Senior Quiz Finals

SENIOR QUIZThe Mathematical Crusade 2011

PROBLEMSRound 1

Q1.A coin is flipped, a 6-sided die numbered through 6 is rolled, and a 10-sided die numbered 0 through 9 is rolled. What is the probability that the coin comes up heads and the sum of the numbers that show on the dice is 8?

Q2.A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to three quarters its previous height. So, after its first bounce it rises to 75 ft, and after its second bounce it rises to 3/4 of 75 ft, and so on forever. What is the total distance the ball travels?

Q3.How many of the first 1000 natural numbers leave a remainder of 1 when divided by 5 and a remainder of 2 when divided by 6?

Q4.We wish to color each of the integers 1, 2, 3….10 red, green, or blue, so that no two numbers a and b with a and b co-prime have the same color. (We do not require that all three colors be used.) In how many ways can this be done?

Q5.How many ways are there to color three sides of a cube red and the other three sides black, calling two colorings the same if we can rotate one to the other?

Q6.A cubic polynomial takes values 1 at x=1, 2 at x=2, 3 at x=3 and 4 at x=5. If P(x) is the polynomial, find P(5).

Q7.Find all n such that n^4 +4 is prime.

Q8.Let n be the smallest positive integer such that Let n be the smallest positive integer such that n is divisible by 20, n2 is a perfect cube, and n3 is a perfect square. What is the number of digits of n?

TRIVIARound 2

Q1.What is common among Isaac Newton, Edward Waring, Charles Babbage, George Gabriel Stokes, Paul Dirac, Stephen Hawking and Michael Green?

Q2.The International Congress of Mathematicians, which meets every 4 years, is organized by the International Mathematical Union. It is the largest meeting of its kind and has been part of many historical events. For example, in 1900 David Hilbert presented his famous 23 unsolved problems to his contemporaries. Further, the IMU prizes such as the Fields medal and the Gauss prize are awarded at the conference. Who inaugurated ICM 2010, and where was it organized?

Q3.X and Y are two contemporary mathematicians. X, an Australian child prodigy, was both the youngest participant and the youngest gold medallist at IMO, at ages 10 and 13 respectively. X recieved his PhD by 20 and was a full professor at UCLA by 24. He is also the youngest mathematician ever to receive the Fields medal Y, a British mathematician, earned his doctorate under Timothy Gowers and held various positions in Princeton, MIT and the Clay Mathematics Institute. He is currently the first Herchel Smith Professor of Pure Mathematics at the University of Cambridge. X and Y, among other things, have both gotten various prizes such as the Clay Research Award, the Salem prize, the SASTRA Ramanujan prize and even Fellowship at the Royal Society. In 2004. they published a paper proving that there exists at least one arithmetic progression of any arbitrary length in the sequence of primes, a result now known as the Y-X theorem. Identify X and Y.

Q4.Game theory is a field of applied mathematics with far-reaching applications in various fields such as Economics, Biology and Sociology. The field is considered to have begun in its modern form after the publishing of the book Theory of Games and Economic Behavior. Name the authors of this book.

Q5.Stephen Cook, a Canadian-American mathematician and computer scientist published in 1971, a paper entitled "The Complexity of Theorem Proving Procedures". What is the historical significance of this paper?

Q6."Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is." Identify the speaker of these lines.

Q7.Connect:•Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?•Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?•Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?•Are there infinitely many primes p such that p − 1 is a perfect square?

Q8.X, a logician and mathematician once famously said “God created the integers, all else is the work of man.” The first half of this quote, “God created the integers” was appropriated by Stephen Hawking for his collection of what he considered the best work by the mathematicians and physicists of previous times. Identify X.

PROBLEMS IIRound 3

Q1.What is the area bounded by the region |x+y| + |x-y| = 4?

Q2.Find all n such that

is prime.

Q3.How many four digit numbers are there beginning with 1 and having exactly two identical digits?

Q4.Two distinct pennies are flipped and one of the pennies turns out to be heads. What is the probability that both pennies are heads?

Q5.Find the sum of the roots, real and non-real, of the equation given that there are no multiple roots.

Q6.

Evaluate:

Q7.Does the polynomial

have any integer roots?

Q8.ON ITS WAY. NOT HERE YET!

VISUALSRound 4

Q1.

Identify

Q2.

Connect

Q3.

Connect

Thurston's Geometrization ConjectureOn 22 December 2006, the journal Science recognized ____________'s proof of the ____________ conjecture as the scientific "Breakthrough of the Year", the first such recognition in the area of mathematics.

Q4.

Connect

Q5.

Connect

Q6.

Connect

Q7.

Identify!

Q8.

Connect

“On formally undecidable propositions of Principia. Mathematica and related systems”

“An Eternal Golden Braid”

RAPID FIRERound 5

SET 1Rapid Fire

Q1.What is the measure of the angle formed by the minute and hour hands of a clock at 1:30?

Q2.For how many ordered pairs of (x,y) is x + 2y = 100?

Q3.Evaluate the following:

Q4.If Richard can paint a living room in 4 hours, and Vanessa can paint the same living room in 5 hours, then how long will it take them to paint the living room working together?

SET 2Rapid Fire

Q1.What is the measure of the angle formed by the minute and hour hands of a clock at 1:30?

Q2.Find the last two digits of20112011

(=2011^2011)

Q3.Find a pair of positive integers m and n such that:

Q4.Find all integer solutions to the Diophantine equation: (6x+15y)(8x+7y)=129

SET 3Rapid Fire

Q1.What is the units digit of 3121 x 4998 ?

Q2.Find the value of the following in real numbers:

Q3.A right triangle has sides of integer length. One side has length 7. What is the area of the triangle?

Q4.Evaluate: tan10º x tan 20º x tan 30º x… x tan 80º

SET 4Rapid Fire

Q1.The sequence 1, x, 6, y is an arithmetic progression. Find x+y.

Q2.A quadratic function:f(x) = x2 + bx + c satisfiesf(0) + f(1) = –21. Find f(1/2)

Q3.After a cyclist has gone 2/3 of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. How many times as fast does he ride as walk?

Q4.What is the square root of the sum of the first 2006 positive odd integers?

SET 5Rapid Fire

Q1.A twelve foot tree casts a five foot shadow. How long is Henry's shadow (at the same time of day) if he is five and a half feet tall?

Q2.Let f(x) = 1 + x + x2 +… + x100

Find f ’(1).

Q3.Find the smallest positive integer n such that 107*n has the same last two digits as n.

Q4.Find integral solutions for a and b for which 10, a, b, a*b is an arithmetic progression.

SET 6Rapid Fire

Q1.Evaluate the sum 1+2-3+4+5-6+7+8-9.......+208+209-210.

Q2.x and y are positive real numbers where x is p percent of y, and y is 4p percent of x. What is p?

Q3.Find the greatest prime factor of the sum of the two largest two-digit prime numbers.

Q4.The product of two positive numbers is equal to their difference. Find their sum.

SET 7Rapid Fire

Q1.If m and n are integers such that 3m + 4n = 100, what is the smallest possible value of |m – n| ?

Q2.What is the largest integer whose prime factors add to 14?

Q3.Find all integers (a,n) such (an+1) – (a + 1)n = 2001.

Q4.In the diagram, AB = AC = 20, AD = AE = 12, and the area of ADFE is 24. Find the area of BFC.

ENDSenior Quiz,

The Mathematical Crusade, 2011