MathDB

Team Round

Part of 2006 JHMT

Problems(1)

2006 JHMT Team Round - Johns Hopkins Mathematics Tournament

Source:

1/18/2022
p1. Evaluate SS. S=100002−1100002−19999S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}
p2. Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves?
p3. Given that (a+b)+(b+c)+(c+a)=18(a + b) + (b + c) + (c + a) = 18 1a+b+1b+c+1c+a=59,\frac{1}{a + b}+\frac{1}{b + c}+ \frac{1}{c + a}=\frac59, determine ca+b+ab+c+bc+a.\frac{c}{a + b}+\frac{a}{b + c}+\frac{b}{c + a}.
p4. Find all primes pp such that 2p+1+p3−p2−p2^{p+1} + p^3 - p^2 - p is prime.
p5. In right triangle ABCABC with the right angle at AA, AFAF is the median, AHAH is the altitude, and AEAE is the angle bisector. If ∠EAF=30o\angle EAF = 30^o , find ∠BAH\angle BAH in degrees.
p6. For which integers aa does the equation (1−a)(a−x)(x−1)=ax(1 - a)(a - x)(x- 1) = ax not have two distinct real roots of xx?
p7. Given that a2+b2−ab−b+13=0a^2 + b^2 - ab - b +\frac13 = 0, solve for all (a,b)(a, b).
p8. Point EE is on side AB‾\overline{AB} of the unit square ABCDABCD. FF is chosen on BC‾\overline{BC} so that AE=BFAE = BF, and GG is the intersection of DE‾\overline{DE} and AF‾\overline{AF}. As the location of EE varies along side AB‾\overline{AB}, what is the minimum length of BG‾\overline{BG}?
p9. Sam and Susan are taking turns shooting a basketball. Sam goes first and has probability PP of missing any shot, while Susan has probability PP of making any shot. What must PP be so that Susan has a 50%50\% chance of making the first shot?
p10. Quadrilateral ABCDABCD has AB=BC=CD=7AB = BC = CD = 7, AD=13AD = 13, ∠BCD=2∠DAB\angle BCD = 2\angle DAB, and ∠ABC=2∠CDA\angle ABC = 2\angle CDA. Find its area.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
algebrageometrycombinatoricsnumber theory