Given the polynomial identity:

  x⁶ + 1 = (x² + 1)(x² + ax + 1)(x² + bx + 1)

what is the value of ab ? 

How many occurrences of the digit 5 are there in the list of numbers 1, 2, 3, ..., 1000 ?

There are 29 people in a room. Of these, 11 speak French, 24 speak English and 3 speak neither French nor English. How many people in the room speak both French and English?

 If the length of each side of a triangle is increased by 20%, then the area of the triangle is increased by

(a) 40%       (b) 44%         (c) 48%            (d) 52%           (e) 60%

 Point P is inside \(\Delta\)ABC. Determine points D on side AB and E on side AC such that BD = CE and PD+ PE is minimum.

Let A, B, C and D be four distinct points on a line, in that order. The circles with diameters AC and BD intersect at the points X and Y: The line XY meets BC at the point Z: Let P be a point on the line XY different from Z. The line CP intersects the circle with diameter AC at the points C and M, and the line BP intersects the circle with diameter BD at the points B and N. Prove that the lines AM, DN and XY are concurrent.

Show that if a convex quadrilateral with side-lengths a, b, c, d and area \(\sqrt{abcd}\)  has an inscribed circle, then it is a cyclic quadrilateral.

 Given an infnite number of points in a plane, prove that if all the distances between every pair are integers, then the points are collinear.

 Let C₁ and C₂ be circles whose centers are 10 units apart, and whose radii are 1 and 3. Find the locus of all points M for which there exists points X on C₁ and Y on C₂ such that M is the midpoint of the line segment XY.

In the figure below,

- Both ABCD and EFGH are squares.

- The points A, B, C, G, H, and D are on a circle of radius 1.

- The points D, E, F, and C are collinear (they lie on the same line).

What is the area of the square EFGH?

 Let O be a point inside the square ABCD such that its distances to the vertices are OA = 10, OB = 9, OC = 5, OD = x. What is the value of x? 

 Simplify the expression (where a, b, and c are different real numbers) :

\(\dfrac{\left(x-a\right)\left(x-b\right)}{\left(c-a\right)\left(c-b\right)}+\dfrac{\left(x-b\right)\left(x-c\right)}{\left(a-b\right)\left(a-c\right)}+\dfrac{\left(x-c\right)\left(x-a\right)}{\left(b-c\right)\left(b-a\right)}\)

 How many positive integers n have the property that when 1,000,063 is divided by n, the remainder is 63?

 A circle passes through two adjacent vertices of a square and is tangent to one side of the square. If the side length of the square is 2, what is the radius of the circle?

Jack and Lee walk around a circular track. It takes Jack and Lee respectively 6 and 10 minutes to finish each lap. They start at the same time, at the same point on the track, and walk in the same direction around the track. After how many minutes will they be at the same spot again (not necessarily at the starting point) for the first time after they start walking?

You play the following game with a friend. You share a pile of chips, and you take turns removing between one and four chips from the pile. (In particular, at least one chip must be removed on each turn.) The game ends when the last chip is removed from the pile; the one who removes it is the loser.

It is your turn, and there are 2014 chips in the pile. How many chips should you remove to guarantee that you win, assuming you then make the best moves until the game is over?

Two armies are advancing towards each other, each one at 1 mph. A messenger leaves the first army when the two armies are 10 miles apart and runs towards the second at 9 mph. Upon reaching the second army, he immediately turns around and runs towards the first army at 9 mph. How many miles apart are the two armies when the messenger gets back to the first army?

What is the largest integer k for which 85! is divisible by 42^k?

A machine was programmed to transmit a certain sequence of five digits, all zeros and ones, five times. One time it did it correctly, one time it did so with one mistake, one time it did so with two mistakes, one time it did so with three mistakes, one time it did so with four mistakes. The five transmissions are listed below. Which is the correct sequence?
(a) 00001    (b) 00100     (c) 01100     (d) 10010     (e) 10011 

Alice and Bill are walking in opposite directions along the same route between A and B. Alice is going from A to B, and Bill from B to A. They start at the same time. They pass each other 3 hours later. Alice arrives at B 2.5 hours before Bill arrives at A. How many hours does it take for Bill to go from B to A?