By Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović

The overseas Mathematical Olympiad (IMO) has inside its virtually 50-year-old heritage develop into the most well-liked and prestigious pageant for high-school scholars drawn to arithmetic. basically six scholars from each one partaking kingdom are given the glory of partaking during this festival each year. The IMO represents not just a good chance to take on attention-grabbing and demanding arithmetic difficulties, it additionally bargains a fashion for prime institution scholars to degree up with scholars from the remainder of the world.

The IMO has sparked off a burst of creativity between lovers in developing new and fascinating arithmetic difficulties. In an exceptionally stiff pageant, merely six difficulties are selected every year to seem at the IMO. the whole variety of difficulties proposed for the IMOs as much as this element is miraculous and, as a complete, this selection of difficulties represents a useful source for all highschool scholars getting ready for the IMO.

Until now it's been nearly very unlikely to procure a whole number of the issues proposed on the IMO in booklet shape. "The IMO Compendium" is the results of a 12 months lengthy collaboration among 4 former IMO individuals from Yugoslavia, now Serbia and Montenegro, to rescue those difficulties from outdated and scattered manuscripts, and convey the final word resource of IMO perform difficulties. This e-book makes an attempt to assemble all of the difficulties and ideas showing at the IMO, in addition to the so-called "short-lists", a complete of 864 difficulties. additionally, the e-book comprises 1036 difficulties from a number of "long-lists" through the years, for a grand overall of 1900 problems.

In brief, "The IMO Compendium" is the final word selection of hard high-school-level arithmetic difficulties. it will likely be a useful source, not just for high-school scholars getting ready for arithmetic competitions, yet for someone who loves and appreciates math.

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**Extra resources for The IMO compendium a collection of problems suggested for the international mathematics olympiads 1959**

**Example text**

2. (POL) Let a, b, and c be the lengths of a triangle whose area is S. Prove that √ a2 + b2 + c2 ≥ 4S 3 . In what case does equality hold? 3. (BUL) Solve the equation cosn x− sinn x = 1, where n is a given positive integer. Second Day 4. (GDR) In the interior of P1 P2 P3 a point P is given. Let Q1 , Q2 , and Q3 respectively be the intersections of P P1 , P P2 , and P P3 with the opposing edges of P1 P2 P3 . Prove that among the ratios P P1 /P Q1 , P P2 /P Q2 , and P P3 /P Q3 there exists at least one not larger than 2 and at least one not smaller than 2.

Construct the fourth point on the circle D such that one can inscribe a circle in ABCD. 6. (GDR) Let ABC be an isosceles triangle with circumradius r and inradius ρ. Prove that the distance d between the circumcenter and incenter is given by d = r(r − 2ρ) . 7. (USS) Prove that a tetrahedron SABC has ﬁve diﬀerent spheres that touch all six lines determined by its edges if and only if it is regular. 1 Contest Problems First Day 1. (CZS) Determine all real solutions of the equation x, where p is a real number.

A) Find the volume of this polyhedron; (b) can this polyhedron be regular, and under what conditions? 21. (BUL) Prove that the volume V and the lateral area S of a right circular cone satisfy the inequality 6V π 2 ≤ 2S √ π 3 3 . When does equality occur? 38 3 Problems 22. (BUL) Assume that two parallelograms P, P of equal areas have sides a, b and a , b respectively such that a ≤ a ≤ b ≤ b and a segment of length b can be placed inside P . Prove that P and P can be partitioned into four pairwise congruent parts.