Freshmens problem

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

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

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

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

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

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

Shortest path

Take a unit sphere S, i.e., a sphere with radius 1. Circumscribe a cube C about S, and inscribe a cube
D in S, so that every edge of cube C is parallel to some edge of cube D. What is the shortest possible
distance from a point on a face of C to a point on a face of D

How many triangles

A dot is marked at each vertex of a triangle ABC. Then, 2, 3, and 7 more dots are marked on the
sides AB, BC, and CA, respectively. How many triangles have their vertices at these dots?

collision probability

The train schedule in Hummut is hopelessly unreliable. Train A will enter Intersection
X from the west at a random time between 9:00 am and 2:30 pm; each moment in that
interval is equally likely. Train B will enter the same intersection from the north at
a random time between 9:30 am and 12:30 pm, independent of Train A; again, each
moment in the interval is equally likely. If each train takes 45 minutes to clear the
intersection, what is the probability of a collision today?

How many hand shakes

Six celebrities meet at a party. It so happens that each celebrity shakes hands with
exactly two others. A fan makes a list of all unordered pairs of celebrities who shook
hands with each other. If order does not matter, how many different lists are possible?

Lenth of shortest path

In the plane, what is the length of the shortest path from (−2, 0) to (2, 0) that avoids
the interior of the unit circle (i.e., circle of radius 1) centered at the origin?

A square and an equaliteral triangle

A square and an equaliteral triangle together have the property that the area of each
is the perimeter of the other. Find the square’s area.

What is the probability that two cards randomly selected

What is the probability that two cards randomly selected (without replacement) from
a standard 52-card deck are neither of the same value nor the same suit?

How many positive integers

How many positive integers x are there such that 3x has 3 digits and 4x has four digits?