1. Coins

There are 12 coins. One of them is false; it weights differently. It is not known, if the false coin is heavier or lighter than the right coins. How to find the false coin by three weighs on a simple scale?
Solution:

2. Bridge crossing
This problem was recently published in MAA on line: Crossing a Rickety Bridge at Night By Flashlight.

A group of four people has to cross a bridge. It is dark, and they have to light the path with a flashlight. No more than two people can cross the bridge simultaneously, and the group has only one flashlight. It takes different time for the people in the group to cross the bridge:
Annie crosses the bridge in 1 minute,
Bob crosses the bridge in 2 minutes,
Volodia Mitlin crosses the bridge in 5 minutes,
Dorothy crosses the bridge in 10 minutes.
How can the group cross the bridge in 17 minutes?

Solution:

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3. Apples delivery

The distance between the towns A and B is 1000 miles. There is 3000 apples in A, and the apples have to be delivered to B. The available car can take 1000 apples at most. The car driver has developed an addiction to apples: when he has apples aboard he eats 1 apple with each mile made. Figure out the strategy that yields the largest amount of apples to be delivered to B.

Generalize the strategy for an arbitrary amount of apples.

Party!

There is a group of people at a party. Show that you can introduce some of them to each other so that after the introduction, no more than two people in the group would have the same number of friends (initial configuration doesn't work because they all initially have 0 friends).

Digits

Show that for any natural n, at least one of two numbers, n or n+1, can be represented in the following form: k + S(k) for a certain k, where S(k) is the sum of all digits in k. For instance, 21 = 15 + (5+1)

Zen problem

A Buddhist monk got an errand from his teacher: to meditate for exactly 45 minutes. He has no watch; instead he is given two inscent sticks, and he is told that each of those sticks would completely burn in 1 hour. The sticks are not identical, and they burn with variant yet unknown rates (they are hand-made). So he has these two inscent and some matches: can he arrange for exactly 45 minutes of meditation?

Lucky tickets

In Russia you get into a bus, take a ticket, and sometimes say : Wow, a lucky number! Bus tickets are numbered by 6-digit numbers, and a lucky ticket has the sum of 3 first digits being equal to the sum of 3 last digits. When we were in high school (guys from math school No. 7 might remember that ) we had to write a code that prints out all the lucky tickets' numbers; at least I did, to show my loyalty to the progammers' clan. Now, if you add up all the lucky tickets' numbers you will find out that 13 (the most unlucky number) is a divisor of the result. Can you prove it (without writing a code)?

Divisor

Prove that for any natural N, 1000^N - 1 cannot be a divisor of 1978^N - 1

Distances

There are 6 points in a rectangle with the sides, 3 and 4. Prove that the distance between at least two of these points is smaller than the square root of 5.

King

A chess king is placed on a 8x8 chessboard. It has to make 64 moves, visiting each (of 64) squares only once, and to return there where it started. The path has to be with no intersections (a path looking like '8' is no good). For a loop, one can count the total number of horizontal + vertical (i.e. excluding diagonal) moves; let's call this number M. 1. Give an example of at least one such loop. 2. Give an example of a loop with the largest possible M. 3. Give an example of a loop with M=28. 4. Prove that 28 is the smallest possible M.

Checkers

Consider a rectangular M x N checker board and some checkers on it. What is the minimum number of checkers you should put on the board for any straight line parallel to any one of two sides of the board would cross some (at least one) checker?

Four numbers

Show that for positive a, b, c, end d, such that abcd=1, a^2 +b^2+c^2+d^2 + ab+ac+ad+bc+bd+cd is not smaller then 10.

Three planets in a galaxy and a market crash

A galaxy consists of three planets, each of them moving along a straight line with its own constant speed. If the centers of all three planets happen to lie on a straight line (some kind of eclipse) the inhabitants of each planet go nuts (they cannot see their two neighbor planets all at once), start talking about the end of the world, and the stock market crashes. Show that there will be no more than two such market crashes on each of these planets.

Fermat, computers, and a smart boy

A computer scientist claims that he proved somehow that the Fermat theorem is correct for the following 3 numbers:

x=2233445566,
y=7788990011,
z=9988776655

He announces these 3 numbers and calls for a press conference where he is going to present the value of N (to show that

x^N + y^N = z^N

and that the guy from Princeton was wrong). As the press conference starts, a 10-years old boy raises his hand and says that the respectable scientist has made a mistake and the Fermat theorem cannot hold for those 3 numbers. The scientist checks his computer calculations and finds a bug.

How did the boy figure out that the scientist was wrong?

Toy Fermat

Does the equation, x^2 + y^3 = z^4 have solutions in prime numbers? Find at least one if yes, give a nonexistence proof otherwise.

A sequence

A sequence of natural numbers is determined by the following formula, A[n+1] = a[n] + f(n) Where f(n) is the product of digits in a[n]. Is there an a[1] such that the above sequence is unbounded?

Intellectual power of a dragon pack

Dragons have to meet for a brainstorm in a convention center. The delegates have to be selected to provide the maximum efficiency of the brainstorm session. A dragon has any amount of heads, and for any N, any amount of N-headed dragons is available if needed. The problem is that the size of the convention center is limited so no more than 1000 heads can fit into the assembly hall. The intellectual power of a dragon pack is the product of head numbers of dragons in the pack. How should an optimum pack look like (the total number of dragons, the head number distribution)?

The vampire slayer

On the surface of the planet lives a vampire , that can move with the speed not greater than u. A vampire slayer spaceship approaches to the planet with its speed v. As soon as the spaceship sees the vampire it shots a silver bullet - the vampire is dead. Prove that if v/u > 10 , the vampire slayer can accomplish his mission, even the vampire is trying to hide.

Projectors

(2D) a projector illuminates a quadrant on a plane. 4 projectors are set in 4 arbitrary points of the plane. Show that they can be turned so that the whole plane would be illuminated.(3D) Show that the whole space can be illuminated with 8 projectors, each illuminating an octant, however the location points are.

A vigilance campaign in Salt Lake City.

Salt Lake City looks like a rectangle crossed with M streets going from North to South and with N streets going from East to West. The city is frequently visited by tourists who suppose to run around in the busses. The Utah governor wants to vigil all moves of the busses. He plans to put policemen at some intersections to watch all the busses moving on the streets visible from that intersections. What is the minimum number of policemen needed for the bus watch?

Blondes (the puzzle from Oldaque P. de Freitas)

Two blondes are sitting in a street cafe, talking about the children. One says that she has three daughters. The product of their ages equals 36 and the sum of the ages coincides with the number of the house across the street. The second blonde replies that this information is not enough to figure out the age of each child. The first agrees and adds that the oldest daughter has the beautiful blue eyes. Then the second solves the puzzle. You might solve it too!

This one comes from Grzegorz Dzierzanowski

There are 12 toothpicks. Find polygons of extremal field, using all toothpicks. While building these polygons follow the rules : - you cannot break the toothpicks, - the length of each edge is 1,2,3,... toothpicks, - the edges of the polygon cannot cross each other.

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