Cezar Lupu

Home » Elementary and Advanced Problems

Elementary and Advanced Problems

A BOOK OF KNOTTY PROBLEMS

\bullet Kleinecke-Shirokov Theorem (Functional Analysis)

1. Let A be a Banach algebra, and for a, b\in A denote the additive commutator [a, b]=ab-ba. Show that if a commutes with [a, b], then [a,b] is quasinilpotent.

Solution. It will be posted \square

\bullet The fractional part of the harmonic sequence (Real Analysis)

2. For positive integer n\geq 1, denote the sequence (H_{n})_{n\geq 1}, H_{n}=1+\frac{1}{2}+\ldots+\frac{1}{n}. Show that \{H_{n}\} diverges, where \{x\} is the fractional part of the real number x.

Solution. It will be posted \square

\bullet An exponential equation for positive reals (Algebra)

3. Find all numbers x, y>0 such that x^y+y^x=1+xy.

Author: Laurentiu Panaitopol, Gazeta Matrmatica (seria B), 1995

Solution. It will be posted \square

\bullet Common values of \pi(n) and n  (Analytic Number Theory)

4. Show that there are many infinitely postive integers n\geq 1 such that the prime counting function \pi(n) divides n.

Author: Solomon Golomb, American Mathematical Monthly,1962

Solution. It will be posted \square

\bullet A Mean Value Theorem for Convex Functions (Real Analysis)

5. Let f:[a, b]\to\mathbb{R} be convex and continuous in a and b. Show that there exists c\in (a, b) such that

\displaystyle\int_a^bf(x)dx=\frac{3}{4}(b-a)f\left(\frac{a+b}{2}\right)+\frac{1}{4}(b-a)f(c).

Author: Tudorel Lupu, Romanian Olympiad Shortlist, 2004

Solution. It will be posted \square

\bullet Cauchy’s Determinant is nonegative (Linear Algebra)

6. Let a_{1}, a_{2}, \ldots, a_{n} be positive integers. Prove that

\displaystyle\det\left(\left(\frac{1}{a_{i}+a_{j}}\right)_{1\leq i, j\leq n}\right)\geq 0.

Solution. It will be posted \square

\bullet Derivative operator is unbounded in the sup norm (Functional Analysis)

7. Show that the derivative operator D: (Df)(x) = f'(x) is an unbounded operator.

Solution. Consider C^\infty([-1,1]) the vector space of functions with derivatives or arbitrary order on the set [-1,1].

This space admits a norm called the supremum norm given by

\displaystyle |f| = \sup \left\{ f(x): x\in [-1,1] \right\}.

This norm makes this vector space into a metric space. All we need to prove is that there exists a succession of functions \displaystyle f_n\in C^\infty([-1,1]) such that \displaystyle \frac{|D(f_n)|}{|f_n|} is divergent as \displaystyle n \to \infty. Consider \displaystyle f_n (x) = \exp(-n^4 x^2) with \displaystyle (Df_n)(x) = -2x n^4 \exp(-n^4 x^2). Clearly |f_n| = f_n(0) = 1
In order to find |Df_n| we need to find the extrema of the derivative of f_n, to do that calculate the second derivative and equal it to zero. However for the task at hand a crude estimate will be enough.

\displaystyle |Df_n| \ge |(Df_n)(\frac{1}{n^2})| = \frac{2n^2}{e}.

So we finally get

\displaystyle \frac{|Df_n|}{|f_n|} \ge \frac{2n^2}{e}.

showing that the derivative operator is indeed unbounded since \frac{2n^2}{e} is divergent as n \to \infty. \square

\bullet Finsler-Hadwiger inequality (Euclidian Geometry)

8. In any triangle ABC with sides of lengths a, b, c, and area K, the following inequality holds true:

\displaystyle a^2+b^2+c^2\geq 4\sqrt{3}K+(a-b)^2+(b-c)^2+(c-a)^2.

Solution. It will be posted \square

\bullet An estimate for L^2-norm (Real Analysis)

9. For u\in C^1([a, b]) prove the estimate

\displaystyle\int_a^b |u(x)|^2dx\leq\frac{2}{b-a}\left|\int_a^bu(y)dy\right|^2+2(b-a)^2\int_a^b|u^{\prime}(t)|^2dt.

Solution. It will be posted \square

\bullet Expanders increase dimension (Real Analysis)

10. Prove that there is no function f from \mathbb{R}^3 to \mathbb{R}^2 with the property that

\displaystyle ||f(x)-f(y)||\geq ||x-y||

for all x, y\in\mathbb{R}^3.

Author: Marius Cavachi, American Mathematical Monthly, 2010

Solution. It will be posted \square

\bullet An intriguing sequence (Real Analysis)

11. Show that

\displaystyle\lim_{n\to\infty}\frac{1^n+2^n+\ldots +n^n}{n^n}=\frac{e}{e-1}.

Author: Isaac Schoenberg, Nieuw Archief voor Wiskunde, 1982

Solution. It will be posted \square

\bullet A “bounded” complex sequence (Algebra)

12. Consider the sequence (a_{n})_{n\geq 1} defined by

\displaystyle a_{n}=\left|z^n+\frac{1}{z^n} \right|, n\geq 1,

where z\in\mathbb{C}^{*} is given. Show that if there exists k\geq 1 such that a_{k}\leq 2, then a_{1}\leq 2.

Author: Romanian Olympiad, 2010

Solution. It will be posted \square

\bullet When n^2 divides a^n+b^n? (Number Theory)

13. For which paris (a, b) of positive integers do there exist infinitely many positive integers n such that n^2 divides a^n+b^n?

Author: Francisc Bozgan and Andrei Ciupan, American Mathematical Monthly (Problem 11587), 2011

Solution. It will be posted \square

\bullet Periodic configurations on a circle (Combinatorics)

14. On a fixed circle, consider a set of n\geq 2 labeled points (say by 1,2,3, \ldots n) together with a choice of direction for each (clockwise, counterclockwise). The configuration evolves by letting each point travel around the circle in its specified direction at constant speed 1; if in this process points collide they bounce off (i.e. both reverse direction) and continue moving around the circle at constant speed 1. A configuration is said to be periodic if it repeats after evolving for some non-zero time. Show that

1. All configurations are periodic.

2. Find a finite time T(n) such that any periodic configuration of n points repeats after evolving for time T(n).

Author: folklore, communicated by Bogdan Ion

Solution. It will be posted \square

\bullet Rational multiple of \pi and laticial vertices of a triangle (Euclidian Geometry)

15. Let ABC be a triangle with vertices of integer coordinates. Show that if M is an interior point of integer coordinates then at least one of the angles MAB, MBC, and MCA respectively cannot be a rational multiple of \pi.

Authors: Cezar Lupu & Stefan Spataru, American Mathematical Monthly (Problem 11871), 2015

Solution. It will be posted \square

\bullet Maximum value of n points on the unit circle (Euclidian Geometry)

16. Given an integer n\geq 2 and a closed unit disc, evaluate the maximum of the product of the lengths of all \frac{n(n-1)}{2} segments determined by n points in that disc.

Author: Romanian selection test for IMO, 2009

Solution. It will be posted \square

\bullet Erdos-Mordell inequality (Euclidian Geometry)

17. Let M be the point inside the triangle ABC. Denote by A_{1}, B_{1} and C_{1} the deet of perpendiculars from M to BC, CA, and AB respectively. Show that

MA+MB+MC\geq 2(MA_{1}+MB_{1}+MC_{1}).

Author: Paul Erdos, American Mathematical Monthly, 1935

Solution. It will be posted \square

\bullet (Euclidian Geometry)

18. Let M be a point in the interior of a convex n-gon A_{1}A_{2}\ldots A_{n}. Show that at least one of the angles \angle MA_{1}A_{2}, \angle MA_{2}A_{3}, \ldots, \angle MA_{n}A_{1} does not exceed \frac{\pi}{2}-\frac{\pi}{n}.

Author: American Mathematical Monthly, 2015

Solution. It will be posted \square

\bullet Wirtinger’s inequality (Real Analysis)

19. Let f be a piecewise smooth, continuous 2\pi-periodic function with 0 mean value, i.e.

\displaystyle\int_0^{2\pi}f(x)dx=0.

Show that

\displaystyle\int_0^{2\pi}(f^{\prime}(t))^2dt\geq \int_0^{2\pi}(f(t))^2dt,

with equality if and only if f(t)=a\cos t+b\sin t where a, b are constants.

Author:

Solution. It will be posted \square

\bullet Zeros in the image of a Volterra operator (Real Analysis)

20. Denote by C([0, 1]) the Banach space of continuous real valued functions on [0, 1], and let C_{null}([0,1]) be the subspace of functions with null integral. Let \phi :[0, 1]\to\mathbb{R} be nondecreasing, continuous at 0 and \phi(0)=0.  Define the Volterra map V_{\phi}: C([0,1])\to C([0, 1]),

\displaystyle V_{\phi}f(x):=\int_0^x\phi(t)f(t)dt.  

Show that all functions in V_{\phi}(C_{null}) have at least one zero in (0, 1).

Author: Radu Gologan & Cezar Lupu, Gazeta Matematica (A-series), 2009

Solution. It will be posted \square

\bullet Convergent trigonometric series which is not a Fourier series (Real Analysis)

21. Show that there exists a convergent trigonometric series which is not a Fourier series.

Author: folklore

Solution. It will be posted \square

\bullet Generalized Riemann-Lebesgue lemma (Real Analysis)

22. Let \phi be a bounded real or complex valued measurable function defined on \mathbb{R}, and assume that \phi is periodic with period p>0. If I is an arbitrary interval and f a real or complex-valued measurable function defined on I and integrable over I, then

\displaystyle\lim_{|\lambda|\to\infty}\int_{I}f(x)\phi(\lambda x)dx=\left(\frac{1}{p}\int_0^p\phi(x)dx\right)\left(\int_{I}f(x)dx\right).

Author: W. A. J. Luxemburg, American Mathematical Monthly, 1962

Solution. It will be posted \square

\bullet A matrix trace inequality (Linear Algebra)

23. If A and B are positive semi-definite matrices, then show that

tr(AB)^2\leq tr(A^2B^2).

Author: folklore

Solution. It will be posted \square

\bullet An information theory inequality (Algebra)

24. For any real numbers a_{1}, a_{2}, \ldots, a_{n}, and b_{1}, b_{2}, \ldots, b_{n}, we have the following inequality

\displaystyle\sum_{1\leq i, j\leq n}\min (a_{i}a_{j}, b_{i}b_{j})\leq\displaystyle\sum_{1\leq i, j\leq n}\min (a_{i}b_{j}, a_{j}b_{i}).

Author: Gheorghita Zbaganu, USAMO, 1999

Solution. It will be posted \square

\bullet Quantitative Bertrand postulate (Analytic Number Theory)

25. For any \epsilon>0 there exists an n_{0} such that for all n>n_{0} there are at least \displaystyle\left(\frac{2}{3}-\epsilon\right)\frac{n}{\log_{2}n} primes between n and 2n.

Author: Paul Erdos

Solution. It will be posted \square

\bullet The Hex game (Combinatorics)

26. If a hex is completely filled with red and blue counters then either red or blue has a winning path. Hence a game of Hex never ends in a tie.

Author: John Nash, 1948

Solution. It will be posted \square

\bullet Buzano’s inequality in inner product spaces (Linear Algebra)

27. Show that in any inner product space (H, \langle\cdot , \cdot\rangle) one has the inequality


\displaystyle \langle x, y\rangle \langle x, z\rangle\leq\frac{1}{2}(\langle y, z\rangle+||y|| ||z||)||x||^2

Author: M. L. Buzano, 1971

Solution. It will be posted \square

\bullet Integer power of a matrix is the identity (Linear Algebra)

28. Let A be a nonsingular square matrix with integer entries. Suppose that for every positive integer k, there is a matrix X with integer entries such that X^k=A. Show that A must be the identity matrix.

Author: Marius Cavachi, American Mathematical Monthly, 2010

Solution. It will be posted \square

\bullet Rational series representations involving \zeta(2n) (Real-Complex Analysis)

29. Show that

\displaystyle\sum_{n=1}^{\infty}\frac{\zeta(2n)}{n(2n+1)4^n}=\log\pi-1,

where \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}, Re(s)>1 is the Riemann zeta function.

Author:  Douglas TylerPaul Chernoff, American Mathematical Monthly, 1985

Solution. It will be posted \square

 

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8 Comments

  1. Chandrasekhar says:

    Hi —

    What happened to Math olympiad problems. I hope you convert that inot the english version and send it to me.

    Chandraskehar

  2. aboubakre says:

    how are you crezarlupu

  3. Moubinool Omarjee says:

    Hello Cezar,

    could you send me your new electronic adress ?

    Moubinool

  4. first t ime i see you, i like your inequlity problems.if you ccan ,please send some inequlity problems and materials.

  5. Marcelo Bongarti says:

    Hello, okay?

    I have a question about one of your articles. I sent it to your email, you received? Is how you respond, please?

    Thank you.

    Att,

    Marcelo Bongarti.

    Universidade Estadual Paulista

  6. beni22sof says:

    For 2. it is enough to see that H_n = \log n + \gamma+o(1) so if the fractional part of H_n converges, then the fractional part of \log n must converge. But for example, the fractional part of \log(2^n) is the same as the fractional part of n \log 2. Since \log 2 is irrational, the sequence (n \log 2) is dense in [0,1] so it cannot be convergent.

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