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2017-18 Meeting Resources and History

Useful Links

Modular Arithmetic

http://www.math.ucla.edu/~radko/circles/lib/data/Handout-868-959.pdf

 

Vieta's

http://www.andrew.cmu.edu/user/daltizio/Vietas%20Formulas.pdf

 

PIE

https://artofproblemsolving.com/wiki/index.php?title=Principle_of_Inclusion-Exclusion

 

Discussion History

9-29-17: Thinking outside the box. Teamwork.

10-3-17: The Factor Theorem. 

10-5-17: Introduction to Modulus. First AIME Problem.

10-10-17: Important and prevalence of Coordinate Geometry.

10-12-17: Coordinate Geometry cont. 

10-17-17: Vieta's

10-24-17: Vieta's cont.

10-31-17: MathCount's Countdown Tournament [Winner: J. Lee]

11-2-17: PIE and Complementary Counting. Introduction to counting and probability.

11-7-17: Combinatorics

11-9-17: More Combinatorics

11-14-17: Guess what. More Combinatorics.

11-16-17: Mathcounts Round 2, dual club meeting with the Coffehouse

11-28-17: 2 Fun AIME Problems

11-30-17: Practice Test

12-5-17: Solutions to Practice Test

12-12-17: Geometric Sequence/AM-GM

Problem History

9-28-17

A five-digit palindrome is a positive integer with respective digits abcba, where a is non-zero. Let S be the sum of all five-digit palindromes. What is the sum of the digits of S?

A. 9

B. 18

C. 27

D. 36

E. 45

 

10-2-17

For some positive integer  n, the number 110n^3 has  110 positive integer divisors, including  1 and the number 110n^3 . How many positive integer divisors does the number 81n^4  have?

 

 

10-5-17

The digits of a positive integer n are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when n is divided by 37?

 

10-10-17

https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_10A_Problems/Problem_19

 

10-12-17

square has sides of length 2. Set S is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set S enclose a region whose area to the nearest hundredth is k. Find 100k.

 

10-24-17

Find all n such that x^3 - 3x + n = 0 has three integer roots.

 

11-2-17

Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000?

 

11-6-17

A coin is altered so that the probability that it lands on heads is less than frac 1  2 and when the coin is flipped four times, the probability of an equal number of heads and tails is frac 1  6. What is the probability that the coin lands on heads?

 

11-14-17

A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure x. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is 14/25. Find the difference between the largest and smallest possible values of x.

 

11-28-17

A right prism with height h has bases that are regular hexagons with sides of length 12. A vertex A of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain A measures 60 degrees. Find h^2.

 

There is a 40 chance of rain on Saturday and a 30 chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is a/b, where a and b are relatively prime positive integers. Find a + b.

 

12-12-17

Tom, Dick, and Harry are playing a game. Starting at the same time, each of them flips a fair coin repeatedly until he gets his first head, at which point he stops. What is the probability that all three flip their coins the same number of times?