Đề thi Toán hình học quốc tế năm 1993-1997

Đề thi Toán hình học quốc tế năm 1993-1997

\ /                                                                                                                                                                                                                                                                                                                                                          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              [...]... row on must be 0, as there are no integers to the right of the last integer in the preceding row The second last integer on each row from the third row on must be 0, as there are only Os to the right of the second last integer in the preceding row Similarly, the third last integer on each row from the fourth row on must be 0, and so on It follows that the eleventh row must consist of only Os ( b ) The... degrees 3 Each of the numbers 1, 2, 3, 25 is arranged in a 5 by 5 table In each row they appear in increasing order (left to right) Find the maximal and minimal possible sum of the numbers in the third column (Folklore, 5 points) 4 Peter wants to make an unusual die having different positive inte­ gers on each of its faces For neighbouring faces the corresponding numbers should differ by at least... Prove that the points K , L and M lie on a straight line (S Markelov , 5 points) 7 Consider an arbitrary "figure" F (non convex polygon) A chord of F is defined to be a segment which lies entirely within F and whose ends are on its boundary (a) Does there always exist a chord of F that divides its area in half? (3 points) (b) Prove that for any F there exists a chord such that the area of each of... ( A Tolpygo, 5 points ) sequence of such moves? Junior Questions 7 Spri ng 1994 (A Level) 1 A schoolgirl forgot to write a multiplication sign between two 3- digit numbers and wrote them as one number This 6-digit re­ sult proved to be 3 times greater than the product (obtained by multiplication ) Find these numbers (A Kovaldzhi, 3 points ) 2 Two circles intersect at the points A and B Tangent lines... (n - 1) + f (n- 2) for n;::: 3 Suppose no object is ever missing We claim that for any integer w, 1 :::; w :::; f(n + 2) - 1 , there is a subset of the first n objects whose total weight is w We prove this by induction on n For n = 1 , /(3) - 1 = 1 , and the claim i s trivial Suppose it holds for some n ;::: 1 Consider the first (n + 1 )st object By the induction hypothesis, there is a subset of the... justified Suppose now that one arbitrary object may be missing We claim that for any integer w, 1 :::; w :::; f (n + 1) - 1 , there is a subset of the first n objects whose total weight is w We prove this by induction on n For n = 1, f (n + 1) - 1 = 0, and the claim is trivial Suppose it holds for some n � 1 Consider the first n + 1 objects If the (n + 1)st object is missing, then none of the first... arbitrary one, having total weight w, where 1 :::; w :::; f ( n + 1) - 1 Adding the ( n + 1 )st object, we have a subset of total weight w, where f (n + 1) :5 w :5 f(n + 1) - 1 + f (n + 1) � f (n + 2 ) - 1 This justifies our claim It follows that if we take 10 objects of respective weights /(1) = 1, /(2) = 1, /(3) = 2, /(4) = 3, !(5) = 5, !(6) = 8, !(7) = 13, f (8) = 21, / (9) = 34 and 20 TO U R N A M E... weight is w Suppose now th at two arbitrary objects may be missing We claim th at for any integer w , 1 � w :=::; g(n + 1) - 1, there is a subset of the first n objects whose total weight is w We prove this by induction on n For n 1, g(2) - 1 = 0 and the claim is trivial Suppose it h olds for some n 2: 1 Consider the first n + 1 objects If the (n + 1 ) st object is missing, then at most one of the first... Storozhev ) 24 TOU RNAM ENT 15 Spring 1994 (A Level) 1 Let x and y be the given numbers Then we can interpret the data in terms of an algebraic equation as 3xy = 1000x + y Since two of the three terms of this equation are divisible by x, y is to be divisible by x That is, y = nx for some integer n Clearly, 1 $ n $ 9 as both x and y are 3-digit numbers The equation 3xy = 1000x + y can be reduced to 3xn... ( 1 + 2 + + 10) - ( i l + i 2 + + i10 ) 0 TO U RNA M ENT 15 26 as all i; are different So we have come to the conclusion that 0 = 5 ( mod 10) which is not true (A Storozhev ) 5 Alternative 1 In this solution we will call a "special" pentagon a pentagon whose sides are parallel to the sides of a regular pentagon It is easily seen that if three consecutive sides of one special pentagon are r�spec­