An empty cube is given, and a grid of square unit cells is drawn on each of its six faces. A beam is a rectangular prism. Several beams are placed inside the cube subject to the following conditions:
[list=]
[*]The two faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are possible positions for a beam.)
[*]No two beams have intersecting interiors.
[*]The interiors of each of the four faces of each beam touch either a face of the cube or the interior of the face of another beam.
[/list]
What is the smallest positive number of beams that can be placed to satisfy these conditions?

Proposed by Alex Zhai

The region consisting of all points in three-dimensional space within units of line segment has volume . What is the length ?

A cube-shaped container has vertices , , , and where and are parallel edges of the cube, and and are diagonals of the faces of the cube. Vertex of the cube is set on a horizontal plane so that the plane of the rectangle is perpendicular to , vertex is meters above , vertex is meters above , and vertex is meters above . The cube contains water whose surface is meters above . The volume of the water is cubic meters, where and are relatively prime positive integers. Find .
IMAGE

Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer no collection of pairs made by the child contains the shoes from exactly of the adults is where and are relatively prime positive integers. Find .

Three concentric circles have radii , , and . An equilateral triangle with one vertex on each circle has side length . The largest possible area of the triangle can be written as , where and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .

Stereotypical AIME problem types:
1. Very bashy arithmetic
2. Simple AMC 10 level problem
3. Intimidating geometry problem that you easily kill using analytic geometry
4. Trick question
5. Logarithm algebra
6. Special-case counting
7. Normal number theory problem
8. Easy but time-consuming and bashy geometry problem
9. An either very easy or very hard counting problem
10. Borderline-impossible geometry that requires you to know a very specific formula
11. Number theory but made very diabolical
12. Advanced counting problem
13. Crazy difficult precalculus problem involving advanced trigonometry and/or complex numbers
14. Algebraic manipulation
15. “Those who know” ahh moment :skull:

Discuss your thoughts in the comments!

A rectangular box has distinct edge lengths and . The sum of the lengths of all edges of is , the sum of the areas of all faces of is , and the volume of is . What is the length of the longest interior diagonal connecting two vertices of ?

A block of wood has the shape of a right circular cylinder with radius and height , and its entire surface has been painted blue. Points and are chosen on the edge on one of the circular faces of the cylinder so that on that face measures . The block is then sliced in half along the plane that passes through point , point , and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is , where , , and are integers and is not divisible by the square of any prime. Find .

IMAGE

Let be the real solution of the system of equations , , . The greatest possible value of can be written in the form , where and are relatively prime positive integers. Find .

A square piece of paper has sides of length . From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance from the corner, and they meet on the diagonal at an angle of (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form , where and are positive integers, , and is not divisible by the th power of any prime. Find .

IMAGE