Home » 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 $a_{1}\leq 2$.

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 $\square$

1. Chandrasekhar says:

Hi —

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

2. aboubakre says:

how are you crezarlupu

3. Moubinool Omarjee says:

Hello Cezar,

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?

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.