Extension: Harder Sequences and Patterns
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- Introduction to Recognizing Recurrences: Recurrences refer to a class of problems that share one trait in common: every instance of a recurrence is deterministically related to …
By Sidhanth Mohanty, Po-Shen Loh, Jacob Imola, Expii Team, Victor Wang
Problems 16
- Let a1,a2,…,a2013a_1,a_2,\dots, a_{2013}a1,a2,…,a2013 be a permutation of the numbers from 111 to 201320132013. Let An=a1+a2+⋯+annA_n = \frac{a_1 + a_2 + \cdots + a_n} {n}An…
- Find the missing number in the following sequence: 3,16,5,36,7,64,?,100,113,16,5,36,7,64,?,100,113,16,5,36,7,64,?,100,11.
- What is the next number in the following sequence? 1,11,21,1211,111221,312211,....1,11,21,1211,111221,312211,....1,11,21,1211,111221,312211,....
- How many positive integers less than 100010001000 have an odd number of factors?
- At most how many regions in the plane do 444 lines create? (Alternative formulation: Using only 444 vertical cuts, how many pieces can you slice a circular pie into?)
- At most how many regions in the plane do 555 lines create? (Alternative formulation: Using only 555 vertical cuts, how many pieces can you slice a circular pie into?)
- At most how many regions in space do 444 planes create? (Alternative formulation: Using only 444 cuts, how many pieces can you slice a spherical watermelon into?) Hint: Look at …
- Let SSS be a subset of the integers 1,2,…,1001,2,\dots,1001,2,…,100 such that 3x∉S3x\notin S3x∈/S whenever x∈Sx\in Sx∈S. What is the largest possible size of SSS? (Hint: Conside…
- Define a sequence recursively by a1=3a_1 = 3a1=3, a2=5a_2 = 5a2=5, and an=an−1an−2a_n = \frac{a_{n-1}}{a_{n-2}}an=an−2an−1 for positive integers n≥3n \ge 3n≥3. What is a20…
- The sequence a0a_0a0, a1a_1a1, a2a_2a2, …\ldots… satisfies the recurrence equation an=2an−1−2an−2+an−3a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}an=2an−1−2an−2+an−3 for eve…
- There is a polynomial PPP such that for every real number xxx, x512+x256+1=(x2+x+1)P(x).x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).x512+x256+1=(x2+x+1)P(x). When PPP is written …
- If −1<x<1-1 < x < 1−1<x<1 and −1<y<1-1 < y < 1−1<y<1, define the "relativistic sum'' x⊕yx \oplus yx⊕y to be x⊕y=x+y1+xy.x \oplus y = \frac{x + y}{1 + xy} .x⊕y=1+xyx+y. The oper…
- What is the probability that at the end of the process, the court has 111 Red and 101010 Blue judges?
- You might recall that we have encountered the famous Fibonacci sequence. There is a world of sequences out there — you can even make your own! To get you started, here is a fun …
- The above triangle is made up with numbers. Now observe closely, how do we get each number? Is there a pattern in this triangle? Can you figure out the 5th number in the next ro…
- 0,3,8,15,24,…0, 3, 8, 15, 24,\dots0,3,8,15,24,… What is the number after 242424?