By Alex Zawaira, Gavin Hitchcock

The significance of arithmetic competitions has been well known for 3 purposes: they assist to strengthen inventive means and pondering abilities whose price a ways transcends arithmetic; they represent the best manner of getting to know and nurturing mathematical expertise; and so they supply a way to wrestle the accepted fake photo of arithmetic held by way of highschool scholars, as both a fearsomely tricky or a lifeless and uncreative topic. This publication presents a accomplished education source for competitions from neighborhood and provincial to nationwide Olympiad point, containing 1000's of diagrams, and graced by way of many light-hearted cartoons. It incorporates a huge selection of what mathematicians name "beautiful" difficulties - non-routine, provocative, attention-grabbing, and not easy difficulties, usually with based options. It gains cautious, systematic exposition of a range of an important issues encountered in arithmetic competitions, assuming little past wisdom. Geometry, trigonometry, mathematical induction, inequalities, Diophantine equations, quantity conception, sequences and sequence, the binomial theorem, and combinatorics - are all constructed in a steady yet vigorous demeanour, liberally illustrated with examples, and regularly influenced by way of appealing "appetiser" difficulties, whose answer appears to be like after the proper conception has been expounded.
Each bankruptcy is gifted as a "toolchest" of tools designed for cracking the issues accumulated on the finish of the bankruptcy. different themes, reminiscent of algebra, co-ordinate geometry, practical equations and chance, are brought and elucidated within the posing and fixing of the massive choice of miscellaneous difficulties within the ultimate toolchest.
An strange function of this booklet is the eye paid all through to the historical past of arithmetic - the origins of the tips, the terminology and a few of the issues, and the social gathering of arithmetic as a multicultural, cooperative human achievement.
As an advantage the aspiring "mathlete" might stumble upon, within the most delightful manner attainable, the various themes that shape the center of the traditional institution curriculum.

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It follows that P1 Q1 Q1 R1 P1 R1 = = . 8) is called a ratio statement and expresses the fact that in triangles P1 Q1 R1 and P2 Q2 R2 , the ratio of the corresponding sides is fixed. If this ratio is 1, the triangles are of course congruent. The converse also applies: that is, if for a pair of triangles it is given that the ratio of the corresponding sides is fixed, then it follows that the triangles have corresponding angles equal, and so they are similar. 10) and that then the two triangles A1 B1 C1 and A2 B2 C2 are similar.

1) The angle which an arc of a circle subtends at the centre is twice that which it subtends at any point on the complementary arc of the circumference. P P x2 A y2 x1 x2 y2 x1 O A Q y1 B O x2 y2 Q O y1 B (a) Q (b) A x1 (c) P y1 B Proof: The three figures above represent the three possible cases, with the ˆ at the centre and APB ˆ at the circumference. arc AB subtending angle AOB In each figure, draw the line PO and extend to any point Q. OA = OP (radii), therefore so x1 = x2 (base angles of isosceles triangle), ˆ = 2x2 .

Alternate segments B B P P Q S A Figure (a) Q T S A Figure (b) T Geometry In each figure above, the line SAT is tangent to the circle at A, and the chord AB divides the circle into two segments APB and AQB. In Figure (a), the ˆ segment APB is the alternate segment to angle T AB, that is, it is on the ˆ Similarly, in Figure (b), the segment AQB is other side of AB from T AB. ˆ the alternate segment to angle SAB. (7) If a straight line is tangent to a circle, and from the point of contact a chord is drawn, each angle which the chord makes with the tangent is equal to the angle in the alternate segment.

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