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Page Modified Jun 6, 2008 2:55 pm

Quiz 3

Here is our quiz from last year. Check back in late January or early February, when Quiz 4 to be included with applications for the Summer 2008 edition of Brainstorming and Barnstorming should be posted

Problem 1.

What is the smallest positive integer that leaves a remainder of 1 when divided by 2, a remainder of 2 when divided by 3, a remainder of 4 when divided by 5,..., and a remainder of 9 when divided by 10?

Problem 2.

You have eight balls in the three-dimensional space: B1 , B2 , B3 , B4 have radius R, and B5 , B6 , B7 , B8 have smaller radius r. Each ball is tangent to three balls of the same radius and three balls of a different radius. Find the ratio R/r.

Hint: Look at the tetrahedrons of the centers. Where are the centers of the small balls located? Observe that the problem is essentially two-dimensional.

Problem 3.

A non-negative function y=f(x), f(0)=0, satisfies the following property: for every a >0, the area in the rectangle with vertices (0,0), (a,0), (a, f(a)), (0, f(a)) above the graph of the function is twice the area in the rectangle below the graph of the function. Find the function.

Problem 4.

Prove that

π
2
 =lim n &rarr ∞
(2 ·4·6····(2n))2
(3 ·5·7····(2n-1))2(2n+1)

Hint: Define Ik=∫0&pi/2( sin x)k dx, k=1, 2, 3, ... , and show that

I2n=
3 ·5·7···· (2n-1)· π
2 ·4·6····(2n) ·2
, I2n+1=
2 ·4·6····(2n)
3 ·5·7···· (2n+1)

What is lim n &rarr ∞ I2n+1/I2n ?

Problem 5.

Let a1, a2, a3, ... ,an, ... be a sequence of real numbers, such that 0 < an < 1, for all n=1, 2, 3, ..., and such that ( an+1 - an ) &rarr 0 as n &rarr ∞. Assume also that the sequence contains two subsequences ank, ajk such that ank &rarr 0, and ajk &rarr 1 as k &rarr ∞. Prove that for any a, 0< a < 1, there exists a subsequence amk converging to a.