The increasing sequence consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000?
A circle with diameter of length 10 is internally tangent at to a circle of radius 20. Square is constructed with and on the larger circle, tangent at to the smaller circle, and the smaller circle outside . The length of can be written in the form , where and are integers. Find .
The function has the property that, for each real number
If what is the remainder when is divided by 1000?
Find the positive integer for which
(For real , is the greatest integer )
Given a positive integer , let be the product of the non-zero digits of . (If has only one digit, then is equal to that digit.) Let
What is the largest prime factor of ?
The graphs of the equations
For certain ordered pairs of real numbers, the system of equations
The points , , and are the vertices of an equilateral triangle. Find the value of .
A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is where and are relatively prime positive integers. Find
In triangle angle is a right angle and the altitude from meets at The lengths of the sides of are integers, and , where and are relatively prime positive integers. Find
Ninety-four bricks, each measuring are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes or or to the total height of the tower. How many different tower heights can be achieved using all 94 of the bricks?
A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?
A beam of light strikes at point with angle of incidence and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments and according to the rule: angle of incidence equals angle of reflection. Given that and determine the number of times the light beam will bounce off the two line segments. Include the first reflection at in your count.
Given a point on a triangular piece of paper consider the creases that are formed in the paper when and are folded onto Let us call a fold point of if these creases, which number three unless is one of the vertices, do not intersect. Suppose that and Then the area of the set of all fold points of can be written in the form where and are positive integers and is not divisible by the square of any prime. What is ?