Number Theory
Geometry
Algebra
Logic
Combinatorics
100
What is 
The set of integers. Derived from the German word "Zahlen", which means "numbers."
100
The type of geometry we do is named after who? (Hint: he wrote the Elements and there is a computational algorithm in number theory named after him as well.)
Euclid. Non-Euclidean Geometry exists where the interior angles of a triangle can add up to more or less than 180 degrees.
100
After taking several practice tests, Brian improved the results of his GRE test by 30%. Given that the first time he took the test Brian had answered 150 questions correctly, how many correct answers did he answer in the second test?
Answer: 195
100
What is e backwards?
∃, the "there exists" symbol.
100
There are 5 red, 6 blue, and 7 green balls in a bag. If I grab one ball at a time, without replacement, what is the minimum number of balls I must grab to ensure there is at least one occurrence of each colour?
Answer: 14
Worst-case scenario: all 7 green balls grabbed, all 6 blue balls grabbed, and 1 red ball grabbed
200
pq + 17 = r , where p,q and r are all prime numbers. What is the value of p?
Answer: p = 2. Simplest solution is (p,q,r) = (2,13,8209)
If p =/= 2, then p is odd and therefore pq is odd, which in turn makes r even. The only way r can be even is if r=2, which is clearly untrue.
200
Of the five Platonic solids, what do you call the one with the most faces?
Icosahedron. It has 20 faces with 5 equilateral triangles forming a vertex (3 triangles = tetrahedron, 4 triangles = octahedron).
200
Given the sequence {1, 10, 11, 101, 1000, 1101}, determine the next term.
101012 = 21. The sequence is the Fibonacci sequence in binary.
200
0+0=2
9+6=2
6+1=1
5+1=0
8+0=?
3, count the number of closed loops in the numerals.
200
How many ways are there to put 2 white balls and 2 black balls into 3 boxes, given that balls of the same colour are indistinguishable, but the boxes are distinguishable?
Answer: 36
n = number of white balls, k = number of boxes.
Stars and Bars => n stars, k bars, n+k symbols
(n + k - 1)C(k - 1) = 6
300
Find the ordered pair of digits (M, N) that make 52MN5 a multiple of 1125.
(M, N) = (8, 7). 1125 = 3253, so 53 divides MN5. Also, 9 divides (5+2+M+N+5).
300
Determine the angle between a diagonal of a cube and one of its edges that meets the diagonal at a vertex.
Answer: 54.7 degrees
sinθ = sqrt(2)/sqrt(3) --> θ = arcsin(sqrt(2)/sqrt(3))
300
A light clock consists of a pair of mirrors with a photon bouncing back and forth in a straight line. The mirrors are initially 10c(s) apart (where c=light speed in m/s) and they each begin to move towards the other at a constant rate of 0.5c. Calculate the total distance the photon travels until the mirrors collide in the middle. (Ignore any relativistic effects)
10s * c
300
Red door - Freedom is behind this door
Blue door - Freedom is not behind this door
Green door - Freedom is not behind the blue door
At LEAST ONE of the three statements on the three doors is true and at LEAST ONE of them is false. Which door would lead to freedom?
Green door has the path to freedom.
300
In a sequence of coin tosses, we record of the number of instances when a tail is immediately followed by a head, a head is immediately followed by a head, etc. We denote these by TH, HH, etc. For example, in the sequence HTTTHHHTHT of 10 coin tosses, there are 2 HH, 3 HT, 2 TH, and 2 TT subsequences. How many different sequences of 15 coin tosses will contain exactly 2 HH, 3 HT, 4 TH and 5 TT subsequences?
ANS: 560
calculate how many H's and T's: 6H, 9T;
arrange THT: THTHTHTH;
using star and bar, arrange the left 2 H's and 5 T's. (2+4-1)C(4-1)*(5+4-1)C(4-1)
400
Find the remainder when 1010 + 10100 + 101000 + ... + 1010,000,000,000 is divided by 7.
1010 ≡ 310 (mod 7) ≡ 25 (mod 7) ≡ 4 (mod 7)
10100 ≡ (1010)10 (mod 7) ≡ 410 (mod 7) ≡ 165 (mod 7) ≡ 25 (mod 7) ≡ 4 (mod 7)
and so on.
Answer: 5
400
What is the maximum number of circles of radius 1 you can fit into a square with side length 14?
4 layers of 7 circles, 4 layers of 6 circles
ANS: 52
400
Points A and B are on the parabola y=4x2 +7x -1, and the origin is the midpoint of line segment AB. What is the length of AB?
ANS: 5(2)1/2
400
There are three people (Alex, Ben and Cody), one of whom is a knight, one a knave, and one a spy. The knight always tells the truth, the knave always lies, and the spy can either lie or tell the truth. Alex says: “Cody is a knave.” Ben says: “Alex is a knight.” Cody says: “I am the spy.” Who is the knight, who the knave, and who the spy?
Alex = Knight, Ben = Spy, Cody = Knave
400
A bug starts at a vertex of an octahedron. On each move, it randomly selects one of the four vertices adjacent to the one where it is currently located, and crawls along an edge of the octahedron to that vertex. What is the probability that the bug moves to its starting vertex on its seventh move?
Answer: 21/128
let An be the probability of the starting vertex on the nth move, Bn be the prob. on a specific one of the four adjacent vertices, Cn be the prob. on the faraway vertex. An = Cn =Bn-1 ,
Bn = 1/2 * An-1+1/2 * Bn-1
500
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed that there is a positive integer n such that 1335 + 1105 + 845 + 275 = n5. Find the value of n.
We have 
, and thus,
500
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? (Polygons are distinct unless they have exactly the same vertices.)
210 - (10C0 + 10C1 + 10C2) = 968