MathDB

9

Part of 2013 Harvard-MIT Mathematics Tournament

Problems(4)

2013 HMMT Algebra #9: Sum of Complex Numbers

Source:

2/17/2013
Let zz be a non-real complex number with z23=1z^{23}=1. Compute ∑k=02211+zk+z2k.\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.
HMMTAMCAIME
2013 Team #9: S_m, the property mansion

Source:

3/1/2013
Let mm be an odd positive integer greater than 11. Let SmS_m be the set of all non-negative integers less than mm which are of the form x+yx+y, where xy−1xy-1 is divisible by mm. Let f(m)f(m) be the number of elements of SmS_m.
(a) Prove that f(mn)=f(m)f(n)f(mn)=f(m)f(n) if mm, nn are relatively prime odd integers greater than 11. (b) Find a closed form for f(pk)f(p^k), where k>0k>0 is an integer and pp is an odd prime.
number theoryrelatively prime
2013 HMMT Guts #9: Building an 2 x 2 x 2 Cube

Source:

3/26/2013
I have 88 unit cubes of different colors, which I want to glue together into a 2×2×22\times 2\times 2 cube. How many distinct 2×2×22\times 2\times 2 cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.
HMMTgeometry3D geometrygeometric transformationrotationreflectionsymmetry
2013 HMMT Geometry # 9

Source:

3/3/2024
Pentagon ABCDEABCDE is given with the following conditions:
(a) ∠CBD+∠DAE=∠BAD=45o\angle CBD + \angle DAE = \angle BAD = 45^o, ∠BCD+∠DEA=300o\angle BCD + \angle DEA = 300^o (b) BADA=223\frac{BA}{DA} =\frac{ 2\sqrt2}{3} , CD=753CD =\frac{ 7\sqrt5}{3} , and DE=1524DE = \frac{15\sqrt2}{4} (c) AD2⋅BC=AB⋅AE⋅BDAD^2 \cdot BC = AB \cdot AE \cdot BD
Compute BDBD.
geometry