IMO预选题1974

IMO ShortList 1974 IMO ShortList/LongList Project Group June 19, 2004 1. (Bulgaria 1)We consider the division of a chess board 8×8 in p disjoint rectangles which satisfy the conditions: a) every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares. b) the numbers a1, a2,..., ap of white squares from p rectangles satisfy a1, a2, . . . , ap. Find the greatest value of p for which there exists such a division and then for that value of p, all the sequences a1, a2,..., ap for which we can have such a division. Remark: This question was chosen as fourth question in the IMO. 2. (Cuba 1) If x,y,z are real positive numbers with the property that x+ y + z = xyz and none of the numbers is 1/ √ 3, prove that: 3x−x 3 1−3x2 + 3y−y3 1−3y2+ 3z−z3 1−3z2 = 3x−x3 1−3x2 · 3y−y 3 1−3y2 · 3z−z 3 1−3z2 3. (Finland 1) Prove that there is a point D on the side AB of the triangle ABC, such that CD is the geometric mean of AD and DB iff sinA sinB ≤ sin2 C2 . Remark: This question was chosen as second question in the IMO. 4. (Finland 2) We consider the triangle ABC. Prove that: sin(A)sin(B)≤sin2(C/2) is a necessary and sufficient condition for the existence of a point D on the segment AB so that CD is the geometrical mean of AD and BD. 5. (Great Britain 1) For any r=1,2,..., we consider a triangle 4r=ArBrCr. It is presumed that: 1) Ar+1 6=Ar; Br+1 6=Br; Cr+1 6=Cr; 2) Ar+1; Br+1; Cr+1 are on the circumscribed circle of the triangle ArBrCr ; 3) Ar+1Ar‖BrCr; Br+1Br‖CrAr; Cr+1Cr‖ArBr; 4) the measures of any of the angles of 41 is an integer number of degrees which is not a multiple of 45 ◦. Prove that in the first 15 triangles ( 4r r =1,2,...,15 ) there exists two which are congruent. 6. (Netherlands 1) a,b,c,d, traverse , independently from each other, the set of positive real values. What are the values which the expression S = a a+ b+ d + b a+ b+ c + c b+ c+ d + d a+ c+ d takes ? Remark: This question was chosen as fifth question in the IMO. 1 7. (Poland 1) Prove that in a square of sides 3/2 we can arrange all the squares Qn from a sequence, n=1,2,..., where the side of Qn is 1/n , so that there will not be any two of them which have common interior points. 8. (Poland 2) Let ai,bi, i =1,2,...,k be positive integers. For any i,ai and bi we have that gcd(i,ai,bi)=1. Let m be the smallest common multiple of b1, b2,..., bk. Prove the equality between the greatest common divisor of ( aim/ bi) i=1,2,...,k and the greatest common divisor of ai, i=1,2,...,k . 9. (Romania 1) Prove that for any n natural, the number n∑ k=0 ( 2n+ 1 2k + 1 ) 23k cannot be divided by 5. Remark: This question was chosen as third question in the IMO. 10. (Sweden 1) Let P(x) be a polynomial with integer coefficients. We denote deg(P) its degree which is ≥ 1. Let n(P) be the number of all the integers k for which we have (P(k))2=1. Prove that n(P)−deg(P)62. Remark: This question was chosen as sixth question in the IMO. 11. (Soviet Union 1) The sum of the squares of 5 real numbers a1, a2, . . . , a5 is 1 . Show that min(ai − aj)2 when 1 ≤ i < j ≤ 5 is smaller or equal to 110 . 12. (Soviet Union 2) An alphabet has three letters. With it we can form different words. Some sequences of words, each formed from at least two letters, are forbidden. Two different forbidden sequences have always different lengths. Prove that we can form words of any length we want which do not contain any forbidden sequences of letters. 13. (USA 1) Three players A,B and C play a game with three cards and on each of these 3 cards it is written a positive integer, all 3 numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number (≥ 2) of games we find out that A has 20 points, B has 10 points and C has 9 points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value). Remark: This question was chosen as first question in the IMO. c© by Orlando Do¨hring, Cristian Baba˜: members of the IMO ShortList/LongList Project Group, page 2/2