Mathematical Olympiad Treasures

Andreescu, Titu; Enescu, Bogdan

doi:10.1007/978-1-4612-2052-7

Cited by 9 publications

(12 citation statements)

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“…One may try to design more problems for which the ASS pseudocongruence theorem is handy. I give below three such problems that I have designed for examinations and assignments in my classes, a fourth problem taken from [2] and [3], and a fifth problem that is taken from [4]. Problem 3 has recently appeared in [5].…”

Section: Popularising the Ass Pseudo-congruence Theoremmentioning

confidence: 99%

“…The proof is given on pages 175-176 of the same book [4] and is based on the simple observation that if M is a point on the bisector of LXOY and if P and Q are points on XO and YO, respectively, such that MP = MQ, then either LOPM = LOQM or LOPM + LOQM = 180°. This is clearly another application of the ASS pseudo-congruence theorem.…”

Section: Popularising the Ass Pseudo-congruence Theoremmentioning

confidence: 99%

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94.35 Popularising the ASS pseudo-congruence theorem

Hajja¹

2010

Math. gaz.

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Section: Popularising the Ass Pseudo-congruence Theoremmentioning

confidence: 99%

Section: Popularising the Ass Pseudo-congruence Theoremmentioning

confidence: 99%

94.35 Popularising the ASS pseudo-congruence theorem

Hajja¹

2010

Math. gaz.

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“…(a) (Σ, T ) = (9,18) or (18,50), (b) (Σ, T ) = (5m 2 , 5m 2 + 5) for some integer m for which there exists integers n, Y, Z such that m 2 − 10n 2 = −1 and (5m 2 − 8)Y 2 = 5 + 8Z 2 .…”

Section: Introductionmentioning

confidence: 99%

“…We examine the two possibilities for (Σ, T ) in Section 9, and study the corresponding extangential LEQs up to Euclidean motions. We explicitly classify all LEQs with (Σ, T ) = (9,18); there is a single infinite family corresponding to solutions of the negative Pell equation x 2 − 2y 2 = −1. For (Σ, T ) = (18, 50), we prove that there is precisely one extangential LEQ; this isolated example has sides (a, b, c, d) = (13,2,5,10) and is shown on the right of Figure 10.…”

Section: Introductionmentioning

confidence: 99%

Lattice Equable Quadrilaterals III: tangential and extangential cases

Aebi¹,

Cairns²

2021

Preprint

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A lattice equable quadrilateral is a quadrilateral in the plane whose vertices lie on the integer lattice and which is equable in the sense that its area equals its perimeter. This paper treats the tangential and extangential cases. We show that up to Euclidean motions, there are only 6 convex tangential lattice equable quadrilaterals, while the concave ones are arranged in 7 infinite families, each being given by a well known diophantine equation of order 2 in 3 variables. On the other hand, apart from the kites, up to Euclidean motions there is only one concave extangential lattice equable quadrilateral, while there are infinitely many convex ones.and suppose the following conditions hold:/s, b = a + c − d and suppose that the above conditions (i) -(iii) hold for t = s s−1 and that b > 0. Then there is a tangential LEQ OABC with successive side lengths a, b, c, d for which (σ, τ ) = (s, t).Furthermore, in both of the above cases, if OABC is concave, then the reflex angle is at B. gives explicit examples: we present calculations of the incenters of LEQs that are kites, and we give an infinite nested family of non-dart concave tangential LEQs. Section 3 gives a series of lemmas on tangential LEQs leading to the definition of the key functions σ and τ , and their properties. In Section 4 we give the proof of Theorem 1 and Corollary 1. Section 5 is the most substantial part of Part 1.Here we prove Theorem 2 and Corollary 2. The final section of Part 1, Section 6, gives more examples. In particular, we show that there are infinitely many LEQs for each of the seven possible choices of (σ, τ ).Part 2 treats extangential LEQs. Sections 7 and 8 follow the general plan adopted in Sections 1 and 3 of Part 1; Section 7 presents some general results for all extangential quadrilaterals, and Section 8 gives a series of lemmas leading to the definition of the functions Σ and T , and their properties. Section 9 treats extangential LEQs in the cases where (Σ, T ) = (9, 18), (18, 50) and (45, 50). Section 10 shows how Theorem 3 can be deduced from Theorem 4. Section 11 is the longest section in the paper; here we prove Theorem 4. This section also contains the proof of Corollary 3, see Remark 31. Finally, in Section 12 we discuss the Open Problem presented above.

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“…The International Mathematical Olympiads have a tradition longer than one hundred years, and the first mathematical competitions were organized in Eastern Europe (Hungary and Romania) by the end of the 19th century. In 1959 the first International Mathematical Olympiad was held in Romania, where seven countries, with a total of 52 students, attended that contest (Andreescu & Enescu, 2011) The "story" of the International Mathematical Olympiads continues and will be continued even in the future society of the "artificial intelligent". Numerous math competitions organized at the level of every country in the world, at the continental or international levels, sustained and sustained concerns of all world organizations dealing with educational policies in which mathematics teaching is given great importance, etc.…”

Section: Introductionmentioning

confidence: 99%