MathDB

10

Part of 2013 Stanford Mathematics Tournament

Problems(6)

2013 General Problem 10

Source:

2/4/2013
Consider a sequence given by an=an−1+3an−2+an−3a_n=a_{n-1}+3a_{n-2}+a_{n-3}, where a0=a1=a2=1a_0=a_1=a_2=1. What is the remainder of a2013a_{2013} divided by 77?
modular arithmetic
SMT 2013 Calculus #10

Source:

2/3/2013
Evaluate lim⁡n→∞[(∏k=1n2k2k−1)∫−1∞(cos⁡x)2n2x dx]\lim_{n\to\infty}\left[\left(\prod_{k=1}^{n}\frac{2k}{2k-1}\right)\int_{-1}^{\infty}\frac{(\cos x)^{2n}}{2^x} \, dx\right].
calculuslimitintegrationtrigonometry
SMT 2013 Geometry #10

Source:

2/18/2013
Let triangle ABCABC have side lengths AB=16,BC=20,AC=26.AB=16, BC=20, AC=26. Let ACDE,ABFG,ACDE, ABFG, and BCHIBCHI be squares that are entirely outside of triangle ABCABC. Let JJ be the midpoint of EHEH, KK be the midpoint of DGDG, and LL be the midpoint of ACAC. Find the area of triangle JKLJKL.
geometry
SMT 2013 Algebra #10

Source:

2/3/2013
Given a complex number zz such that z13=1z^{13}=1, find all possible value of z+z3+z4+z9+z10+z12z+z^3+z^4+z^9+z^{10}+z^{12}.
Gauss
SMT 2013 Team #10

Source:

2/4/2013
A unit circle is centered at the origin and a tangent line to the circle is constructed in the first quadrant such that it makes an angle 5π/65\pi/6 with the yy-axis. A series of circles centered on the xx-axis are constructed such that each circle is both tangent to the previous circle and the original tangent line. Find the total area of the series of circles.
geometry
SMT 2013 Advanced Topics #10

Source:

2/3/2013
Compute the number of positive integers bb where b≤2013b \le 2013, b≠17b \neq 17, and b≠18b \neq 18, such that there exists some positive integer NN such that N17\dfrac{N}{17} is a perfect 1717th power, N18\dfrac{N}{18} is a perfect 1818th power, and Nb\dfrac{N}{b} is a perfect bbth power.