MathDB

5

Part of 2022 Stanford Mathematics Tournament

Problems(4)

SMT 2022 Algebra #5

Source:

3/27/2023
xx, yy, and zz are real numbers such that xyz=10xyz=10. What is the maximum possible value of x3y3z3−3x4−12y2−12z4x^3y^3z^3-3x^4-12y^2-12z^4?
inequalitiesalgebra
SMT 2022 Calculus #5

Source:

3/29/2023
A net for hexagonal pyramid is constructed by placing a triangle with side lengths xx, xx, and yy on each side of a regular hexagon with side length yy. What is the maximum volume of the pyramid formed by the net if x+y=20x+y=20?
SMT 2022 Discrete #5

Source:

4/1/2023
A classroom has 3030 seats arranged into 55 rows of 66 seats. Thirty students of distinct heights come to class every day, each sitting in a random seat. The teacher stands in front of all the rows, and if any student seated in front of you (in the same column) is taller than you, then the teacher cannot notice that you are playing games on your phone. What is the expected number of students who can safely play games on their phone?
SMT 2022 Geometry #5

Source:

4/1/2023
In quadrilateral ABCDABCD, AB=20AB=20, BC=15BC=15, CD=7CD=7, DA=24DA=24, and CA=25CA=25. Let the midpoint of ACAC be MM, and let ACAC and BDBD intersect at NN. Find the length of MNMN.