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Bài tập toán thi quốc tế Olympiads 2002
[...]... centre at the midpoint of AB and bounding diameter lying on A B touches the other three sides BC, CD and DA Show that AB 2 = 4BC · AD 67 Let A BC be an acute-angled triangle For any point Plying within this triangle, let D, E, F denote the feet of the perpendiculars from P onto the sides of BC, CA , AB respectively Determine the set of all possible positions of the point P for which the triangle DEF... of p dividing p' where r is given by n! is Note that this is in fact a finite sum (�) 8 Let denote the number of combinations of 11 distinct taken r at a time Then For a prime r�p- 1 (; ) - n r!( n � r)! · p, we have that p divides (�) for all r satisfying 9 For a pri me number I' and any integer I � a, we have that p always divides (Cll'- ll) This is called the Fermat's little theorem ( Hint for... area o f triangle A BC composite of the functions f and g� /f!f:(X) = /(g(x)) Chapter 1 Problems 1.1 Number Theory I Find the least number whose last digi t i s 7 and which becomes 5 ti mes larger when thi s last digit i s carried to the beginni ng of the number 2 All the 2 -di git numbers from 1 9 to 93 are written consecutively to form the number N = 1 9 2 0 2 1 2 2 9 1 9 2 93 Find the largest... I Prove that for any m, amam+1 is also a tenn in the sequence 49 Suppose a and b are two positive real numbers such that the roots of the cubic equation xl - ax + b = 0 are all real I (a is a root of this cubic with minimal absolute value prove that 3b b a�2a ;;< 50 Let a, b, c be th ree real numbers such that I 2:: a 2:: b 2:: c 2:: 0 Prove that if A is a root of the cubic equation xl +axl +bx+c =... of any wonh-while problem very rarely comes to us easily and without hard work; it is rather the result of intel lectual effon of days or weeks or months Why should the young mind be willing to make this supreme effon? The ex planation is probably the instinctive preference for cenain values, that is, the attitude which rates intellectual effon and spiritual achievement higher than material advantage... write the product of the numbers written at the four comers of that face Add al l the fourteen numbers so written down Is it possible to arrange the numbers + I and - I at the comers initially so that this final sum is zero ? {a, b, f, gl c, d, e, find a collection T 1 1 0 Given the 7-element set A = of 3-element subsets of A such that each pair of elements from A occurs exactly in one of the subsets... [x] n! (;) integer part of x, i e , the greatest integer less than or equal to x (read as 'n factorial') or nc, the binomial coefficient� the number of = I · 2 · 3 · · · (11- I ) · 11 combinations of n things taken r at a time n! · r!(n - r)! II l:: a; the sum a 1 + a2 + · · · + a,• i=l n n a; the product a 1 a2 • • • tl11• i= l lA I [A BC] fog cardinality of a set A i e , the number o f elements... Polynomials I Remainder theorem: The remainder after dividing a polynomial p{x) by (x - a) is p(a) 2 Factorization of a polynomial: If a is a zero of a polynomial p (x), then (x- a) is a factor of p(x) This follows from ( I ) above since p(a) = 0 If a1,a2, an the zeros of an nth degree polyno mial, then are p(x) = a(x- aJ)(x - a2) · · · (x - a,.) where a is the leading coefficient of p(x) 3 Fundamental... canceling on both sides of an inequality one aa a � must look at the sign of the quantity that is being canceled 4 If a � I , then ( I + x)a � I + ax for x > - I If 0 < + + ( I x)a � I ax for x > - I This is known as Bernoulli's inequality 2.3 a < I then Geometry I The areas of two triangles having equal bases(heights ) are in the ratio of their heights(bases) 2 If ABC and DEF are two triangles, then... may lie inside or outside the circle), then AO OB CO OD Conversely, if AB and CD are two line segments intersecting at 0 such that AO OB = CO OD, then the four points A, B, C, D are concyclic · · · · 7 (This may be considered as a limiting case of 6, in which A and coincide and the chord AB becomes the tangent at A) B If OA is a tangent to a circle at A from a point 0 outside the circle and OCD is any