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grade 6 problems (IV Soros Olympiad 1997-98 Round 2)

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May 31, 2024
algebrageometrycombinatoricsnumber theorySoros Olympiad

Problem Statement

p1. The numerator of the fraction was increased by 20%. By what percentage should its denominator be reduced so that the resulting fraction doubles?
p2. From point OO on the plane there are four rays OAOA, OBOB, OCOC and ODOD (not necessarily in that order). It is known that ∠AOB=40o\angle AOB =40^o, ∠BOC=70o\angle BOC = 70^o, ∠COD=80o\angle COD = 80^o. What values can the angle between rays OAOA and ODOD take? (The angle between the rays is from 0o0^o to 180o180^o.)
p3. Three equal circles have a common interior point. Prove that there is a circle of the same radius containing the centers of these three circles.
p4. Two non-leap years are consecutive. The first one has more Mondays than Wednesdays. Which of the seven days of the week will occur most often in the second year?
p5. The difference between two four-digit numbers is 77. How much can the sums of their digits differ?
p6. The numbers 1,2,3,4,5,6,7,8,91, 2, 3, 4, 5, 6, 7, 8, 9 are written on the board. In one move you can increase any of the numbers by 33 or 55. What is the minimum number of moves you need to make for all the numbers to become equal?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
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