This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 130

2024 Regional Olympiad of Mexico Southeast, 3
Tags: combinatorics , cube , coloring
A large cube of size \(4 \times 4 \times 4\) is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition: In each block of \(1 \times 1 \times 4\), \(1 \times 4 \times 1\), and \(4 \times 1 \times 1\) cubes, there must be exactly one red cube. Determine how many different ways it is possible to choose the 16 small cubes to be colored red. Note: Two colorings are considered different even if one can be obtained from the other by rotations or symmetries of the cube.

1960 IMO, 5
Tags: geometry , 3d geometry , cube , locus , locus problems
Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

Kyiv City MO 1984-93 - geometry, 1990.11.1
Tags: geometric inequality , geometry , 3d geometry , cube
Prove that the sum of the distances from any point in space from the vertices of a cube with edge $a$ is not less than $4\sqrt3 a$.

1959 AMC 12/AHSME, 1
Tags: cube , geometry , 3d geometry , percent , percent increase
Each edge of a cube is increased by $50 \%$. The percent of increase of the surface area of the cube is: $ \textbf{(A)}\ 50 \qquad\textbf{(B)}\ 125\qquad\textbf{(C)}\ 150\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 750 $

2016 Saint Petersburg Mathematical Olympiad, 2
Tags: geometry , combinatorics , max , cube , 3-dimensional geometry , combinatorial geometry , 3d geometry
The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube $50 \times 50 \times 50$?