MathDB

round 4

Part of 2012 BMT Spring

Problems(1)

2012 BMT Tournament Round 4 - Berkley Math Tournament

Source:

1/28/2022
p1. Denote Sn=1+12+13+...+1nS_n = 1 + \frac12 + \frac13 + ...+ \frac{1}{n}. What is 144169⋅S144169−(S1+S2+...+S144168)144169\cdot S_{144169} - (S_1 + S_2 + ... + S_{144168})?
p2. Let A,B,CA,B,C be three collinear points, with AB=4AB = 4, BC=8BC = 8, and AC=12AC = 12. Draw circles with diameters ABAB, BCBC, and ACAC. Find the radius of the two identical circles that will lie tangent to all three circles.
p3. Let s(i)s(i) denote the number of 11’s in the binary representation of ii. What is ∑x=1314(∑i=02576−2xs(i))  mod  629?\sum_{x=1}{314}\left( \sum_{i=0}^{2^{576}-2} x^{s(i)} \right) \,\, mod \,\,629 ?
p4. Parallelogram ABCDABCD has an area of SS. Let k=42k = 42. EE is drawn on AB such that AE=ABkAE =\frac{AB}{k} . FF is drawn on CDCD such that CF=CDkCF = \frac{CD}{k} . GG is drawn on BCBC such that BG=BCkBG = \frac{BC}{k} . HH is drawn on ADAD such that DH=ADkDH = \frac{AD}{k} . Line CECE intersects BHBH at MM, and DGDG at NN. Line AFAF intersects DGDG at PP, and BHBH at QQ. If S1S_1 is the area of quadrilateral MNPQMNPQ, find S1S\frac{S_1}{S}.
p5. Let ϕ\phi be the Euler totient function. What is the sum of all nn for which nϕ(n)\frac{n}{\phi(n)} is maximal for 1≤n≤5001 \le n \le 500?
p6. Link starts at the top left corner of an 12×1212 \times 12 grid and wants to reach the bottom right corner. He can only move down or right. A turn is defined a down move immediately followed by a right move, or a right move immediately followed by a down move. Given that he makes exactly 66 turns, in how many ways can he reach his destination?
PS. You had better use hide for answers.
algebrageometrycombinatoricsnumber theoryBmtBerkeley Math Tournament