WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- lim

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1 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- INSTRUCTOR: CEZAR LUPU Problem Let < x < and x n+ = x n ( x n ), n =,, 3, Show that nx n = Putnam B3, 966 Question? Problem E 334 from the American Mathematical Monthly, 986 asks us to prove the following: n( nx n ) log n = Problem (i) Let (x n ) n be a sequence such that x (, ) and Evaluate x n+ = x n ( x n), n nxn (ii) Let (x n ) n be a sequence such that x = x > and American Mathematical Monthly, 967 Evaluate x n+ = x n + x n, n 4x n n n log n Romanian National contest, Problem 3 Consider the harmonic sequence (H n ) n, H n = + = + n, n Show that the sequence x n = {H n } diverges Here {x} is the fractional part of the real number x

2 INSTRUCTOR: CEZAR LUPU Problem 4 Let (a n ) n be a decreasing sequence of positive reals such that a n converges Show that na n = n= Problem 5 Evaluate the following its: (i) (( + n e ) n ) n Olivier s lemma Vojtech Jarnik Cat I, 998 (ii) ( n k= ) (e 999/n ) k k + n Vojtech Jarnik Cat I, 999 Problem 6 Define the sequence x, x, inductively by x = 5 and x n+ = x n for each n Compute x x x n x n+ International Mathematical Competition Problem 3, Remark Solve problem 59 from the American Mathematical Monthly 99, which has the following statement: Question Let (r n ) n be a sequence defined by r = 3 and r n+ = r n Evaluate n n r k k= Problem 7 Suppose that a = and that a n+ = a n + e an, n =,,, Does the sequence (a n log n) n have finite it? Putnam B4,

3 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 3 Remark At the RNMO (Romanian National Mathematical Olympiad) in 986, the students were asked to show that a n log n = Remark Problem 837 (April, 5) from American Mathematical Monthly asks to prove that the sequence b n = a n log n, satisfies < b n+ < b n and b n = Problem 8 Let k be an integer greater than Suppose that a >, and define the sequence (a n ) n, Evaluate a n+ = a n + k an, n a k+ n n k Putnam B6, 6 Remark The case k = was given at the Romanian National Mathematical Olympiad (District level) in 4 Try to solve this particular case first! It gives some insight how to approach the general case Problem 9 Does there exist a sequence (a n ) n of positive reals such that the n series converges, and a k n n for all n? a n n= k= American Mathematical Monthly (Problem 748), 4 Remark At the Balkan Mathematical Olympiad, 8 the following easier problem was given: Probem 9 Does there exist a sequence (a n ) n of positive reals such that the n n series 8, and a k n for all n? a k k= k= Problem Let (a n ) n be a sequence of positive reals such that the series n= a n converges Show that the series converges also n= a n n+ n Putnam B4, 988

4 4 INSTRUCTOR: CEZAR LUPU Problem Let Let (a n ) n be a decreasing sequence of positive reals Let s n = a + a + + a n and b n = a n+ a n, n Prove that if (s n ) n is convergent, then (b n ) n is unbounded Mathematical Reflections (Problem U49), 6 Problem Let (x n ) n ) be a sequence of real numbers such that x > and Show that: x n+ = + n + nx n, n (i) x n = ; (ii) x n+ x n = and nx n = RNMO 99, Vojtech Jarnik (Cat I) 5 Problem 3 Let (a n ) n be a sequence of positive reals Show that and ( ) + an+ sup n, a n sup ( ) n a + a n+ e a n Problem 4 Let a n be a sequence such that x n = y n = n a k k= Putnam A4, 963 converges, while k= n a k is unbounded Show that the sequence (b n ) n, b n = {y n }, n diverges Here {x} is the fractional part of the real number x Romanian National Mathematical Olympiad, 998 Problem 5 Show that if the series are positive real numbers, then the series p n= n is convergent, where p, p,, p n

5 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 5 is also convergent n= n (p + p + + p n ) p n Putnam B3, 966 Problem 6 Determine, with proof, whether the series converges or diverges n= n 8+sin n University of Illinois at Urbana-Champaign Math Contest, 5 Problem 7 [Carleman s inequality] Let a, a,, a n be a sequence of nonnegative real numbers Prove that n= n a a a n e a n n= Problem 8 [Hardy s inequality] Let a, a,, a n be a sequence of nonnegative real numbers and p > Prove that ( ) p a + a + + a n n n= ( ) p p a p p n n= Problem 9 Let f : [, ) R be a function satisfying Prove that: f(x)e f(x) = x, x (a) f is monotone; (b) x f(x) = (c) f(x) tends to as x log x Problem Let f : (, ) R be a function satisfying and f(x) = a, k x x k f(x + ) f(x) = b R x x k

6 6 INSTRUCTOR: CEZAR LUPU Prove that b = ka RNMO, 985 Problem Let f : (, ) R be a function satisfying x + f(x) = and such that there exists < λ < with x +(f(x) f(λx))/x = Prove that f(x) x + x = Problem Let f : (a, b) R be a function such that x x f(x) exists for any x [a, b] Prove that f is bounded if and only for all x [a, b], x x f(x) is finite Problem 3 Let a, b (, ) and f : R R is a continuous function such that f(f(x)) = af(x) + bx, for all x Prove that f() = and and find all such functions RNMO 983 and Putnam 99 Problem 4 Prove that there is no continuous function f : R R such that f(x) Q f(x + ) R Q RNMO, 979 Problem 5 Let f : R R be a continuous function such that ( f(x) f x + ), n for every real x and positive integer n Show that f is nondecreasing Problem 6 Find all functions f : (, ) (, ) subject to the conditions: (i) f(f(f(x))) + x = f(3x) for all x > (ii) x (f(x) x) = RNMO SHL 3 Problem 7 Let f(x) be a continuous function such that f(x ) = xf(x) for all x Show that f(x) = for x Putnam B4, Problem 8 Find all functions f : R R such that for any a < b, f([a, b]) is an interval of length b a IMC Problem (Day ), 6

7 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 7 Problem 9 Calculate n= ( ln + ) ( ln + ) ( ln + ) n n n + IMC Problem 3 (Day ), Problem 3 Prove that if f : [, ] [, ] is a continuous function then the sequence of iterates x n+ = f(x n ) converges if and only if (x n+ x n ) = IMC Problem (Day ), 996 n Problem 3 Let f(x) = a k sin kx, with a, a,, a n k= n Prove that if f(x) sin x, for all x R, then n ka k k= real numbers and Putnam A, 967 Problem 3 Let f : [a, b] R be a function, continuous on [a, b] and twice differentiable on (a, b) If f(a) = f(b) and f (a) = f (b), prove that for every real number λ, the equation f (x) λ(f (x)) = has at least one zero in the interval (a, b) Problem 33 Let f be a real function with continuous third derivative such that f(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) < f(x) for all x Putnam B4, 999 Problem 34 Let α > be a real number, and let (u n ) n be a sequence of pos- u n u n+ itive numbers such that u n = and u α n Prove that n= u n converges if and only if α < exists and is nonzero American Mathematical Monthly, Problem 35 Prove that, for any two bounded functions g, g : R [, ), there exist functions h, h : R R such that for every x R, sup(g (s) x g (s)) = max (xh (t) + h (t)) s R t R Putnam B5, Problem 36 Is there a strictly increasing function f : R R such that

8 8 INSTRUCTOR: CEZAR LUPU for all reals x? f (x) = f(f(x)), Putnam Problem 37 Does there exist a continuously differentiable function f : R R satisfying f(x) > and for all reals x? f (x) = f(f(x)), IMC, Problem 38 Let f and g be (real-valued) functions defined on an open interval containing, with g nonzero and continuous at If fg and f/g are differentiable at, must f be differentiable at? Problem 39 Find all differentiable functions f : R R such that Putnam B3, f f(x + n) f(x) (x) = n for all real numbers x and all positive integers n Putnam A, Problem 4 Let f be a real function on the real line with continuous third derivative Prove that there exists a point a such that f(a) f (a) f (a) f (a) Putnam A3, 998 Problem 4 Find the set of all real numbers k with the following property: For any positive, differentiable function f that satisfies f (x) > f(x) for all x, there is some number N such that f(x) > e kx for all x > N Putnam B3, 994 Problem 4 Let f : (, ) R be a differentiable function such that Prove that x f(x) = f (x) = x (f(x)) x ( (f(x)) ) for all x > + Putnam B5, 9 Problem 43 Let f : (, ) R be a twice continuously differentiable such that f (x) + xf (x) + (x + )f(x),

9 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 9 for all x Prove that x + f(x) = Problem 44 Define f : R R by { x f(x) = xf(ln x) Does n= if x e if x > e IMC Problem 5(Day ), 5 f(n) converge? Putnam A4, 8 Problem 45 Find all sequences a, a,, a n of real numbers such that a n, for which the following statement is true: If f : R R is an n times differentiable function and x < x < < x n are real numbers such that f(x ) = f(x ) = = f(x n ) = then there is h (x, x n ) for which a f(h) + a f (h) + + a n f (n) (h) = IMC Problem 6 (Day ), 6 Problem 46 Let f : R R be a continuously differentiable function that satisfies f (t) > f(f(t)) for all t R Prove that f(f(f(t))) for all t IMC Problem 4 (Day ), Problem 47 Let f : (, ) R be a differentiable function Assume that Prove that x f(x) = ( f(x) + f ) (x) = x x Vojtech Jarnik Competition, 4 Problem 48 Let f be twice continuously differentiable function on (, ) such that f (x) = and f (x) = + Show that x + x + x + f(x) f (x) = IMC Problem 3 (Day ), 995 Problem 49 Determine all Riemann integrable functions f : R R such that

10 INSTRUCTOR: CEZAR LUPU x+/n f(t)dt = x for all reals x and all positive integers n f(t)dt + n f(x), Problem 5 For each continuous function f : [, ] R, let I(f) = and J(f) = functions RNMO SHL, 6 x f(x)dx xf (x)dx Find the maximum value of I(f) J(f) over all such Putnam B5, 6 Problem 5 Let f : [, ] [, ) be a continuous function Show that ( ) ( ) f 3 (x)dx 4 x f(x)dx xf (x)dx Problem 5 Compute the following integrals: (i) (ii) (iii) log( + x) dx x arctan x x + dx American Mathematical Monthly, 5 log( + x) x + dx Putnam A5, 5 (iv) (v) (vi) π/4 log( + x ) dx x + log( + x ) dx + x log( + tan x)dx Problem 5 Compute the following its: ( n /n (i) (n + k )) n k=

11 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- (ii) (iii) n 4 (iv) ( n ) /n k= kk n n k= k n n k= k= (n + k ) /n ( + k n) /k Putnam B, 97 Problem 53 Let f be a real-valued continuous function on [, ] such that x k f(x)dx =, for k n and x [, ] such that f(x ) n (n + ) x n f(x)dx = Show that there exists Putnam A6, 97 Problem 53 (a) Find all differentiable functions f : [, ] R with continuous derivative such that f() = 6 and (f (x)) dx f(x)dx RNMO, 9 (b) Problem 53 Let f : [, ] R be a differentiable function with a continuous derivative such that f() = f() = Prove that 6 (f (x)) dx f(x)dx + 4 Mathematics Magazine, Problem 54 (a) (a) Let f : [, ] R be continuously differentiable on [, ] and satisfy f() = Show that f(x) dx 4 x (f (x)) dx University of Pittsburgh Preinary Exam, (b) Let f : [, ] R be a differentiable function with continuous derivative such that f() = Show that 4 x f (x) dx ( f(x) dx + f(x)dx) College Mathematics Journal,

12 INSTRUCTOR: CEZAR LUPU Problem 55 Let f : [, ] [, ) be a continuous function such that f(x n )dx Show that f(x) = for all x [, ] f n (x)dx, n Problem 56 Find all continuous functions f : R [, ) for which there exists a R and k positive integer such that f(x)f(x) f(nx) an k, for every real number x and positive integer n Problem 57 (i) Compute the integral I n = π Use the answer to prove Wallis formula: (ii) (a) Prove that sin n xdx [ ] 4 6 (n) = π n 3 5 (n ) RNMO, 999 (b) Prove that π sin n+ xdx = 4 (n) 3 5 (n + ) arcsin x = n= c) Deduce the Euler celebrated series, 3 (n ) 4 (n) n= n = π 6 x n+, x < n + American Mathematical Monthly, 988 Problem 58 Let f : [, ] R + be an integrable function Compute the following its: n k= ( + ( )) k n f, n

13 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 3 ( n k= ( exp n f ( )) ) k n n Problem 59 Let f : [, ] R be a continuous function having finite derivative at, and I(h) = h h f(x) dx, h [, ] Prove that a) there exists M > such that I(h) f()h Mh, for any h [, ] b) the sequence (a n ) n, defined by a n = n k= k I(/k), is convergent if and only if f() = Problem 6 Prove that n ( π 4 n ) x n dx = + xn f(x) dx, RNMO, where f(x) = arctan x x if x (, ] and f() = Problem 6 Let f : [, ] R be a continuous function such that Prove that there is c (, ) such that c f(x)dx = xf(x)dx = RNMO, 6 Problem 6 Let f : [, ] R be a continuous function Show that: a) if b) if RNMO, 6 f(sin(x + α)) dx =, for every α R, then f(x) =, x [, ] f(sin(nx)) dx =, for every n Z, then f(x) =, x [, ] RNMO, Problem 63 Let f : [, ) R be a periodic function, with period, integrable on [, ] For a strictly increasing and unbounded sequence (x n ) n, x =, with (x n+ x n ) =, we denote r(n) = max{k x k n}

14 4 INSTRUCTOR: CEZAR LUPU a) Show that: b) Show that: r(n) (x k x k+ )f(x k ) = n k= r(n) f(log k) log n k k= = f(x) dx f(x) dx RNMO, Problem 64 Let f and g be two continuous, distinct functions from [, ] (, + ) such that f(x)dx = g(x)dx Let (y n ) n be a sequence defined by f n+ (x) y n = g n (x) dx, for all n Prove that (y n ) is an increasing and divergent sequence RNMO 3 Problem 65 Let f, g : [a, b] [, ) be two continuous and non-decreasing functions such that each x [a, b] we have x x b b f(t) dt g(t) dt and f(t) dt = g(t) dt a a a a Prove that b b + f(t) dt + g(t) dt a a IMC Problem (Day ), 4 Problem 66 Let f : R R be a continuous and bounded function such that x x+ f(t) dt = x x f(t) dt, for any x R Prove that f is a constant function RNMO, Problem 67 Let f : [, ] R be an integrable function such that: < f(x) dx

15 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 5 Show that there exists x x, x, x [, ], such that: x x f(x) dx = (x x ) RNMO, Problem 68 Let f : R (, ) be a continously differentiable function Prove that f 3 (x)dx f () ( f(x)dx max f (x) f(x)dx) [,] IMC Problem 3 (Day ), 5 Problem 69 Let f : [, ] [, ) be an integrable function Show that and f(x)dx f (x)dx f 3 (x)dx, xf(x)dx x f(x)dx f(x)dx x 3 f(x)dx Problem 7 Let f : [, ] R be an integrable function such that Show that f(x)dx = xf(x)dx = f (x)dx 4 RNMO 4 Problem 7 Show that for any continuous function f : [, ] R, 4 ( f (x)dx + f(x)dx) 3 f(x)dx xf(x)dx Problem 7 Let f : [, ] R be an integrable function such that RNMO SHL, 7

16 6 INSTRUCTOR: CEZAR LUPU Show that f(x)dx = xf(x)dx = x f(x)dx = f (x)dx 9 Jozeph Wildt Competition, 5 Problem 73 Let f : [, ] R be a differentiable function with continuous derivative such that Show that f(x)dx = xf(x)dx = (f (x)) dx 3 RNMO SHL, 5 Problem 74 (a) Consider f : [, ) R and g : [, ] R be two continuous functions such that f(x) = L R Show that x n n f(x)g ( x n) dx = L g(x)dx (b) Find RNMO, 3 n n x log( + x/n) dx + x American Mathematical Monthly, 6 Problem 75 (a) Let f : [, ] R be a continuous function such that Show that there exists c (, ) such that f(c) = c f(x)dx f(x)dx = Gazeta Matematică (B-series), 99 (b) Let f : [, ] R be a continuous function such that f() = Show that there exists c (, ) such that

17 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 7 f(c) = c f(x)dx Mathematical Reflections, 7 Problem 76 (a) Let f C ([, ]) such that f() = f() = Show that ( (f (x)) dx f(x)dx) (b) Let f C ([, ]) such that f(/) = Show that ( (f (x)) dx f(x)dx) Elemente der Mathematik, 983 RNMO, 8 Problem 77 (a) Suppose that f : [, ] R has a continuous derivative and that f(x) dx = Prove that for every α (, ), α f(x) dx 8 max f (x) x Putnam B, 7 (b) Let f : [, ] R be a differentiable function for which there exists a, b (, ), a < b, such that and a f(x)dx = b f(x)dx = Let M = sup f (x) Show that x [,] f(x)dx a + b M 4 (c) Let f : [, ] [, ) be differentiable on [, ] such that there exists a (, ] with a f(x)dx = Show that f(x)dx a sup f (x) x (,)

18 8 INSTRUCTOR: CEZAR LUPU RNMO, 984 Problem 78 (a) Let f : [, ] R be a continuous function Define the sequence of functions f n : [, ] R by for all integers n f n (x) = x f n (t)dt, Prove that the series x [, ] and find an explicit formula for the sum of the series f n (x) is convergent for every n= (b) Let F = ln x For n and x >, let F n+ (x) = x SEEMOUS, F n (t) dt Evaluate n!f n () ln n Putnam B, 8 Problem 79 Let f : [, ) R be a strictly decreasing continuous function f(x) f(x + ) such that x f(x) = Prove that dx diverges f(x) Putnam A6, Problem 8 Prove that x dx dy + log y b a IMC Problem 5 (Day ), 4 Problem 8 Let < a < b and let f : [a, b] R be a continuous function with f(t)dt = Show that b b a a f(x)f(y) log(x + y)dxdy Problem 8 Let us define the sequence (a n ) n a n = 3 log n log( t + t + + n t )dt Vojtech Jarnik Competition, 4 (i) Show that the sequence a n is convergent and find its it a (ii) Show that < n(a a n ) < 3 RNMO, 8

19 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 9 Problem 83 Let f : R R be a continuous and periodic function of period T > Show that: (i) x x x f(t)dt = T T f(t)dt (ii) 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 + ) = f(x) and g(x + ) = g(x) for all real numbers x Prove that f(x)g(nx)dx = f(x)dx g(x)dx Putnam B3, 967 Problem 84 Let a, a, be real numbers Suppose there is a constant A such that for all n, ( n ) dx An + (x a i ) Prove there is a constant B > such that for all n, n ( + (ai a j ) ) Bn 3 i= i,j= Putnam B5, Problem 85 Let f(x) be a continuous real-valued function dened on the interval [, ] Show that f(x) + f(y) dx dy f(x) dx Putnam B6, 3 Problem 86 Suppose that f(x, y) is a continuous real-valued function on the unit square x, y Show that ( f(x, y)dx) dy + ( f(x, y)dxdy) + ( f(x, y)dy) dx [f(x, y)] dxdy Putnam A6, 4 Problem 87 Find all r > such that when f : R R is differentiable, grad f(, ) =, grad f(u) grad f(v) u v, then the max of f on the disk u r, is attained at exactly one point

20 INSTRUCTOR: CEZAR LUPU IMC Problem 5 (Day ), 5 Problem 88 (i) [Hermite-Hadamard] Let f : [a, b] R be a convex function Show that ( ) a + b (b a)f (ii) Show that b a f(x)dx (b a) f(a) + f(b) and k+ k f(x)dx log k + log(k + ), k, k+/ k / log xdx log k, k (iii) Consider the sequence (a n ) n defined by a n = n f(x)dx log log(n ) log n, n Show that a n is increasing and a n log 5 4 (iv) Prove that 4 ( e n 5 e ( e ) n n n! e n, n n) n (v) [Stirling] Show that n! ( n e ) n πn = Problem 89 Show that for any continuous function f : [, ] R we have the following inequalities: f (x)dx ( ) f(x)dx + 3 ( xf(x)dx ) f (x)dx 5 ( ) x f(x)dx + 3 ( xf(x)dx) RNMO 997 Gazeta Matematică (A-series), 9 Problem 9 (i) Let f : [, ] R be a continuous function Show that

21 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- x n f(x)dx = (ii) Let f : [, ] R be a continuous function Show that and n x n f(x)dx = f(), n x n f(x n )dx = f(x)dx (iii) Let f, g : [, ] R be two continuous functions Show that and xn f(x)dx xn g(x)dx = f() g(), xn f(x n )dx xn g(x n )dx = f(x)dx g(x)dx (iv) Find a real number c and a positive number L for which r r c π/ x r sin xdx π/ x r cos xdx = L Putnam A3, Problem 9 Let (a n ) n N be the sequence defined by Find the it a =, a n+ = n + n k= a k n k + n k= a k k IMC Problem 6 (Day ), 3 Problem 9 For any continuous real-valued function f defined on the interval [, ], let µ(f) = f(x)dx, Var(f) = (f(x) µ(f)) dx, and M(f) = max x f(x) Show that if f and g are continuous real-valued functions on the interval [, ], then

22 INSTRUCTOR: CEZAR LUPU Var(fg) Var(f)M (g) + Var(g)M (f) Problem 93 Prove that for any real numbers a, a,, a n we have i,j n ( n ) ij i + j a ia j a i i= Putnam B4, 3 Problem 94 American Mathematical Monthly, 99 [Hardy] Let p > and let f : [, ) R be a differentiable increasing function such that f() = and (f (x)) p dx is finite Show that ( ) p p x p f(x) p dx (f (x)) p dx p Problem 95 Let u, u,, u n C([, ] n ) be nonnegative and continuous functions, and let u j do not depend on the j-th variable for j =,,, n Show that ( [,] n j= ) n n u j n j= [,] n u n j Vojtech Jarnik Competition, 999 Problem 96 Let f : [, ] [, ] R be a continuous function Find the it ( ) (n + )! (n!) (xy( x)( y)) n f(x, y)dxdy Vojtech Jarnik Competition, 5 Problem 97 [Zagier] Prove that for all a, a,, a n, b, b, b n are nonnegative, then the following inequality holds true: ( ) ( ) ( ) min(a i, a j ) min(b i, b j ) min(a i, b j ) i,j n i,j n i,j n American Mathematical Monthly

23 WORKSHEET FOR THE PUTNAM COMPETITION -REAL ANALYSIS- 3 Problem 98 Let {D, D,, D n } be a set of disks in the Euclidian plane and a ij = S(D i D j ) be the area of D i D j Prove that for any real numbers x, x,, x n, the following inequality holds true: n i= n a ij x i x j j= Vojtech Jarnik Competition, 3 Problem 99 Let D{(x, y) R : x + y } and u C (R ) Suppose that u(x, y) = for all (x, y) D Show that D u(x, y da 8 π ( D u(x, y)da) Problem Let u C (D) and u = on D, where D is an open unit ball in R 3 Prove that the following inequality u dv ɛ D D( u) dv + u dv, 4ɛ D holds true for all ɛ > Vojtech Jarnik Competition, 997 Problem Let a, b R, f : [a, b] [, ] an increasing function and define the sequence (c n ) n by Show that: (i) (c n+ c n ) = c n+ (ii) f(b ) c n c n = n b a f n (x)dx, n Romanian contest, 5 Problem Let f : [, ] R be a C function Show that [ ( n ) f(x)dx n i,j n (f() f()) + f ( ) ( ) ] i j f = n n f(x)dx f (x)dx+ Problem 3 Find all continuously differentiable functions f : R R, such that for every a the following relation holds:

24 4 INSTRUCTOR: CEZAR LUPU D(a) xf ( ) ay dxdydz = πa3 (f(a) + sin a ), x + y 8 where D(a) = {(x, y, z) : x + y + z a, y x 3 } Problem 4 (a) Find the it (b) Prove that Vojtech Jarnik Competition, 5 ( /n ( + x n ) dx) n Mathematics Magazine, ( n ) n + xn dx = π Vojtech Jarnik (Cat II), Problem 5 Let f : [, ] R be a differentiable function such that f is bounded Show that ( f (x)dx ) f(x)dx ( sup f (x) ) x [,] Problem 6 Show that n + n + + n n = e n n e