MathDB

Sets 6-9

Part of 2019 MOAA

Problems(1)

2019 MOAA Gunga Bowl - Math Open At Andover - last 4 sets - 12 problems

Source:

2/9/2022
Set 6
p16. Let n!=n×(n−1)×...×2×1n! = n \times (n - 1) \times ... \times 2 \times 1. Find the maximum positive integer value of xx such that the quotient 160!160x\frac{160!}{160^x} is an integer.
p17. Let △OAB\vartriangle OAB be a triangle with ∠OAB=90o\angle OAB = 90^o . Draw points C,D,E,F,GC, D, E, F, G in its plane so that △OAB∼△OBC∼△OCD∼△ODE∼△OEF∼△OFG,\vartriangle OAB \sim \vartriangle OBC \sim \vartriangle OCD \sim \vartriangle ODE \sim \vartriangle OEF \sim \vartriangle OFG, and none of these triangles overlap. If points O,A,GO, A, G lie on the same line, then let xx be the sum of all possible values of OGOA\frac{OG}{OA }. Then, xx can be expressed in the form m/nm/n for relatively prime positive integers m,nm, n. Compute m+nm + n.
p18. Let f(x)f(x) denote the least integer greater than or equal to xxx^{\sqrt{x}}. Compute f(1)+f(2)+f(3)+f(4)f(1)+f(2)+f(3)+f(4).
Set 7
The Fibonacci sequence {Fn}\{F_n\} is defined as F0=0F_0 = 0, F1=1F_1 = 1 and Fn+2=Fn+1+FnF_{n+2} = F_{n+1} + F_n for all integers n≥0n \ge 0.
p19. Find the least odd prime factor of (F3)20+(F4)20+(F5)20(F_3)^{20} + (F_4)^{20} + (F_5)^{20}.
p20. Let S=1F3F5+1F4F6+1F5F7+1F6F8+...S = \frac{1}{F_3F_5}+\frac{1}{F_4F_6}+\frac{1}{F_5F_7}+\frac{1}{F_6F_8}+... Compute 420S420S.
p21. Consider the number Q=0.000101020305080130210340550890144...,Q = 0.000101020305080130210340550890144... , the decimal created by concatenating every Fibonacci number and placing a 0 right after the decimal point and between each Fibonacci number. Find the greatest integer less than or equal to 1Q\frac{1}{Q}.
Set 8
p22. In five dimensional hyperspace, consider a hypercube C0C_0 of side length 22. Around it, circumscribe a hypersphere S0S_0, so all 3232 vertices of C0C_0 are on the surface of S0S_0. Around S0S_0, circumscribe a hypercube C1C_1, so that S0S_0 is tangent to all hyperfaces of C1C_1. Continue in this same fashion for S1S_1, C2C_2, S2S_2, and so on. Find the side length of C4C_4.
p23. Suppose △ABC\vartriangle ABC satisfies AC=102AC = 10\sqrt2, BC=15BC = 15, ∠C=45o\angle C = 45^o. Let D,E,FD, E, F be the feet of the altitudes in △ABC\vartriangle ABC, and let U,V,WU, V , W be the points where the incircle of △DEF\vartriangle DEF is tangent to the sides of △DEF\vartriangle DEF. Find the area of △UVW\vartriangle UVW.
p24. A polynomial P(x)P(x) is called spicy if all of its coefficients are nonnegative integers less than 99. How many spicy polynomials satisfy P(3)=2019P(3) = 2019?
The next set will consist of three estimation problems.
Set 9
Points will be awarded based on the formulae below. Answers are nonnegative integers that may exceed 1,000,0001,000,000.
p25. Suppose a circle of radius 2019201920192019 has area AA. Let s be the side length of a square with area AA. Compute the greatest integer less than or equal to ss.
If nn is the correct answer, an estimate of ee gives max⁡{0,⌊1030(min{ne,en}18⌋−1000}\max \{ 0, \left\lfloor 1030 ( min \{ \frac{n}{e},\frac{e}{n}\}^{18}\right\rfloor -1000 \} points.
p26. Given a 50×5050 \times 50 grid of squares, initially all white, define an operation as picking a square and coloring it and the four squares horizontally or vertically adjacent to it blue, if they exist. If a square is already colored blue, it will remain blue if colored again. What is the minimum number of operations necessary to color the entire grid blue?
If nn is the correct answer, an estimate of ee gives ⌊1805∣n−e∣+6⌋\left\lfloor \frac{180}{5|n-e|+6}\right\rfloor points.
p27. The sphere packing problem asks what percent of space can be filled with equally sized spheres without overlap. In three dimensions, the answer is π32≈74.05%\frac{\pi}{3\sqrt2} \approx 74.05\% of space (confirmed as recently as 2017!2017!), so we say that the packing density of spheres in three dimensions is about 0.740.74. In fact, mathematicians have found optimal packing densities for certain other dimensions as well, one being eight-dimensional space. Let d be the packing density of eight-dimensional hyperspheres in eightdimensional hyperspace. Compute the greatest integer less than 108×d10^8 \times d.
If nn is the correct answer, an estimate of e gives max⁡{⌊30−10−5∣n−e∣⌋,0}\max \left\{ \lfloor 30-10^{-5}|n - e|\rfloor, 0 \right\} points.

PS. You had better use hide for answers. First sets have be posted [url=https://artofproblemsolving.com/community/c4h2777330p24370124]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
algebrageometrycombinatoricsnumber theoryMOAA