A farmer use four straight fences, with respective lengths , , and units to form a quadrilateral. What is the maximum area of the quadrilateral the farmer can enclose?

Find the largest positive integer such that is divisible by .

The diagram shows two identical triangular pieces of paper and . The side lengths of each triangle are , and . Each triangle is folded along a line through a vertex, so that the two sides meeting at this vertex coincide. The regions not covered by the folded parts have respective areas and . If , find the area of the original triangular piece of paper .

The sum of consecutive positive integers is a perfect square. What is the minimum value of the largest of these integers?

A regiment had soldiers but only half of them had uniforms. During inspection, they form a rectangle, and it was just enough to conceal in its interior everyone without a uniform. Later, some new soldiers joined the regiment, but again only half of them had uniforms. During the next inspection, they used a different rectangular formation, again just enough to conceal in its interior everyone without a uniform. How many new soldiers joined the regiment?

I am trying to find a problem from within the last 20 years or so that I believe was on an AIME (though I could be wrong). I recall the basics of the problem, which involved calculating the volume of space that was a given distance away from a given rectangular prism. For example (recreating another version): A unique fly trap projects a 3-dimensional rectangular prism of light that measures 3 inches by 4 inches by 5 inches. The light attracts flies, and once "trapped", they cannot get any further away than 1 inch from the surface of the rectangular prism. Calculate the volume of space, including the rectangular prism of light in which "trapped" bugs can fly.
I feel in my heart that there was a problem of this nature on some AIME, but I am struggling to find it again. Please help.

I flip fair coins. of them land heads. Find the expected value of .


I've seen this in two mock contests I'm taking in a row. I'm curious: is there a scalable method for doing these types of problems?

Let be integers and is not a perfect square. Prove that if then .

The figure below shows a net of a tetrahedron with given angles , , , and equal lengths .
Find the sum of the 6 internal dihedral angles of formed by each pair of adjacent faces of .

Given 7 arbitrary points on a plane. Prove that there do not exist 7 circles such that each circle passes through exactly 4 of those 7 points.

So I solved this problem regarding intervals, and the answer was all real numbers. I was wondering, whether Alcumus would accept (simply typing R) as an answer, instead of typing , and if it doesn't, could it be a suggestion for AoPS devs as a feature in future updates?