MathDB

10

Part of 2019 Harvard-MIT Mathematics Tournament

Problems(4)

HMMT Team 2019/10: All roots on unit circle

Source:

2/17/2019
Prove that for all positive integers nn, all complex roots rr of the polynomial P(x)=(2n)x2n+(2n−1)x2n−1+⋯+(n+1)xn+1+nxn+(n+1)xn−1+⋯+(2n−1)x+2nP(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n lie on the unit circle (i.e. ∣r∣=1|r| = 1).
HMMTalgebra
HMMT Combinatorics 2019/10: I wish I could walk for 40 minutes

Source:

2/17/2019
Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position (a1,a2,a3,a4)(a_1, a_2, a_3, a_4) to some position (x1,x2,x3,x4)(x_1, x_2, x_3, x_4) with integer coordinates satisfying (x_1-a_1)^2 + (x_2-a_2)^2 + (x_3-a_3)^2 + (x_4-a_4)^2 = 4   \text{and}   |(x_1 + x_2 + x_3 + x_4) - (a_1 + a_2 + a_3 + a_4)| = 2. In how many ways can Fred reach (10,10,10,10)(10, 10, 10, 10) after exactly 40 minutes, if he is allowed to pass through this point during his walk?
HMMTcombinatorics
HMMT Algebra/NT 2019/10: How hard could it be?

Source:

2/17/2019
The sequence of integers {ai}i=0∞\{a_i\}_{i = 0}^{\infty} satisfies a0=3a_0 = 3, a1=4a_1 = 4, and an+2=an+1an+⌈an+12−1an2−1⌉a_{n+2} = a_{n+1} a_n + \left\lceil \sqrt{a_{n+1}^2 - 1} \sqrt{a_n^2 - 1}\right\rceil for n≥0n \ge 0. Evaluate the sum ∑n=0∞(an+3an+2−an+2an+an+1an+3−anan+1).\sum_{n = 0}^{\infty} \left(\frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}}\right).
HMMTalgebra
HMMT Geometry 2019/10: Coordinate basher's dream

Source:

2/17/2019
In triangle ABCABC, AB=13AB = 13, BC=14BC = 14, CA=15CA = 15. Squares ABB1A2ABB_1A_2, BCC1B2BCC_1B_2, CAA1B2CAA_1B_2 are constructed outside the triangle. Squares A1A2A3A4A_1A_2A_3A_4, B1B2B3B4B_1B_2B_3B_4 are constructed outside the hexagon A1A2B1B2C1C2A_1A_2B_1B_2C_1C_2. Squares A3B4B5A6A_3B_4B_5A_6, B3C4C5B6B_3C_4C_5B_6, C3A4A5C6C_3A_4A_5C_6 are constructed outside the hexagon A4A3B4B3C4C3A_4A_3B_4B_3C_4C_3. Find the area of the hexagon A5A6B5B6C5C6A_5A_6B_5B_6C_5C_6.
HMMTgeometry