Number problem

For how many integer values of i, 1 ≤ i ≤ 1000, does there exist an integer j, 1 ≤ j ≤ 1000, such that i is a divisor of 2^j−1?

Can U Solve It

Singapore has dollar notes in denominations of 1, 2 and 5. How many ways are there to form exactly $100 using just multiples of these notes?

Geometry

ABCD is a square. Γ1 is a circle that circumscribes ABCD (i.e. Γ1 passes through points A,B,C and D). Γ2is a circle that is inscribed in ABCD (i.e. Γ2 is tangential to sides AB,BC,CD and DA). If the area of Γ1 is 100, what is the area of Γ2?

A prime number problem

The biggest Prime Number

Can you tell which is the biggest prime number? 

Note: You also have to prove your answer. 

Winner Gets a Prize!

Vampire and 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.

Lucky ticket

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)?

A Brilliant Problem

Two candles are placed on a table between two parallel walls that are perpendicular to the ground. The line that passes through the bottom of these candles (which are on the ground) is also perpendicular to the walls.

These two candles separate the space between the walls into three regions of length d1=20 cm, d2=30 cm, d3=10 cm (from the left to right).

The candle on the left is Candle 1, and the other is Candle 2. The height of Candle 1 is h1=15 cm and Candle 2 is h2=10 cm.

Now light these candles - their height shrinks with time. The rate at which Candle 1's height shrinks is 2 times faster than Candle 2's rate.

The question:

On the left wall, we see the shadow of Candle 1; on the right wall we see the shadow of Candle 2. Find the ratio of the speed of the Candle 1 shadow's top to Candle 2 shadow's top.
Source

Crossing the bridge

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?

A bit awkard

On which date will you have 1st January in 2014 , 2015 , 2020 and 2050? Don't think it to be easy. 99.9% get it wrong Let us see can you do it?

The worst Player

A woman, her brother, her son and her daughter (all related by birth) are chess players. The worst player's twin and the best player are of opposite sex. The worst player and the best player are the same age, who is the worst player? 

The Exchange Problem

The Waiter

Three men in a cafe order a meal the total cost of which is $15. They each contribute $5. The waiter takes the money to the chef who recognizes the three as friends and asks the waiter to return $5 to the men.

The waiter is not only poor at mathematics but dishonest and instead of going to the trouble of splitting the $5 between the three he simply gives them $1 each and pockets the remaining $2 for himself.

Now, each of the men effectively paid $4, the total paid is therefore $12. Add the $2 in the waiters pocket and this comes to $14.....where has the other $1 gone from the original $15?

A woman and her bicycle

Cycling

A woman rides her bicycle from her house to and from a lake. It takes her 112 hours to ride to the lake and 134hours to ride back. She rides downhill at 12 kilometers per hour, on level ground at 8 kilometers per hour, and uphill at 6 kilometers per hour. What is the distance in kilometers from her house to the lake?

Details and assumptions

Let's assume she doesn't get tired on her way back!

The two cars

Find the speed?

Two cars run at constant speeds around a one-mile racetrack. The faster car passes the slower car every two minutes if the cars run in the same direction. If they run in opposite directions, they meet every 15 seconds. Find the speed of the faster car in miles per hour.

Details and assumptions

The racetrack is 'circular' in the sense that the start and end points are the same, and the cars are running laps around the racetrack

A maths problem

The Complex Numbers

If complex numbers α and β satisfy

αα¯¯=ββ¯=2 and α+β=1+3√i,

what is the value of (1α+1β)2058?

Details and assumptions

i is the imaginary unit, where i2=−1, and z¯ denotes the complex conjugate of the complex number z.