MathDB

3

Part of 2023 Stanford Mathematics Tournament

Problems(6)

SMT 2023 Algebra #3

Source:

5/3/2023
Nathan has discovered a new way to construct chocolate bars, but it’s expensive! He starts with a single 1×11\times1 square of chocolate and then adds more rows and columns from there. If his current bar has dimensions w×hw\times h (ww columns and hh rows), then it costs w2w^2 dollars to add another row and h2h^2 dollars to add another column. What is the minimum cost to get his chocolate bar to size 20×2020\times20?
SMT 2023 Algebra Tiebreaker #3

Source:

5/3/2023
Let f(x)=x3−6x2+252x−7f(x)=x^3-6x^2+\tfrac{25}{2}x-7. There is an interval [a,b][a,b] such that for any real number xx, the sequence x,f(x),f(f(x)),…x,f(x),f(f(x)),\dots is bounded (i.e., has a lower and upper bound) if and only if x∈[a,b]x\in[a,b]. Compute (a−b)2(a-b)^2.
SMT 2023 Discrete #3

Source:

5/3/2023
How many trailing zeros does the value 300⋅305⋅310…1090⋅1095⋅1100300\cdot305\cdot310\dots1090\cdot1095\cdot1100 end with?
SMT 2023 Discrete Tiebreaker #3

Source:

5/3/2023
What is the least positive integer xx for which the expression x2+3x+9x^2 + 3x + 9 has 33 distinct prime divisors?
SMT 2023 Geometry #3

Source:

8/9/2023
Consider an equilateral triangle △ABC\vartriangle ABC of side length 44. In the zeroth iteration, draw a circle Ω0\Omega_0 tangent to all three sides of the triangle. In the first iteration, draw circles Ω1A\Omega_{1A},Ω1B \Omega_{1B}, Ω1C\Omega_{1C} such that circle Ω1v\Omega_{1v} is externally tangent to Ω0\Omega_0 and tangent to the two sides that meet at vertex vv (for example, Ω1A\Omega_{1A} would be tangent to Ω0\Omega_0 and sides ABAB, ACAC). In the nth iteration, draw circle Ωn,v\Omega_{n,v} externally tangent to Ωn−1,v\Omega_{n-1,v} and the two sides that meet at vertex vv. Compute the total area of all the drawn circles as the number of iterations approaches infinity.
geometry
SMT 2023 Geometry Tiebreaker #3

Source:

8/9/2023
Triangle △ABC\vartriangle ABC has side lengths AB=5AB = 5, BC=8BC = 8, and CA=7CA = 7. Let the perpendicular bisector of BCBC intersect the circumcircle of △ABC\vartriangle ABC at point DD on minor arc BCBC and point EE on minor arc ACAC, and ACAC at point FF. The line parallel to BCBC passing through FF intersects ADAD at point GG and CECE at point HH. Compute [CHF][DGF]\frac{[CHF]}{[DGF]} . (Given a triangle △ABC\vartriangle ABC, [ABC][ABC] denotes its area.)
geometryperpendicular bisector