Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing)
has made only six copies of today’s handout. No freshman should get more than one handout, and any
freshman who does not get one should be able to read a neighbor’s. If the freshmen are distinguishable
but the handouts are not, how many ways are there to distribute the six handouts subject to the above
conditions
Saturday, July 23, 2011
The clocks
Tim has a working analog 12-hour clock with two hands that run continuously (instead of, say, jumping
on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At
noon one day, both clocks happen to show the exact time. At any given instant, the hands on each
clock form an angle between 0 and 180 inclusive. At how many times during that day are the angles
on the two clocks equal
on the minute). He also has a clock that runs really slow—at half the correct rate, to be exact. At
noon one day, both clocks happen to show the exact time. At any given instant, the hands on each
clock form an angle between 0 and 180 inclusive. At how many times during that day are the angles
on the two clocks equal
Smallest positive integer
What is the smallest positive integer n such that n^2 and (n + 1)^2 both contain the digit 7 but (n + 2)^2 does not
Octagon
Octagon ABCDEF GH is equiangular. Given that AB = 1, BC = 2, CD = 3, DE = 4, and
EF = F G = 2, compute the perimeter of the octagon
EF = F G = 2, compute the perimeter of the octagon
Jack of all sports
Neerja modi school has 85 seniors, each of whom plays on at least one of the school’s three varsity
sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are
on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and
football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors
playing all three sports, given that twice this number are members of the lacrosse team
sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are
on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and
football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors
playing all three sports, given that twice this number are members of the lacrosse team
Swim it
Lolinsan can swim from nowhere to everywhere (with the current of the empty river) in 40 minutes, or back
(against the current) in 45 minutes. How long does it take him to row from nowhere to everywhere, if he
rows the return trip in 15 minutes? (Assume that the speed of the current and Lolinsan’s swimming and
rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss
(against the current) in 45 minutes. How long does it take him to row from nowhere to everywhere, if he
rows the return trip in 15 minutes? (Assume that the speed of the current and Lolinsan’s swimming and
rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss
The flippant number
A positive integer n is called “flippant” if n does not end in 0 (when written in decimal notation) and,
moreover, n and the number obtained by reversing the digits of n are both divisible by 7. How many
flippant integers are there between 10 and 1000
moreover, n and the number obtained by reversing the digits of n are both divisible by 7. How many
flippant integers are there between 10 and 1000
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