## Mar 4, 2014

1. [PDF]

### Some B1 Putnam Problems

Some B1 Putnam Problems. 18.S34, Fall 2009. (1995) For a partition π of {1, 2, 3, 4, 5, 6, 7, 8, 9}, let π(x) be the number of elements in the part containing x.
2. [PDF]

### Some A1 Putnam Problems

Some A1 Putnam Problems. 18.S34, Fall 2009. (1995) Let S be a set of real numbers which is closed under multiplication. (that is, if a and b are in S, then so is ...
3. [PDF]

### mini putnam exam

Here is a mini-exam for you, based on previous Putnam problems. Give yourself a solid 3 uniterrupted hours - no more - and do what you can. 1. For what region ...
4. [PDF]

### Mini Putnam Exam I These are all taken from prevvious Putnam ...

Mini Putnam Exam I. These are all taken from prevvious Putnam Exams. 1. Is. √. 2 the limit of a sequence of numbers of the form 3. √ n − 3. √ m, (n, m = 0, 1, 2, ...
5. [PDF]

### For the Putnam Group

For the Putnam Group. Big Oh Notation. Professor Mel Hausner. This important notation is used to estimate the relative size of functions. f(x) = O(g(x)) as.
6. [PDF]

### Formulas for power series and some integrals.

For the Putnam Group. Important Series You Gotta Know! Professor Mel Hausner. 1. 1 − x. =1+ x + x2 + ···. = ∞. ∑ n=0 xn. Geometric Series (|x| < 1). 1. 1 + x.
7. [PDF]

### Mini Putnam Exam II

These are all taken from Putnam Exams given from 1986–1988. Give yourself 3 hours and write up your solutions as best you can. Submit, and I'll constructively ...
8. [PDF]

### Solutions

www.math.nyu.edu/~bellova/putnam/putnam08_1s.pdf
The Sixtieth William Lowell Putnam Mathematical Competition. December 4, 1999. A1. (124. 1?. 34. 0. 0, 0.0. 0. 10.4.11'5). Find polynomials f (.r), g(.i:), and h(.lf), ...
9. [PDF]

### Solutions of A1 Putnam Problems 12/2/2009 1. [1995] Suppose on ...

www.math.nyu.edu/~bellova/putnam/putnam09_8.pdf
Solutions of A1 Putnam Problems. 12/2/2009. 1. [1995] Suppose on the contrary that there exist t1,t2 ∈ T with t1t2 ∈ U and u1,u2 ∈ U with u1u2 ∈ T.
10. [PDF]

### Putnam Exam: Combinatorics Problems 1985A1. Determine, with ...

www.math.nyu.edu/faculty/hausner/PutnamCombinatorics.pdf
Putnam Exam: Combinatorics Problems. 1985A1. Determine, with proof, the number of ordered triples {A1,A2,A3} of sets which have the property that.
11. [PDF]

### Putnam Exam: Integration problems 1987B1. Evaluate ln(9 − x)dx ln ...

www.math.nyu.edu/faculty/hausner/Putnamintegrals.pdf
Putnam Exam: Integration problems. 1987B1. Evaluate. ∫ 4. 2. √ ln(9 − x)dx. √ ln(9 − x) +. √ ln(x + 3) . 1989A2 Evaluate. ∫ a. 0. ∫ b. 0 emax{b2x2,a2y2}dy dx ...
12. [PDF]

### Putnam Exam: Number Theory problems These are from 1985 ...

Putnam Exam: Number Theory problems. These are from 1985 through 2002. 2002B5. A palindrome in base b is a positive integer whose base-b digits read the ...
13. [PDF]

### Solutions of B1 Putnam Problems 12/2/2009 1. [1995] For a given π ...

www.math.nyu.edu/~bellova/putnam/putnam09_8b.pdf
Solutions of B1 Putnam Problems. 12/2/2009. 1. [1995] For a given π, no more than three different values of π(x) are possible (four would require one part each ...
14. [PDF]

### Putnam Integration Problems. These are all from 1985–2000. A4 ...

Putnam Integration Problems. These are all from 1985–2000. A4-2000. Show that the improper integral lim. B→∞. ∫ B. 0 sin(x) sin(x2)dx converges. A5-1999.
15. [PDF]

### Putnam Exam: Number Theory Problems 1988B1. A composite ...

www.math.nyu.edu/faculty/hausner/PutnamNumberTheory.pdf
Putnam Exam: Number Theory Problems. 1988B1. A composite (positive integer) is a product ab with a and b not necessarily distinct in {2, 3, 4,...}. Show that ...
16. [PDF]

### Putnam Exam: Series problems These are from 1985 through 2002 ...

Putnam Exam: Series problems. These are from 1985 through 2002. 2002A6. Fix an integer b ≥ 2. Let f(1) = 1, f(2) = 2, and for each n ≥ 3, define f(n) = nf(d), ...
17. [PDF]

### Putnam Exam: Sequence problems 1985A3. Let d be a real number ...

www.math.nyu.edu/faculty/hausner/PutnamSequences.pdf
Putnam Exam: Sequence problems. 1985A3. Let d be a real number. For each integer m ≥ 0 define a sequence {am(j)}, j = 0, 1, 2,... by the condition.
18. [PDF]

### Putnam Group 2004 Mel Hausner Solution problem B4, 2003. We ...

Putnam Group 2004. Mel Hausner. Solution problem B4, 2003. We have P(z)=(z − r1)(z − r2)(z − r3)(z − r3) has rationalrcoefficients. We are given that r1 + r2 is ...
19. [PDF]

### For the Putnam Group 2003 Melvin Hausner The length PA, PB, and ...

For the Putnam Group 2003. Melvin Hausner. The length PA, PB, and PC are |1−z|, |ω−z|, and |ω2 −z|. Here we use complex numbers to represent points. z is ...
20. [PDF]

### Putnam Exam: Series problems 1986A3. Evaluate ∑∞ n=0 Arccot ...

Putnam Exam: Series problems. 1986A3. Evaluate ∑∞ n=0 Arccot(n2 + n + 1), where Arccot t for t ≥ 0 denotes the number θ in the interval 0 < θ ≤ π/2 with cot ...
21. [PDF]

### Putnam 1991, B5 Here is an alternate approach to the problem ...

Putnam 1991, B5. Here is an alternate approach to the problem: Given f(x + y) = f(x)f(y) − g(x)g(y) g(x + y) = g(x)f(y) + f(x)g(y) with f,g non-constant, real valued ...
22. [PDF]

### The Putnam Group - 2004 Mel Hausner Problems on symmetric ...

The Putnam Group - 2004. Mel Hausner. Problems on symmetric functions of roots of a polynomial equation. In what follows, a1,a2,...an are the roots of the ...
23. [PDF]

### Solutions (part 2)

www.math.nyu.edu/~bellova/putnam/putnam08_2sp.pdf
THE THIRTY-SECOND WILLIAM LOWELL PUTNAM. MATHEMATICAL COMPETITION. December 4, I971. A—l The set of all lattice points can be divided into ...
24. [PDF]

### see problems 1

www.math.nyu.edu/~bellova/putnam/putnam08_4sp.pdf
N U l'iiﬁEk THEOK'i COMTW 05B. ® i0i 54. Do there exist 1.000.000 consecutive integers each of which contains a repeated prime factor? First Solution.
25. [PDF]

### Problem Set 3 (10/12-10/21) - Number Theory

www.math.nyu.edu/~bellova/putnam/putnam09_3.pdf
Number Theory. 10/21/2009. Easy problems, appeared in previous years' problem sets: 1. Find the last two digits of 77.. .7 where the tower contains seven 7's. 2.
26. [PDF]

### Solutions (part 1)

www.math.nyu.edu/~bellova/putnam/putnam08_7s.pdf
favorite of physics Nobelist (and Putnam Fellow) Richard Feynman. Noam Elkies notes that this integral is number 2.491#8 in Gradshteyn and Ryzhik, Table of ...
27. [PDF]

### Problem Set 2 (10/05) - Polynomials

www.math.nyu.edu/~bellova/putnam/putnam09_2.pdf
Recent Putnam problems: 1.[2007-B1] Let f be a polynomial with positive integer coefficients. Prove that if n is a positive integer, then f(n) divides f(f(n) + 1) if and ...
28. [PDF]

### ATchentsov - resume - Department of Mathematics - New York ...

Awards: Top 1% nationwide in 2009 US William L. Putnam Mathematical Competition (among 20,000 candidates), awarded annual UCLA Basil Gordon Prize, ...
29. [PDF]

### Problem Set 1 (09/10) - Some Putnam Problems

www.math.nyu.edu/~bellova/putnam/putnam08_1.pdf
[1999-A1]. Find polynomials f(x),g(x), and h(x), if they exist, such that for all x,. |f(x)|−|g(x)| + h(x) = −1 if x < −1. = 3x +2 if − 1 ≤ x ≤ 0. = −2x +2 if x > 0. [1995-A1].
30. [PDF]

### Problems 12 - 16: see problems 5, 7

www.math.nyu.edu/~bellova/putnam/putnam08_3sp.pdf
1%5. @ lags,. N U H \s e R "1-. Ba H 56 R 7 divisible by 2 211g SFZHHSIZQtfnCtOIAiOl two of such numbers would have to be. _ _ . . . 1 IS su ment to prove that.
31. [PDF]

### Problems 1, 7, 11

www.math.nyu.edu/~bellova/putnam/putnam09_3s.pdf
Solutions to Number Theory problems. 1. The question is, what is the remainder of 77.. . 7 after division by 100. Note that 74 = 2401 ≡ 1(mod 100); so it suffices ...
32. [PDF]

### Solutions (part 2)

www.math.nyu.edu/~bellova/putnam/putnam08_9sp.pdf
Soluligh of Reﬁurrehceg Pmblemg. \, DQOO'AITE (150,9,1,0,0,0,0,o,1,145.33). Let. T9=2, T1 =3, T2=6, and for n 2 3,. T, = (n + 4)T,,_1 — 4nT,,_g + (4n — 8)T,,_3.
33. [PDF]

### Solutions (part 1)

www.math.nyu.edu/~bellova/putnam/putnam08_2s.pdf
Solutions to the problems on induction and pigeonhole principle. 1. Summing up the first n odd numbers for small n, we get 1 = 1, 1+3 = 4, 1+3+5 = 9, 1+3+5+7 ...
34. [PDF]

### Solutions (part 1)

www.math.nyu.edu/~bellova/putnam/putnam08_3s.pdf
Solutions to Number Theory problems. 1. The question is, what is the remainder of 77.. . 7 after division by 100. Note that 74 = 2401 ≡ 1(mod 100); so it suffices ...
35. [PDF]

### Solutions to ”Summation of Series” problems 1. (a) By binomial ...

www.math.nyu.edu/~bellova/putnam/putnam08_6s.pdf
Solutions to ”Summation of Series” problems. 1. (a) By binomial theorem,. 1 − (n1) + (n2) − (n3) + ··· + (−1)n (nn) = (1 − 1)n = 0. (b). 1 · 2 (n2) + 2 · 3 (n3) + ··· + (n ...
36. [PDF]

### CV - New York University > Courant Institute > Department of ...

Putnam Exam Practice Sessions, NYU, Fall 2009, 2008 and 2007. • Math Patterns in Nature, NYU, Spring 2008. • Calculus Proseminar, Charles University, Fall ...
37. [PDF]

### Solutions (part 2)

www.math.nyu.edu/~bellova/putnam/putnam08_8sp.pdf
\q_ L.\o|qL|-BZ.] (2a, a, 49, o, o, o, o, o, 56, 10, 39, 16). For which real numbers 0 is there a straight line that intersects the curve y= =4+913 +cr2+9z+4.
38. [PDF]

### YUE YU - New York University > Courant Institute > Department of ...

Lowell Putnam Competition; Co-organizer of Brown University "Year of China". EXPERIENCE. Taichi Capital. New York, NY. Fund of Funds Analyst (July 2012 ...
39. [PDF]

### also available in PDF - New York University > Courant Institute ...

1998 Highest score on the Ohio State team for the Putnam math competition. 1997 First place, national trig-star championship. 1st in United States among all ...
40. [PDF]

### On Binomial Coeffcients.

Putnam Notes. Melvin Hausner. Basic Facts about Binomial Coefficients. There are many equivalent ways of defining. (n r. ) . (Read this as “n choose r.”) Here ...
41. [PDF]

### ATchentsov - resume - New York University > Courant Institute ...

activities), May 2007; Top 5% in 2007 William L. Putnam Mathematical Competition. ▫ Coursework in macro and microeconomics, financial accounting, ...
42. [PDF]

### Number Theory 1. Find the last two digits of 77.. where the tower ...

www.math.nyu.edu/~bellova/putnam/putnam08_3.pdf
Number Theory. 1. Find the last two digits of 77.. . 7 where the tower contains seven 7's. 2. Several positive integers are written on a chalk board. One can ...
43. [PDF]

### Problem Set 6 (11/11) - Inequalities

www.math.nyu.edu/~bellova/putnam/putnam09_6.pdf
Inequalities. 11/11/2009. 1. For a, b, c ≥ 0, prove. (a + b)(b + c)(c + a) ≥ 8abc. 2. For all x, prove x2 + 2. √x2 + 1 ≥. 2. 3. Show that. 1 +. 1. √2 +. 1. √3 + ... +. 1.
44. [PDF]

### Limits and infinite series 10/27/2009 0. Compute ∑ n=0arctan( 1 1 + ...

www.math.nyu.edu/~bellova/putnam/putnam09_4.pdf
Limits and infinite series. 10/27/2009. 0. Compute. ∞. ∑ n=0arctan(. 1. 1 + n + n2 ) . 1.[1982-A2] For positive real x, let. Bn(x)=1x + 2x + 3x + ... + nx. Prove or ...
45. [PDF]

### Problem Set 1 (09/28) - The Invariance Principle

www.math.nyu.edu/~bellova/putnam/putnam09_1.pdf
The Invariance Principle. 09/28/2009. 1. Suppose the positive integer n is odd. First Al writes the numbers 1, 2,..., 2n on the blackboard. Then he picks any two ...
46. [PDF]

### Number Theory (continued) 10/01/2008 1.[1955-B4] Do there exist 1 ...

www.math.nyu.edu/~bellova/putnam/putnam08_4.pdf
Number Theory (continued). 10/01/2008. 1.[1955-B4] Do there exist 1, 000, 000 consecutive integers each of which contains a repeated prime factor? 2.
47. [PDF]

### Solutions (part 1)

www.math.nyu.edu/~bellova/putnam/putnam08_9s.pdf
Solutions to Recurrences problems (11/12/2008). 2.[1996-B1] Let [n] denote the set {1, 2,...,n}, and let fn denote the number of minimal selfish subsets of [n].
48. [PDF]

### Summation of Series 10/15/2008 1. Sum each of the following: 1 ...

www.math.nyu.edu/~bellova/putnam/putnam08_6.pdf
Summation of Series. 10/15/2008. 1. Sum each of the following: 1 − (n1) + (n2) − (n3) + ... + (−1)n (nn) ;. 1 · 2 (n2) + 2 · 3 (n3) + ... + (n − 1)n (nn) ;. (n1) + 22 (n2) + ...
49. [PDF]

### Algebraic identities, polynomials 10/08/2008 1.[2004-B1] Let P(x) = c ...

www.math.nyu.edu/~bellova/putnam/putnam08_5.pdf
Algebraic identities, polynomials. 10/08/2008. 1.[2004-B1] Let P(x) = cnxn + cn−1xn−1 + ··· + c0 be a polynomial with integer coefficients. Suppose that.
50. [PDF]

### Problem Set 7 (10/29) - Calculus & Analysis

www.math.nyu.edu/~bellova/putnam/putnam08_7.pdf
Calculus/Analysis. 10/29/2008. 0. Prove or disprove: Every infinite sequence of real numbers has either a nondecreasing subsequence or a nonincreasing ...
51. [PDF]

### Calculus & Analysis continued

www.math.nyu.edu/~bellova/putnam/putnam08_8w.pdf
Calculus/Analysis. 11/5/2008. A few more calculus/analysis problems: 16.[1997-B2] Let f be a twice-differentiable real-valued function satisfying f(x) + f′′(x) ...
52. [PDF]

### Solutions (part 1)

www.math.nyu.edu/~bellova/putnam/putnam08_8s.pdf
Solutions to Calculus problems (11/5/2008). 16.[1997-B2] It suffices to show that |f(x)| is bounded for x ≥ 0, since f(−x) satisfies the same equation as f(x).
53. [PDF]

### Problem Set 10 (11/19) - Inequalities

www.math.nyu.edu/~bellova/putnam/putnam08_10.pdf
Inequalities. 11/19/2008. 1.[2004-A2] For i = 1, 2 let Ti be a triangle with side lengths ai,bi,ci, and area Ai. Suppose that a1 ≤ a2,b1 ≤ b2,c1 ≤ c2, and that T2 is ...
54. [PDF]

### The Courant Institute - New York University > Courant Institute ...

activities), May 2007; Top 5% in 2007 William L. Putnam Mathematical Competition. ▫ Coursework in macro and microeconomics, financial accounting, ...
55. [PDF]

### Problem Set 7 (11/18) - Pigeonhole Principle

www.math.nyu.edu/~bellova/putnam/putnam09_7.pdf
Pigeonhole Principle. 11/18/2009. From last year: 1. Let n > 1 and let P(x) be a polynomial with integer coefficients and degree at most n. Suppose that. |P(x)| < n ...
56. [PDF]

### Problem Set 11 (12/03) - Probability

www.math.nyu.edu/~bellova/putnam/putnam08_11.pdf
Probability. 12/3/2008. 0. Two evenly matched teams play in the world series, a best of seven competition in which the competition stops as soon as one team ...
57. [PDF]

### utmdabtr-005-04 - MD Anderson Cancer Center

by LD Broemeling - ‎Cited by 1 - ‎Related articles
Translation by C.G. Putnam.Boston, Hilliard Gray. MATTHEWS, R.J. (1995). Quantification and the Quest for Medical Certainty. Princeton, University of Princeton ...
58. [PDF]

### Solutions of Inequalities problems (11/19/2008) 1.[2004-A2] First ...

www.math.nyu.edu/~bellova/putnam/putnam08_10s.pdf
Solutions of Inequalities problems (11/19/2008). 1.[2004-A2] First solution: (partly due to Ravi Vakil) Yes, it does follow. For i = 1, 2, let Pi,Qi,Ri be the vertices of ...
59. [PDF]

### Induction and Pigeonhole Principle 1. Find a formula for the sum of ...

www.math.nyu.edu/~bellova/putnam/putnam08_2.pdf
Induction and Pigeonhole Principle. 1. Find a formula for the sum of the first n odd numbers. 2. Let f(n) be the number of regions which are formed by n lines in ...
60. [PDF]

### Solutions to ”Algebraic identities, polynomials” problems 1.[2004-B1 ...

www.math.nyu.edu/~bellova/putnam/putnam08_5s.pdf
Solutions to ”Algebraic identities, polynomials” problems. 1.[2004-B1] Let k be an integer, 0 ≤ k ≤ n − 1. Since P(r)/rk = 0, we have cnrn−k + cn−1rn−k+1 + .
61. [PDF]

### Completing Book II of Archimedes's On Floating Bodies - New York ...

by C RORRES - ‎Cited by 18 - ‎Related articles
Brunetti), Penguin Putnam Inc., New York, 1969, p. 368. 22. E. S. Stamatis, The Complete Archimedes, (In Greek: Ε. Σ. ΣΤΑΜΑΤΗΣ,. ΑΡΧΙΜΗ∆ΟΥΣ ΑΠΑΝΤΑ) ...
62. [PDF]

### CourseNum ShortTitle CourseType SectionNum InstructorUniqueId ...

V121. CALC I. LEC. 61. Sundarajan, Arun. 4:55 P. 6:10 P. V111. Putnam. LEC. 1. TBA ,. 7:00 P. 8:00 P. V121. CALC I. RCT. 43. TBA, TA,. 4:00 A. 4:50 A. V255.
63. [PDF]

### Interview

（Thomas J.Putnam）. Q：总统图书馆和普通图书馆的使命有什么. 不同？ A：它的使命有两部分。一方面，总统图书馆. 是总统在任期间的原始档案和函件的官方. 存放点 ...