Algebra
The lines x+y=a, 3x+2y=a+15, and 4x+5y=a+19, when graphed in the xy-plane, all intersect at the same point. What is the value of a? Express your answer as a common fraction.
34/5
Triangle ABC has AB=3, AC=10, and angle A=30 degrees. D is the foot of the angle bisector of A. Find the area of ABD. Express your answer as a common fraction.
45/13
Two points are randomly placed on a number line from -0.4 to +0.6. What is the probability that the origin lies between the two points? Express your answer as a common fraction.
12/25
If a,b, and c are positive integers such that a - bc = 19 and a + bc = 99, what is the least possible value of a + b + c?
72
If (72+242)4 * (52+102) * (752+1002)6=5n, what is the value of n?
67
Triangle ABC has AB=10 and BC=6. How many different integer lengths are possible for side AC?
11
How many of the first 1024 positive integers are neither perfect squares nor perfect fifth-powers?
990
What is the greatest integer k such that 80! is divisible by 45k?
18
The graph of the equation [(x-3)(y-7)] / [(x+1)(2y-5)] = 1/2 is a line missing two points. What is the slope of the line? Express your answer as a common fraction.
-9/8
Suppose PQRS is inscribed in circle O such that SP is a diameter of circle O, PQ=2, and QR=RS. What is the area of quadrilateral PQRS? Express your answer in simplest radical form.
8sqrt2
How many different terms are in the simplified expansion of (u+v+w)8 when like terms are combined?
45
If m and n are positive integers where m2+14m-32=3n, what is the value of m+n?
16
Suppose x+y+xy=3 and x2+y2=5 for positive numbers x and y. What is the value of x+y+1? Express your answer in simplest radical form.
2sqrt3
Point P is the intersection of 3 chords of different length in circle O. If the distance from P to the circle along each segment of the chords is an integer, what is the least possible integer radius of circle O?
7
Janice creates an alphabetical list containing all the arrangements of the letters in the word FELINES. Word 1 is EEFILNS. What number is the word NIEELFS?
Word 1983
For how many positive integers n from 1 to 2100, inclusive, is the value of the expression n3+n2+n divisible by 7?
900 integers