light years. (The sizes of the space stations and the base are negligible.) Determine the volume, in cubic light years, of the set of all possible locations for the Admiral’s base.

1983 IMO Shortlist, 25

Tags: geometry , 3D geometry , sphere , trigonometry , combinatorics , partition , IMO Shortlist

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

2000 IMO Shortlist, 2

Tags: 3D geometry , combinatorics , packing , Coloring , IMO Shortlist

A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers $ n$ for which it is possible to build a cube of side $ n$ using such bricks.

2013 Baltic Way, 5

Tags: geometry , 3D geometry , algebra unsolved , algebra

Numbers $0$ and $2013$ are written at two opposite vertices of a cube. Some real numbers are to be written at the remaining $6$ vertices of the cube. On each edge of the cube the difference between the numbers at its endpoints is written. When is the sum of squares of the numbers written on the edges minimal?