- Andrei Frimu, Yimin Ge, Ofir Gorodetsky, Daniel Kohen, David Kotik, Hojoo Lee, Soo-Hong Lee, Cosmin Pohoata, Ho Chung Siu, Peter Vandendriessche, Problems in Elementary Number Theory 1 (Fall 2008) (39 pages; link to the PDF file).
In this volume, you will meet the solutions of the problems from the first season of PEN biweek project. Additional details about this project can be read from http://projectpen.wordpress.com/.
- Hojoo Lee, Tom Lovering, Cosmin Pohoata, INFINITY, 2008. (204 pages; both TeX and PDF are available for download)
In this never-ending project, which bears the name Infinity, we offer a delightful playground for young mathematicians and try to continue the beautiful spirit of IMO and MathLinks. Infinity begins with a chapter on elementary number theory and mainly covers Euclidean geometry and inequalities. We re-visit beautiful well-known theorems and present heuristics for elegant problem-solving. Our aim in this weblication is not just to deliver must-know techniques in problem-solving. Young readers should keep in mind that our aim in this project is to present the beautiful aspects of Mathematics. Eventually, Infinity will admit bridges between Olympiads Mathematics and undergraduate Mathematics.
Additional information related to this project can be found on Hojoo Lee’s blog at: http://ideahitme.wordpress.com/infinity/.
- Cosmin Pohoata, A combinatorial proof of Lindstr”om’s theorem on unions of subsets of a finite set. (Link to the PDF file)
In this note, we give a combinatorial proof of Bernt Lindstr\”om’s theorem that if A_{1}, A_{2},…, A_{n+1} are nonempty subsets of an n-element set, then we can find two disjoint and nonempty groups of indices \left\{i_{1},\ i_{2},\ \ldots,\ i_{k}\right\} and \left\{j_{1},\ j_{2},\ \ldots,\ j_{m}\right\} such that
A_{i_{1}} \cup A_{i_{2}} \cup \ldots \cup A_{i_{k}} = A_{j_{1}} \cup A_{j_{2}} \cup \ldots \cup A_{j_{m}}.
- Cosmin Pohoata, Geometric constructions of the mixtilinear incircles, Crux Mathematicorum with Mathematical Mayhem, to appear, 2008. (Link to the submitted PDF file)
The term mixtilinear incircles of a triangle was introduced by L. Bankoff [1] naming the three circles each tangent to two sides and to the circumcircle internally. In the same paper, he establishes the “fundamental formula”, as P. Yiu [2] reminds it, expressing the radius of a mixtilinear incircle in terms of the triangle’s inradius. More precisely, consider a triangle ABC and its mixtilinear incircle in the angle A, with center K_{A}, and radius \rho_{A}. Then, r=\rho_{A} \cdot \cos^{2}{\frac{\alpha}{2}}, where r is the inradius of the triangle, and \alpha is the magnitude of the angle at A. In this paper, we present both Bankoff and Yiu’s geometric (ruller-and-compass) constructions of these mixtilinear incircles, and moreover, we shall give three new ones.
- Cosmin Pohoata, A new method of trisection, Gazeta Matematica, 7-8 (2008), 350-351. (in Romanian) - an English version will be attached
In this short note, we present D. A. Brooks’ trisection method (of a given angle) from an elementary point of view.
- Cosmin Pohoata, On a theorem regarding lattice pentagons, Mathematical Reflections, 5 (2008). (Link to the journal’s PDF file)
My original variant can be downloaded from here.
In this note, we give a short proof of a Russian refinement of Arkinstall’s lattice pentagon theorem.
- Gabriel Dospinescu, Mircea Lascu, Cosmin Pohoata, Marian Tetiva, An elementary proof of Blundon’s Inequality, Journal of Inequalities in Pure and Applied Mathematics, 2008, to appear. (Link to the submitted PDF file)
In this note, we give an elementary proof of Blundon’s Inequality. We make use of a simple auxiliary result, provable by only using the Arithmetic Mean - Geometric Mean Inequality.
- Cosmin Pohoata, Generalizing a curious combinatorial identity, submitted. (Link to the PDF file)
This (very short) note is rather a consequence of founding Theorem 1 from my post from http://www.mathlinks.ro/viewtopic.php?p=1101215#1101215.)
In [5] Simons proved a binomial coefficient identity using repeated differentiation which can be equivalently written as
\sum_{k=0}^{n}{\binom{n}{k}\binom{n+k}{k}(-1)^{n-k}(1+x)^{k}}==\sum_{k=0}^{n}{\binom{n}{k}}{\binom{n+k}{k}x^{k}}.
Proofs of this identity have been given by Chapman [1] using generating functions and by Prodinger [3] using Cauchy’s integral formula. Using Prodinger’s approach Munarini [2] established some generalizations of Simons’ identity and other neat identities, some of them proved by Shattuck recently in [4], using elegant combinatorial arguments. In the following we present an alternate generalization, different from the ones given by Munarini. The proof is very elementary, based on a simple algebraic manipulation.
- Cosmin Pohoata, The Euler reflection point, The Harvard College Mathematics Review, 2009, to appear. (Link to the PDF file)
The Euler reflection point of a triangle is known in literature as the common point of the reflections of its Euler line OH into each of its sidelines, where O, and H are the circumcenter and the orthocenter of the triangle, respectively. Known as X_110 in Kimberling’s list of triangle centers [5], the Euler reflection point proved six years ago to be also the focus of the Kiepert hyperbola (see [8]). In this note, we give a new theorem which connects this beautiful point to two triads of circles associated with the triangles of Napoleon.
- Cosmin Pohoata, Son Hong Ta, A short proof of Lamoen’s generalization of the Droz-Farny line theorem, preprint. (Link to the PDF file)
In 1899, Arnold Droz-Farny discovered the following beautiful result, known nowadays as the Droz-Farny line theorem: If two perpendicular straight lines are drawn through the orthocenter of a triangle, they intercept a segment on each of the sidelines. The midpoints of these three segments are collinear. Lamoen’s slightly more general version says that if the midpoints of the intercepted segments are replaced by three points dividing into the same ratio the corresponding segments, then these new points remain collinear. We give a short “halfway” synthetic proof of this fact.
- Cosmin Pohoata, A short proof of Lemoine’s theorem, Forum Geometricorum 8 (2008), 97-98. (Link to the paper’s abstract)
We give a short proof of Lemoine’s theorem that the Lemoine point of a triangle is the unique point which is the centroid of its own pedal triangle. We make use of a theorem by Daneels and Dergiades on orthologic triangles.
- Cosmin Pohoata, From Neuberg-Pedoe back to Hadwiger-Finsler. (Link to the PDF file)
In this note, we give a new parametrized version of the Hadwiger-Finsler Inequality. Within our proof, we make use of a preliminary result, which can be interpreted as a corollary of the Neuberg-Pedoe Inequality.
- Cezar Lupu, Cosmin Pohoata, Sharpening Hadwiger-Finsler’s inequality, Crux Mathematicorum with Mathematical Mayhem, 2 (2008), 97-101. (Link to the extracted PDF file); based on this article, in 2008, the authors were awarded with the Traian Lalescu Medal for their undergraduate research activity (though technically I am not an undergraduate yet).
The Hadwiger-Finsler inequality is known in the literature as a generalization of the following: In any triangle ABC with side lengths a, b, c and area S, we have that a^2+b^2+c^2 \geq 4S \sqrt{3}. This inequality is due to Weitzenbock (1991) [1], but also appeared in the International Mathematical Olympiad in 1961. Finsler and Hadwiger’s stronger version is as follows: a^2+b^2+c^2 \geq 4S \sqrt{3} + (a-b)^2+(b-c)^2+(c-a)^2.
In this note, we give a refinement of this inequality: In any triangle ABC with side lengths a, b, c, area S, inradius r, and circumradius R, the next inequality is valid: a^2+b^2+c^2 \geq 4S \sqrt{3+\frac{4(R-2r)}{4R+r}} + (a-b)^2+(b-c)^2+(c-a)^2.
We give a rather algebraic proof using an equivalent form of Schur’s inequality. In the last section, we give some basic applications of our result.
- Cosmin Pohoata, On the Parry reflection point, Forum Geometricorum 8 (2008), 65-70. (Link to the paper’s abstract)
We give a synthetic proof of C. F. Parry’s theorem that the reflections in the sidelines of a triangle of three parallel lines through the vertices are concurrent if and only if they are parallel to the Euler line, the point of concurrency being the Parry reflection point. We also show that the Parry reflection point is common to a triad of circles associated with the tangential triangle and the triangle of reflections (of the vertices in their opposite sides). A dual result is also given, emerging a new triangle center, which lies on the circumcircle of the triangle formed by the Parry reflection point, the orthocenter and the circumcenter of the triangle.
- Cosmin Pohoata, On a remarkable concurrency, Gazeta Matematica, 2 (2008), 65-70. (in Romanian) - an English version will be attached
We give five different proofs of a concurrency due to Jean-Pierre Ehrmann (see Hyacinthos message #6966): Let ABC be a triangle and let D, E, F be the tangency points of its incircle \rho with the sides BC, CA, and AB, respectively. Draw the tangents to \rho from D, E, F (the ones different from the sidelines of ABC) and let X, Y, Z be their corresponding intersections with \rho. Then, the lines AX, BY, CZ are concurrent.
- Cosmin Pohoata, Paul Yiu, On a product of two points induced by their cevian triangles, Forum Geometricorum, 7 (2007), 169-180. (Link to the paper’s abstract)
The intersections of the corresponding sidelines of the cevian triangles of two points P_0 and P_1form the anticevian triangle of a point T(P_0, P_1). We prove a number of interesting results relating the pair of inscribed conics with perspectors (Brianchon points) P_0 and P_1, in particular, a simple description of the fourth common tangent of the conics. We also show that the corresponding sides of the cevian triangles of points are concurrent if and only if the points lie on a circumconic. A characterization is given of circumconics whose centers lie on the cevian circumcircles of points on them (Brianchon - Poncelet theorem). We also construct a number of new triangle centers with very simple coordinates.
- Cosmin Pohoata, Harmonic Division and Its Applications, Mathematical Reflections, 4 (2007). (Link to the journal’s PDF file)
Let A, B, C, D be four points lying in this order on a given line. The quadruple (A, B, C, D) is called a harmonic division if and only if latex BA/BC=-DA/DC, or in terms of cross-ratios (A, B, C, D)=-1. In this article, we exemplify the use of harmonic divisions and polarity by solving some “olympiad-style problems”, appreciated for their difficulty on the MathLinks forum.
Note: Most of the problems presented here do not appear in V. Nicula, C. Pohoata, The Harmonic Division, GIL, 2007.
- Cezar Lupu, Cosmin Pohoata, About a nice inequality, Mathematical Reflections, 1 (2007). (Link to the journal’s PDF file)
We consider two proofs for Darij Grinberg’s a^2+b^2+c^2 + 2abc +1 \geq 2(ab+bc+ca), which lead us to solving several other three-variable inequalities: one from the Romanian National Mathematics Olympiad from 2004, one from the Romanian BMO 2005 Team Selection Tests, one from the Asian Pacific Mathematics Olympiad from 2004 and also one from the USA IMO 2000 Team Selection Tests.
- Cosmin Pohoata, Remarks on the Japanese Theorem, Arhimede Magazine, 1 (2007), 6-9. (in Romanian) - an English version will be attached; this article was awarded with the First Prize and Gold Medal at the International Arhimede Symposium in Pure and Applied Mathematics, held in Bucharest, 2008.
The Japanese theorem states that the incenters of triangles ABC, BCD, CDA, DAB of a cyclic quadrilateral ABCD are the vertices of a rectangle. We discuss some results related to Wilfred Reyes’ proof from An Application of Thebault’s Theorem, Forum Geom., 2 (2002), 183-185.
