WORKSHEET FOR THE PUTNAM COMPETITION · WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 3 Remark 1. At the RNMO (Romanian National mathematical olympiad) in 1986, the students

WORKSHEET FOR THE PUTNAM COMPETITION-REAL ANALYSIS-

INSTRUCTOR: CEZAR LUPU

Problem 1. Let 0 < x1 < 1 and xn+1 = xn(1− xn), n = 1, 2, 3, . . .. Show that

limn→∞

nxn = 1.

Putnam B3, 1966

Question? Problem E 3034 from the American Mathematical Monthly, 1986asks us to prove the following:

limn→∞

n(1− nxn)

log n= 1.

Problem 2. (i) Let (xn)n≥1 be a sequence such that x1 ∈ (0, 1) and

xn+1 = xn(1− x2n), n ≥ 1.

Evaluatelimn→∞

√nxn.

American Mathematical Monthly, 1967

(ii) Let (xn)n≥1 be a sequence such that x1 = x > 1 and

xn+1 = xn +√xn − 1, n ≥ 1.

Evaluate

limn→∞

4an − n2

n log n.

Romanian National contest, 2002

Problem 3. Consider the harmonic sequence (Hn)n≥1, Hn = 1 +1

2= . . . +

1

n,

n ≥ 1. Show that the sequence xn = {Hn} diverges. Here {x} is the fractional partof the real number x.

1

2 INSTRUCTOR: CEZAR LUPU

Problem 4. Let (an)n≥1 be a decreasing sequence of positive reals such that∞∑n=1

an converges. Show that limn→∞

nan = 0.

Olivier’s lemma

Problem 5. Evaluate the following limits:(i)

limn→∞

((1 + 1

n

)ne

)n

Vojtech Jarnik Cat. I, 1998

(ii)

limn→∞

(n∏k=1

k

k + n

)(e1999/n−1)

.

Vojtech Jarnik Cat. I, 1999

Problem 6. Define the sequence x1, x2, . . . inductively by x1 =√

5 and xn+1 =x2n − 2 for each n ≥ 1.

Compute

limn→∞

x1x2 . . . xnxn+1

.

International Mathematical Competition Problem 3, 2010

Remark. Solve problem 10259 from the American Mathematical Monthly 1992,which has the following statement:

Question. Let (rn)n≥0 be a sequence defined by r0 = 3 and rn+1 = r2n − 2.Evaluate

limn→∞

2n

√√√√n−1∏k=0

rk.

Problem 7. Suppose that a0 = 1 and that an+1 = an + e−an , n = 0, 1, 2, . . ..Does the sequence (an − log n)n≥1 have finite limit?

Putnam B4, 2012

WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 3

Remark 1. At the RNMO (Romanian National mathematical olympiad) in 1986,the students were asked to show that

limn→∞

xnlog n

= 1.

Problem 8. Let k be an integer greater than 1. Suppose that a0 > 0, and definethe sequence (an)n≥0,

an+1 = an +1

k√an, n ≥ 0.

Evaluate

limn→∞

ak+1n

nk.

Putnam B6, 2006

Remark. The case k = 2 was given at the Romanian National MathematicalOlympiad (District level) in 2004. Try to solve this particular case first! It givessome insight how to approach the general case.

Problem 9. Does there exist a sequence (an)n≥1 of positive reals such that the

series∞∑n=1

an converges, andn∏k=1

ak ≤ nn for all n ≥ 1?

American Mathematical Monthly (Problem 11748), 2014

Remark. At the Balkan Mathematical Olympiad, 2008 the following ”easier”problem was given:

Probem 9′. Does there exist a sequence (an)n≥1 of positive reals such that the

seriesn∑k=1

ak ≤ 2008, andn∑k=1

ak ≤ n2 for all n ≥ 1?

Problem 10. Let (an)n≥1 be a sequence of positive reals such that the series∑∞n=1 an converges. Show that the series

∞∑n=1

ann+1n converges also.

Putnam B4, 1988

Problem 11. Let Let (an)n≥1 be a decreasing sequence of positive reals. Letsn = a1 + a2 + . . .+ an and


bn =1

an+1

− 1

an, n ≥ 1.

Prove that if (sn)n≥1 is convergent, then (bn)n≥1 is unbounded.

Mathematical Reflections (Problem U249), 2006

Problem 12. Let (xn)n≥2) be a sequence of real numbers such that x2 > 0 and

xn+1 = −1 + n√

1 + nxn, n ≥ 2.

Show that:

(i) limn→∞

xn = 0;

(ii) limn→∞

xn+1

xn= 1 and lim

n→∞nxn = 0.

RNMO 1991, Vojtech Jarnik (Cat. I) 2005

Problem 13. Let (an)n≥1 be a sequence of positive reals. Show that

lim supn→∞

n

(1 + an+1

an− 1

)≥ 1,

and

lim supn→∞

(a1 + an+1

an

)n≥ e.

Putnam A4, 1963

Problem 14. Let an be a sequence such that xn =n∑k=1

a2k converges, while

yn =n∑k=1

ak is unbounded. Show that the sequence (bn)n≥1, bn = {yn}, n ≥ 1 di-

verges. Here {x} is the fractional part of the real number x.

Romanian National Mathematical Olympiad, 1998

Problem 15. Show that if the series∞∑n=1

1

pnis convergent, where p1, p2, . . . , pn

are positive real numbers, then the series

∞∑n=1

n2

(p1 + p2 + . . .+ pn)2· pn


is also convergent.

Putnam B3, 1966

Problem 16. Determine, with proof, whether the series

∞∑n=1

1

n1.8+sinn

converges or diverges.

University of Illinois at Urbana-Champaign Math Contest, 2005

Problem 17. [Carleman’s inequality] Let a1, a2, . . . , an be a sequence of nonneg-ative real numbers. Prove that

∞∑n=1

n√a1a2 . . . an ≤ e

∞∑n=1

an.

Problem 18. [Hardy’s inequality] Let a1, a2, . . . , an be a sequence of nonnegativereal numbers and p > 1. Prove that

∞∑n=1

(a1 + a2 + . . .+ an

n

)p≤(

p

p− 1

)p ∞∑n=1

apn.

Problem 19. Let f : [0,∞)→ R be a function satisfying

f(x)ef(x) = x, x ≥ 0.

Prove that:

(a) f is monotone;(b) limx→∞ f(x) =∞.

(c)f(x)

log xtends to 1 as x→∞.

Problem 20. Let f : (0,∞)→ R be a function satisfying

limx→∞

f(x)

xk= a, k 6= 1.

and

limx→∞

f(x+ 1)− f(x)

xk−1= b ∈ R.

Prove that b = ka.RNMO, 1985


Problem 21. Let f : (0, 1) → R be a function satisfying limx→0+ f(x) = 0and such that there exists 0 < λ < 1 with lim

x→0+(f(x) − f(λx))/x = 0. Prove that

limx→0+

f(x)

x= 0.

Problem 22. Let f : (a, b) → R be a function such that limx→x0 f(x) exists forany x0 ∈ [a, b]. Prove that f is bounded if and only for all x0 ∈ [a, b], limx→x0 f(x)is finite.

Problem 23. Let a, b ∈(0, 1

2

)and f : R→ R is a continuous function such that

f(f(x)) = af(x) + bx,

for all x. Prove that f(0) = 0 and and find all such functions.

RNMO 1983 and Putnam 1991

Problem 24. Prove that there is no continuous function f : R→ R such that

f(x) ∈ Q⇒ f(x+ 1) ∈ R−Q.RNMO, 1979

Problem 25. Let f : R→ R be a continuous function such that

f(x) ≤ f

(x+

1

n

),

for every real x and positive integer n. Show that f is nondecreasing.

Problem 26. Find all functions f : (0,∞)→ (0,∞) subject to the conditions:

(i) f(f(f(x))) + 2x = f(3x) for all x > 0.(ii) lim

x→∞(f(x)− x) = 0.

RNMO SHL 2003

Problem 27. Let f(x) be a continuous function such that f(2x2 − 1) = 2xf(x)for all x. Show that f(x) = 0 for −1 ≤ x ≤ 1.

Putnam B4, 2000

Problem 28. Find all functions f : R → R such that for any a < b, f([a, b]) isan interval of length b− a.

IMC Problem 2(Day 2), 2006

Problem 29. Calculate∞∑n=1

ln

(1 +

1

n

)ln

(1 +

1

2n

)ln

(1 +

1

2n+ 1

).



Problem 30. Prove that if f : [0, 1] → [0, 1] is a continuous function then thesequence of iterates xn+1 = f(xn) converges if and only if lim

n→∞(xn+1 − xn) = 0.


Problem 31. Let f(x) =n∑k=1

ak sin kx, with a1, a2, . . . , an real numbers and

n ≥ 1. Prove that if f(x) ≤ | sinx|, for all x ∈ R, then∣∣∣∣∣n∑k=1

kak

∣∣∣∣∣ ≤ 1.

Putnam A1, 1967

Problem 32. Let f : [a, b] → R be a function, continuous on [a, b] and twicedifferentiable on (a, b). If f(a) = f(b) and f ′(a) = f ′(b), prove that for every realnumber λ, the equation

f ′′(x)− λ(f ′(x))2 = 0

has at least one zero in the interval (a, b).

Problem 33. Let f be a real function with continuous third derivative such thatf(x), f ′(x), f ′′(x), f ′′′(x) are positive for all x. Suppose that f ′′′(x) ≤ f(x) for all x.Show that f ′(x) < 2f(x) for all x.

Putnam B4, 1999

Problem 34. Let α > 1 be a real number, and let (un)n≥1 be a sequence of pos-

itive numbers such that limn→∞ un = 0 and limn→∞

un − un+1

uαnexists and is nonzero.

Prove that∑∞

n=1 un converges if and only if α < 2.


Problem 35. Prove that, for any two bounded functions g1, g2 : R → [1,∞),there exist functions h1, h2 : R→ R such that for every x ∈ R,

sups∈R

(g1(s)xg2(s)) = max

t∈R(xh1(t) + h2(t)).

Putnam B5, 2012

Problem 36. Is there a strictly increasing function f : R→ R such that

f ′(x) = f(f(x)),

for all reals x?Putnam 2010


Problem 37. Does there exist a continuously differentiable function f : R → Rsatisfying f(x) > 0 and

f ′(x) = f(f(x)),

for all reals x?IMC, 2001

Problem 38. Let f and g be (real-valued) functions defined on an open intervalcontaining 0, with g nonzero and continuous at 0. If fg and f/g are differentiableat 0, must f be differentiable at 0?

Putnam B3, 2011

Problem 39. Find all differentiable functions f : R→ R such that

f ′(x) =f(x+ n)− f(x)

nfor all real numbers x and all positive integers n.

Putnam A2, 2010

Problem 40. Let f be a real function on the real line with continuous thirdderivative. Prove that there exists a point a such that

f(a) · f ′(a) · f ′′(a) · f ′′′(a) ≥ 0.

Putnam A3, 1998

Problem 41. Find the set of all real numbers k with the following property: Forany positive, differentiable function f that satisfies f ′(x) > f(x) for all x, there issome number N such that f(x) > ekx for all x > N.

Putnam B3, 1994

Problem 42. Let f : (1,∞)→ R be a differentiable function such that

f ′(x) =x2 − (f(x))2

x2((f(x))2 + 1

) for all x > 1.

Prove that limx→∞

f(x) =∞.Putnam B5, 2009

Problem 43. Let f : (0,∞) → R be a twice continuously differentiable suchthat

|f ′′(x) + 2xf ′(x) + (x2 + 1)f(x)| ≤ 1,

for all x.Prove that limx→+∞ f(x) = 0



Problem 44. Define f : R→ R by

f(x) =

{x if x ≤ e

xf(lnx) if x > e

Does∞∑n=1

1

f(n)converge?

Putnam A4, 2008

Problem 45. Find all sequences a0, a1, . . . , an of real numbers such that an 6= 0,for which the following statement is true:

If f : R→ R is an n times differentiable function and x0 < x1 < . . . < xn are realnumbers such that f(x0) = f(x1) = . . . = f(xn) = 0 then there is h ∈ (x0, xn) forwhich

a0f(h) + a1f′(h) + . . .+ anf

(n)(h) = 0.


Problem 46. Let f : R → R be a continuously differentiable function thatsatisfies f ′(t) > f(f(t)) for all t ∈ R. Prove that f(f(f(t))) ≤ 0 for all t ≥ 0.


Problem 47. Let f : (0,∞)→ R be a differentiable function. Assume that

limx→∞

(f(x) +

f ′(x)

x

)= 0.

Prove that limx→∞

f(x) = 0.

Vojtech Jarnik Competition, 2014

Problem 48. Let f be twice continuously differentiable function on (0,∞) suchthat lim

x→0+f ′(x) = −∞ and lim

x→0+f ′′(x) = +∞. Show that

limx→0+

f(x)

f ′(x)= 0.


Problem 49. Determine all Riemann integrable functions f : R→ R such that∫ x+1/n

0

f(t)dt =

∫ x

0

f(t)dt+1

nf(x),

for all reals x and all positive integers n.

RNMO SHL, 2006


Problem 50. For each continuous function f : [0, 1]→ R , let I(f) =

∫ 1

0

x2f(x)dx

and J(f) =

∫ 1

0

xf 2(x)dx. Find the maximum value of I(f) − J(f) over all such

functions.

Putnam B5, 2006

Problem 51. Compute the following integrals:

(i)

∫ 1

0

log(1 + x)

xdx.

(ii)

∫ 1

0

arctanx

x+ 1dx.

(iii)

∫ 1

0

log(1 + x)

x2 + 1dx.

Putnam A5, 2005

(iv)

∫ 1

0

log(1 + x2)

x2 + 1dx.

(v)

∫ 1

0

log(1 + x2)

1 + xdx.

(vi)

∫ π/4

0

log(1 + tanx)dx.

Problem 52. Compute the following limits:

(i) limn→∞

1

n2

(n∏k=1

(n2 + k2)

)1/n

.

(ii) limn→∞

(∏nk=1 k

k

n∑nk=1 k

)1/n2

.

(iii) limn→∞

1

n4

2n∏k=1

(n2 + k2)1/n.

Putnam B1, 1970

(iv) limn→∞

n∏k=1

(1 +

k

n

)1/k

.

Problem 53. Let f be a real-valued continuous function on [0, 1] such that∫ 1

0

xkf(x)dx = 0, for 0 ≤ k ≤ n− 1 and

∫ 1

0

xnf(x)dx = 1. Show that there exists

x0 ∈ [0, 1] such that |f(x0)| ≥ 2n(n+ 1).


Putnam A6, 1972

Problem 54. Let f : [0, 1] → R be a differentiable function with continuousderivative such that f(1) = 0. Show that

4

∫ 1

0

x2|f ′(x)|2dx ≥∫ 1

0

|f(x)|2dx+

(∫ 1

0

f(x)dx

)2

.

College Mathematics Journal, 2012

Problem 55. Let f : [0, 1]→ [0, 1) be a continuous function such that∫ 1

0

f(xn)dx ≤∫ 1

0

fn(x)dx, n ≥ 1.

Show that f(x) = 0 for all x ∈ [0, 1].

Problem 56. Find all continuous functions f : R→ [1,∞) for which there existsa ∈ R and k positive integer such that

f(x)f(2x) . . . f(nx) ≤ ank,

for every real number x and positive integer n.

RNMO, 1999

Problem 57. Compute the integral

In =

∫ π2

0

sinn xdx.

Use the answer to prove Wallis formula:

limn→∞

1

n

[2 · 4 · 6 . . . (2n)

1 · 3 · 5 . . . (2n− 1)

]2= π.

Problem 58. Let f : [0, 1] → R+ be an integrable function. Compute thefollowing limits:

limn→∞

n∏k=1

(1 +

1

nf

(k

n

)),

limn→∞

(n∑k=1

exp

(1

nf

(k

n

))− n

).

Problem 59. Let f : [−1, 1]→ R be a continuous function having finite deriva-tive at 0, and

I(h) =

∫ h

−hf(x) dx, h ∈ [0, 1].


Prove that a) there exists M > 0 such that |I(h)−2f(0)h| ≤Mh2, for any h ∈ [0, 1].

b) the sequence (an)n≥1, defined by an =∑n

k=1

√k|I(1/k)|, is convergent if and only

if f(0) = 0.

RNMO, 2010

Problem 60. Prove that

limn→∞

n

(π

4− n

∫ 1

0

xn

1 + x2ndx

)=

∫ 1

0

f(x) dx,

where f(x) = arctanxx

if x ∈ (0, 1] and f(0) = 1.

RNMO, 2006

Problem 61. Let f : [0, 1]→ R be a continuous function such that∫ 1

0

f(x)dx = 0.

Prove that there is c ∈ (0, 1) such that∫ c

0

xf(x)dx = 0.

RNMO, 2006

Problem 62. Let f : [−1, 1]→ R be a continuous function. Show that:

a) if

∫ 1

0

f(sin(x+ α)) dx = 0, for every α ∈ R, then f(x) = 0, ∀x ∈ [−1, 1].

b) if

∫ 1

0

f(sin(nx)) dx = 0, for every n ∈ Z, then f(x) = 0, ∀x ∈ [−1, 1].

RNMO, 2001

Problem 63. Let f : [0,∞)→ R be a periodic function, with period 1, integrableon [0, 1]. For a strictly increasing and unbounded sequence (xn)n≥0, x0 = 0, withlimn→∞

(xn+1 − xn) = 0, we denote r(n) = max{k | xk ≤ n}.a) Show that:

limn→∞

1

n

r(n)∑k=1

(xk − xk+1)f(xk) =

∫ 1

0

f(x) dx

b) Show that:

limn→∞

1

log n

r(n)∑k=1

f(log k)

k=

∫ 1

0

f(x) dx


RNMO, 2001

Problem 64. Let f and g be two continuous, distinct functions from [0, 1] →

(0,+∞) such that

∫ 1

0

f(x)dx =

∫ 1

0

g(x)dx. Let (yn)n≥1 be a sequence defined by

yn =

∫ 1

0

fn+1(x)

gn(x)dx,

for all n ≥ 0.Prove that (yn) is an increasing and divergent sequence.

RNMO 2003

Problem 65. Let f, g : [a, b] → [0,∞) be two continuous and non-decreasingfunctions such that each x ∈ [a, b] we have∫ x

a

√f(t) dt ≤

∫ x

a

√g(t) dt and

∫ b

a

√f(t) dt =

∫ b

a

√g(t) dt.

Prove that ∫ b

a

√1 + f(t) dt ≥

∫ b

a

√1 + g(t) dt.

IMC Problem 2 (Day 2), 2004

Problem 66. Let f : R→ R be a continuous and bounded function such that

x

∫ x+1

x

f(t) dt =

∫ x

0

f(t) dt, for any x ∈ R.

Prove that f is a constant function.

RNMO, 2002

Problem 67. Let f : [0, 1]→ R be an integrable function such that:

0 <

∣∣∣∣∫ 1

0

f(x) dx

∣∣∣∣ ≤ 1.

Show that there exists x1 6= x2, x1, x2 ∈ [0, 1], such that:∫ x2

x1

f(x) dx = (x1 − x2)2002

.

RNMO, 2002


Problem 68. Let f : R→ (0,∞) be a continously differentiable function. Provethat ∣∣∣∣∫ 1

0

f 3(x)dx− f 2(0)

∫ 1

0

f(x)dx

∣∣∣∣ ≤ max[0,1]|f ′(x)|

(∫ 1

0

f(x)dx

)2

.


Problem 69. Let f : [0, 1]→ [0,∞) be an integrable function. Show that∫ 1

0

f(x)dx

∫ 1

0

f 2(x)dx ≤∫ 1

0

f 3(x)dx,

and ∫ 1

0

xf(x)dx

∫ 1

0

x2f(x)dx ≤∫ 1

0

f(x)dx

∫ 1

0

x3f(x)dx.

Problem 70. Let f : [0, 1]→ R be an integrable function such that∫ 1

0

f(x)dx =

∫ 1

0

xf(x)dx = 1.

Show that ∫ 1

0

f 2(x)dx ≥ 4.

RNMO 2004

Problem 71. Show that for any continuous function f : [0, 1]→ R,

1

4

∫ 1

0

f 2(x)dx+ 2

(∫ 1

0

f(x)dx

)2

≥ 3

∫ 1

0

f(x)dx

∫ 1

0

xf(x)dx.

RNMO SHL, 2007

Problem 72. Let f : [0, 1]→ R be an integrable function such that∫ 1

0

f(x)dx =

∫ 1

0

xf(x)dx =

∫ 1

0

x2f(x)dx = 1.

Show that ∫ 1

0

f 2(x)dx ≥ 9.

Jozeph Wildt Competition, 2005


Problem 73. Let f : [0, 1] → R be a differentiable function with continuousderivative such that ∫ 1

0

f(x)dx =

∫ 1

0

xf(x)dx = 1.

Show that ∫ 1

0

(f ′(x))2dx ≥ 30.

RNMO SHL, 2005

Problem 74. Find

limn→∞

1

n

∫ n

0

x log(1 + x/n)

1 + xdx.


Problem 75. (a) Let f : [0, 1]→ R be a continuous function such that

∫ 1

0

f(x)dx =

0. Show that there exists c ∈ (0, 1) such that

f(c) =

∫ c

0

f(x)dx.

(b) Let f : [0, 1] → R be a continuous function such that f(1) = 0. Show thatthere exists c ∈ (0, 1) such that

f(c) =

∫ c

0

f(x)dx.

Problem 76. (a) Let f ∈ C1([0, 1]) such that f(0) = f(1) = 0. Show that∫ 1

0

(f ′(x))2dx ≥ 12

(∫ 1

0

f(x)dx

)2

.

(b) Let f ∈ C1([0, 1]) such that f(1/2) = 0. Show that∫ 1

0

(f ′(x))2dx ≥ 12

(∫ 1

0

f(x)dx

)2

.

Elemente der Mathematik, 1983 & RNMO, 2008

Problem 77. Suppose that f : [0, 1] → R has a continuous derivative and that∫ 1

0f(x) dx = 0. Prove that for every α ∈ (0, 1),∣∣∣∣∫ α

0

f(x) dx

∣∣∣∣ ≤ 1

8max0≤x≤1

|f ′(x)|


Putnam B2, 2007

Problem 78. Let F0 = lnx. For n ≥ 0 and x > 0, let Fn+1(x) =

∫ x

0

Fn(t) dt.

Evaluate limn→∞

n!Fn(1)

lnn.

Putnam B2, 2008

Problem 79. Let f : [0,∞) → R be a strictly decreasing continuous function

such that limx→∞ f(x) = 0. Prove that

∫ ∞0

f(x)− f(x+ 1)

f(x)dx diverges.

Putnam A6, 2010

Problem 80. Prove that∫ 1

0

∫ 1

0

dx dy1x

+ | log y| − 1≤ 1.


Problem 81. Let 0 < a < b and let f : [a, b]→ R be a continuous function with∫ b

a

f(t)dt = 0. Show that∫ b

a

∫ b

a

f(x)f(y) log(x+ y)dxdy ≤ 0.


Problem 82. Let us define the sequence (an)n≥1.

an =3

2log n−

∫ 1

0

log(1t + 2t + . . .+ nt)dt.

(i) Show that the sequence an is convergent and find its limit a.

(ii) Show that 0 < n(a− an) <3

2.

RNMO, 2008

Problem 83. Let f : R → R be a continuous and periodic function of periodT > 0. Show that:

(i) limx→∞

1

x

∫ x

0

f(t)dt =1

T

∫ T

0

f(t)dt.

(ii) limn→∞

∫ b

a

f(nx)dx =(b− a)

T

∫ b

a

f(t)dt.


(iii) Let f, g : R → R be continuous functions such that f(x + 1) = f(x) andg(x+ 1) = g(x) for all real numbers x. Prove that

limn→∞

∫ 1

0

f(x)g(nx)dx =

∫ 1

0

f(x)dx

∫ 1

0

g(x)dx.

Putnam B3, 1967

Problem 84. Let a1, a2, . . . be real numbers. Suppose there is a constant A suchthat for all n, ∫ ∞

−∞

(n∑i=1

1

1 + (x− ai)2

)2

dx ≤ An.

Prove there is a constant B > 0 such that for all n,n∑

i,j=1

(1 + (ai − aj)2

)≥ Bn3.

Putnam B5, 2011

Problem 85. Let f(x) be a continuous real-valued function dened on the interval[0, 1]. Show that ∫ 1

0

∫ 1

0

|f(x) + f(y)|dx dy ≥∫ 1

0

|f(x)|dx

Putnam B6, 2003

Problem 86. Suppose that f(x, y) is a continuous real-valued function on theunit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Show that∫ 1

0

(∫ 1

0

f(x, y)dx

)2

dy +

∫ 1

0

(∫ 1

0

f(x, y)dy

)2

dx

≤(∫ 1

0

∫ 1

0

f(x, y)dxdy

)2

+

∫ 1

0

∫ 1

0

[f(x, y)]2 dxdy.

Putnam A6, 2004

Problem 87. Find all r > 0 such that when f : R2 → R is differentiable,‖grad f(0, 0)‖ = 1, ‖grad f(u) − grad f(v)‖ ≤ ‖u − v‖, then the max of f on thedisk ‖u‖ ≤ r, is attained at exactly one point.


Problem 88. (i) [Hermite-Hadamard] Let f : [a, b] → R be a convex function.Show that

(b− a)f

(a+ b

2

)≤∫ b

a

f(x)dx ≤ (b− a)f(a) + f(b)

2.

(ii) Show that


∫ k+1

k

f(x)dx ≥ log k + log(k + 1)

2, k ≥ 1,

and ∫ k+1/2

k−1/2log xdx ≤ log k, k ≥ 1.

(iii) Consider the sequence (an)n≥1 defined by

an =

∫ n

1

f(x)dx− log 2− . . .− log(n− 1)− 1

2log n, n ≥ 1.

Show that an is increasing and 0 ≤ an ≤1

2log

5

4.

(iv) Prove that √4

5e√n( en

)n≤ n! ≤ e

√n( en

)n, n ≥ 1.

(v) [Stirling] Show that

limn→∞

n!

nnen√

2πn= 1.

Problem 89. Show that for any continuous function f : [−1, 1] → R we havethe following inequalities:

∫ 1

−1f 2(x)dx ≥ 1

2

(∫ 1

−1f(x)dx

)2

+3

2

(∫ 1

−1xf(x)dx

)2

∫ 1

−1f 2(x)dx ≥ 5

2

(∫ 1

−1x2f(x)dx

)2

+3

2

(∫ 1

−1xf(x)dx

)2

.

RNMO 1997, Gazeta Matematica-A series, 2009

Problem 90. (i) Let f : [0, 1]→ R be a continuous function. Show that

limn→∞

∫ 1

0

xnf(x)dx = 0.

(ii) Let f : [0, 1]→ R be a continuous function. Show that

limn→∞

n

∫ 1

0

xnf(x)dx = f(1),

and

limn→∞

n

∫ 1

0

xnf(xn)dx =

∫ 1

0

f(x)dx.


(iii) Let f, g : [0, 1]→ R be two continuous functions. Show that

limn→∞

∫ 1

0xnf(x)dx∫ 1

0xng(x)dx

=f(1)

g(1),

and

limn→∞

∫ 1

0xnf(xn)dx∫ 1

0xng(xn)dx

=

∫ 1

0f(x)dx∫ 1

0g(x)dx

.

(iv) Find a real number c and a positive number L for which

limr→∞

rc∫ π/20

xr sinxdx∫ π/20

xr cosxdx= L.

Putnam A3, 2011

Problem 91. Let (an)n∈N be the sequence defined by

a0 = 1, an+1 =1

n+ 1

n∑k=0

akn− k + 2

.

Find the limit

limn→∞

n∑k=0

ak2k.


Problem 92. For any continuous real-valued function f defined on the interval[0, 1], let

µ(f) =

∫ 1

0

f(x)dx,Var(f) =

∫ 1

0

(f(x)− µ(x))2dx,

and M(f) = max0≤x≤1

|f(x)|.Show that if f and g are continuous real-valued functions on the interval [0, 1],

then

Var(fg) ≤ 2 Var(f)M2(g) + 2 Var(g)M2(f).

Putnam B4, 2013

Problem 93. Prove that for any real numbers a1, a2, . . . , an we have

∑1≤i,j≤n

ij

i+ j − 1aiaj ≥

(n∑i=1

ai

)2

.



Problem 94. [Hardy] Let p > 1 and let f : [0,∞) → R be a differentiable

increasing function such that f(0) = 0 and

∫ ∞0

(f ′(x))pdx is finite. Show that∫ ∞0

x−p|f(x)|pdx ≤(

p

p− 1

)p ∫ ∞0

(f ′(x))pdx.

Problem 95. Let u1, u2, . . . , un ∈ C([0, 1]n) be nonnegative and continuousfunctions, and let uj do not depend on the j-th variable for j = 1, 2, . . . , n. Showthat (∫

[0,1]n

n∏j=1

uj

)n−1

≤n∏j=1

∫[0,1]n

un−1j .


Problem 96. Let f : [0, 1]× [0, 1]→ R be a continuous function. Find the limit

limn→∞

((2n+ 1)!

(n!)2

)2 ∫ 1

0

∫ 1

0

(xy(1− x)(1− y))nf(x, y)dxdy.


Problem 97. [Zagier] Prove that for all a1, a2, . . . , an, b1, b2, . . . bn are nonnega-tive, then the following inequality holds true:( ∑

1≤i,j≤n

min(ai, aj)

)( ∑1≤i,j≤n

min(bi, bj)

)≥

( ∑1≤i,j≤n

min(ai, bj)

).

American Mathematical Monthly

Problem 98. Let {D1, D2, . . . , Dn} be a set of disks in the Euclidian planeand aij = S(Di ∩ Dj) be the area of Di ∩ Dj. Prove that for any real numbersx1, x2, . . . , xn, the following inequality holds true:

n∑i=1

n∑j=1

aijxixj ≥ 0.


Problem 99. Let D{(x, y) ∈ R2 : x2 + y2 ≤ 1} and u ∈ C∞(R2). Suppose thatu(x, y) = 0 for all (x, y) ∈ D. Show that∫ ∫

D

|∇u(x, y|2dA ≥ 8

π

(∫ ∫D

u(x, y)dA

)2

.


Problem 100. Let u ∈ C2(D) and u = 0 on ∂D, where D is an open unit ballin R3. Prove that the following inequality∫

D

|∇u|2dV ≤ ε

∫D

(∆u)2dV +1

4ε

∫D

u2dV,

holds true for all ε > 0.Vojtech Jarnik Competition, 1997

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WORKSHEET FOR THE PUTNAM COMPETITION · WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 3 Remark 1. At the RNMO (Romanian National mathematical olympiad) in 1986, the students