Oil is pumped into a non-empty tank at a changing rate. The volume of oil in the tank doubles every minute and the tank is filled in 10 minutes. How many minutes did it take for the tank to be half full?

 A calculator is broken so that the only keys that still work are the sin, cos, tan, sin^-1, cos^-1 , and tan^-1 buttons. The display initially shows 0: Given any positive rational number q; show that pressing some finite sequence of buttons will yield q: Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

 Two players play a game on an in nite board that consists of 1 x 1 squares. Player I chooses a square and marks it with an O. Then, player II chooses another square and marks it with X. They play until one of the players marks a row or a column of 5 consecutive squares, and this player wins the game. If no player can achieve this, the game is a tie. Show that player II can prevent player I from winning.

Show that any continuous curve of unit length can be covered by a closed rectangles of area 1/4:

Given an alphabet with three letters a, b, c. Find the number of words of n letters which contain an even number of a's. 

The set {1, 2, ..., 49} is partitioned into three subsets. Show that at least one of the subsets contains three different numbers a , b , c such that a + b = c.

Find all integer solutions of the system of equations x + y + z =3 and x³ + y³ + z³ =3 .

Find all sets of positive integers x , y and z such that x \(\le\) y \(\le\) z and \(x^y+y^z=z^x\)

Find all positive integral solutions of \(3^x+4^y=5^z\).

(1995 IMO shortlisted problem)

Let k be a positive integer. Prove that there are infinitely many perfect squares of the form n2^k -7; where n is a positive integer.

Let a;b;c be integers such that \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}=3\). Prove that abc is the cube of an integer.

 A store bought a number of tractors for $800 each. Its regular selling price is $1000. The store decides to discount the price to stimulate sales and to earn a minimum profit of 5%. What is the maximum discount rate?

A worker in a sunglass factory can make 50 frames or 100 lenses per day. There are 90 workers. How many workers should make lenses?

A mother horse and a baby horse walk 10 miles from point A to point B. The mother’s
speed is 10 miles per hour and the baby’s 5 miles per hour. The mother rests for
2 minutes after every x minutes of walking. The baby keeps walking. They arrive
together at the destination. Find the range of x