**Kleinecke-Shirokov Theorem (Functional Analysis)**

*1.* Let be a Banach algebra, and for denote the additive commutator . Show that if commutes with , then is quasinilpotent.

*Solution.* It will be posted

**The fractional part of the harmonic sequence (Real Analysis)**

*2.* For positive integer , denote the sequence , . Show that diverges, where is the fractional part of the real number .

*Solution.* It will be posted

**An exponential equation for positive reals (Algebra)**

*3.* Find all numbers such that .

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

*Solution.* It will be posted

** Common values of and (Analytic Number Theory)**

*4.* Show that there are many infinitely postive integers such that the prime counting function divides

Author:* Solomon Golomb, American Mathematical Monthly,1962*

*Solution.* It will be posted

**A Mean Value Theorem for Convex Functions (Real Analysis)**

*5.* Let be convex and continuous in and . Show that there exists such that

Author:* Tudorel Lupu, Romanian Olympiad Shortlist, 2004*

*Solution.* It will be posted

**Cauchy’s Determinant is nonegative (Linear Algebra)**

*6.* Let be positive integers. Prove that

*Solution.* It will be posted

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

*7.* Show that the derivative operator is an unbounded operator.

*Solution.* Consider the vector space of functions with derivatives or arbitrary order on the set .

This space admits a norm called the supremum norm given by

.

This norm makes this vector space into a metric space. All we need to prove is that there exists a succession of functions such that is divergent as . Consider with . Clearly

In order to find we need to find the extrema of the derivative of , to do that calculate the second derivative and equal it to zero. However for the task at hand a crude estimate will be enough.

.

So we finally get

.

showing that the derivative operator is indeed unbounded since is divergent as .

**Finsler-Hadwiger inequality (Euclidian Geometry)**

*8. *In any triangle with sides of lengths , and area , the following inequality holds true:

*Solution.* It will be posted

**An estimate for -norm (Real Analysis)**

*9.* For prove the estimate

.

*Solution.* It will be posted

**Expanders increase dimension (Real Analysis)**

*10.* Prove that there is no function from to with the property that

for all .

Author:* Marius Cavachi, American Mathematical Monthly, 2010*

*Solution.* It will be posted

**An intriguing sequence (Real Analysis)**

*11. *Show that

.

Author:* Isaac Schoenberg, Nieuw Archief voor Wiskunde, 1982*

*Solution.* It will be posted

** A “bounded” complex sequence (Algebra)**

*12. *Consider the sequence defined by

, ,

where is given. Show that if there exists such that , then .

Author:* Romanian Olympiad, 2010*

*Solution.* It will be posted

**When** **divides** **? (Number Theory)**

*13. *For which paris of positive integers do there exist infinitely many positive integers such that divides ?

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

*Solution.* It will be posted

** Periodic configurations on a circle (Combinatorics)**

*14.* On a fixed circle, consider a set of labeled points (say by ) 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 ; if in this process points collide they bounce off (i.e. both reverse direction) and continue moving around the circle at constant speed . 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 such that any periodic configuration of points repeats after evolving for time .

Author:* folklore, communicated by Bogdan Ion*

*Solution.* It will be posted

** Rational multiple of and laticial vertices of a triangle (Euclidian Geometry)**

*15.* Let be a triangle with vertices of integer coordinates. Show that if is an interior point of integer coordinates then at least one of the angles , and respectively cannot be a rational multiple of .

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

*Solution.* It will be posted

** Maximum value of points on the unit circle (Euclidian Geometry)**

*16.* Given an integer and a closed unit disc, evaluate the maximum of the product of the lengths of all segments determined by points in that disc.

Author: *Romanian selection test for IMO, 2009 *

*Solution.* It will be posted

** Erdos-Mordell inequality (Euclidian Geometry)**

*17.* Let be the point inside the triangle . Denote by , and the deet of perpendiculars from to , and respectively. Show that

.

Author: * Paul Erdos, American Mathematical Monthly, 1935 *

*Solution.* It will be posted

** (Euclidian Geometry)**

*18.* Let be a point in the interior of a convex -gon . Show that at least one of the angles does not exceed .

Author: *American Mathematical Monthly, 2015 *

*Solution.* It will be posted

** Wirtinger’s inequality (Real Analysis)**

*19.* Let be a piecewise smooth, continuous -periodic function with mean value, i.e.

.

Show that

with equality if and only if where are constants.

Author:

*Solution.* It will be posted

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

*20.* Denote by the Banach space of continuous real valued functions on , and let be the subspace of functions with null integral. Let be nondecreasing, continuous at and . Define the Volterra map ,

* *

Show that all functions in have at least one zero in .

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

*Solution.* It will be posted

** 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

** Generalized Riemann-Lebesgue lemma (Real Analysis)**

*22.* Let be a bounded real or complex valued measurable function defined on , and assume that is periodic with period . If is an arbitrary interval and a real or complex-valued measurable function defined on and integrable over , then

.

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

*Solution.* It will be posted

** A matrix trace inequality (Linear Algebra)**

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

.

Author: *folklore*

*Solution.* It will be posted

** An information theory inequality (Algebra)**

*24.* For any real numbers , and , we have the following inequality

.

Author: *Gheorghita Zbaganu, USAMO, 1999*

*Solution.* It will be posted

** Quantitative Bertrand postulate (Analytic Number Theory)**

*25.* For any there exists an such that for all there are at least primes between and .

Author: *Paul Erdos*

*Solution.* It will be posted

** 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

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

*27. *Show that in any inner product space one has the inequality

Author: *M. L. Buzano, 1971
*

*Solution.* It will be posted

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

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

Author: *Marius Cavachi, American Mathematical Monthly, 2010
*

*Solution.* It will be posted

** Rational series representations involving (Real-Complex Analysis)**

*29. Show that*

where , is the Riemann zeta function.

Author: *Douglas Tyler* & *Paul Chernoff**, American Mathematical Monthly, 1985
*

*Solution.* It will be posted

Hi —

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

Chandraskehar

how are you crezarlupu

Hello Cezar,

could you send me your new electronic adress ?

Moubinool

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

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

For 2. it is enough to see that so if the fractional part of converges, then the fractional part of must converge. But for example, the fractional part of is the same as the fractional part of . Since is irrational, the sequence is dense in so it cannot be convergent.

For 6: https://mathproblems123.wordpress.com/2009/09/27/interesting-inequality-in-advanced-algebra/ 🙂