MathDB

9

Part of 2014 ASDAN Math Tournament

Problems(5)

2014 Advanced #9

Source:

7/1/2022
Compute how many permutations of the numbers 1,2,…,81,2,\dots,8 have no adjacent numbers that sum to 99.
2014Advanced Topics Test
2014 General #9

Source:

7/4/2022
The operation ⊘\oslash, called "reciprocal sum," is useful in many areas of physics. If we say that x=a⊘bx=a\oslash b, this means that xx is the solution to 1x=1a+1b\frac{1}{x}=\frac{1}{a}+\frac{1}{b} Compute 4⊘2⊘4⊘3⊘4⊘4⊘2⊘3⊘2⊘4⊘4⊘34\oslash2\oslash4\oslash3\oslash4\oslash4\oslash2\oslash3\oslash2\oslash4\oslash4\oslash3.
2014General Test
2014 Algebra #9

Source:

7/8/2022
A sequence {an}n≥0\{a_n\}_{n\geq0} obeys the recurrence an=1+an−1+αan−2a_n=1+a_{n-1}+\alpha a_{n-2} for all n≥2n\geq2 and for some α>0\alpha>0. Given that a0=1a_0=1 and a1=2a_1=2, compute the value of α\alpha for which ∑n=0∞an2n=10\sum_{n=0}^{\infty}\frac{a_n}{2^n}=10
2014Algebra Test
2014 Geometry #9

Source:

7/12/2022
We have squares ABCDABCD and EFGHEFGH. Square ABCDABCD has points with coordinates A=(1,1,−1)A=(1,1,-1), B=(1,−1,−1)B=(1,-1,-1), C=(−1,−1,−1)C=(-1,-1,-1) and D=(−1,1,−1)D=(-1,1,-1). Square EFGHEFGH has points with coordinates E=(2,0,1)E=(\sqrt{2},0,1), F=(0,−2,1)F=(0,-\sqrt{2},1), G=(−2,0,1)G=(-\sqrt{2},0,1), and H=(0,2,1)H=(0,\sqrt{2},1). Consider the solid formed by joining point AA to HH and EE, point BB to EE and FF, point CC to FF and GG, and point DD to GG and HH. Compute the volume of this solid.
2014Geometry Test
2014 Team #9

Source:

7/1/2022
Find the sum of all real numbers xx such that x4−2x3+3x2−2x−2014=0x^4-2x^3+3x^2-2x-2014=0.
2014team test