Logo for LIVE by Po-Shen Loh

LIVE

English

1979 AMC 12 Problem 27

Attempt Problem 27 of the 1979 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1979 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

27.

An ordered pair (b,c)(b,c) of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation x2+bx+c=0x^2+bx+c=0 will not have distinct positive real roots?

106121\frac{106}{121}

108121\frac{108}{121}

110121\frac{110}{121}

112121\frac{112}{121}

none of these

Answer: E
Concepts:quadraticcomplementary probabilitysystematic listing
Difficulty rating: 2200
Small Hint:

Distinct positive roots require b<0,b\lt0, c>0,c\gt0, and b2>4cb^2\gt4c

Big Hint:

Count the qualifying cc values for each b=−1,b=-1, b=−2,b=-2, …,\ldots, b=−5,b=-5, then take the complement among 121121 pairs

Solution:

There are 112=12111^2=121 ordered pairs. Distinct positive roots require b<0,b\lt0, c>0,c\gt0, and b2>4c.b^2\gt4c. For b=−1b=-1 and b=−2b=-2 there are no choices; for b=−3,b=-3, b=−4,b=-4, and b=−5b=-5 there are respectively 2,2, 3,3, and 55 choices of c.c. Thus only 1010 pairs produce distinct positive roots, so the requested probability is 1−10121=111121, 1-\frac{10}{121}=\frac{111}{121}, which is not listed.

Therefore, the correct answer is E.

← Problem 26#26
Full Exam
Problem 28#28 →

Problem 27 in Other Years

1950 AMC 12 · 1951 AMC 12 · 1952 AMC 12 · 1953 AMC 12 · 1954 AMC 12 · 1955 AMC 12 · 1956 AMC 12 · 1957 AMC 12 · 1958 AMC 12 · 1959 AMC 12 · 1960 AMC 12 · 1961 AMC 12 · 1962 AMC 12 · 1963 AMC 12 · 1964 AMC 12 · 1965 AMC 12 · 1966 AMC 12 · 1967 AMC 12 · 1968 AMC 12 · 1969 AMC 12 · 1970 AMC 12 · 1971 AMC 12 · 1972 AMC 12 · 1973 AMC 12 · 1974 AMC 12 · 1975 AMC 12 · 1976 AMC 12 · 1977 AMC 12 · 1978 AMC 12 · 1980 AMC 12 · 1981 AMC 12 · 1982 AMC 12 · 1983 AMC 12 · 1984 AMC 12 · 1985 AMC 12 · 1986 AMC 12 · 1987 AMC 12 · 1988 AMC 12 · 1989 AMC 12 · 1990 AMC 12 · 1991 AMC 12 · 1992 AMC 12 · 1993 AMC 12 · 1994 AMC 12 · 1995 AMC 12 · 1996 AMC 12 · 1997 AMC 12 · 1998 AMC 12 · 1999 AMC 12