Algebra
Geometry
C & P
Number Theory
Misc
100
If a ball and a bat together cost $1.10, and the bat costs $1.00 more than the ball, how much does the ball cost.
Answer: $0.05, pls tell me you got this right.
100
What is the interior angle measure of a regular heptagon?
Answer: 128.57 or 900/7, this better be common sense.(180(n-2)/n, where n is the number of sides)
100
How many people are in the meeting at this moment in time?
COUNT PPL
100
Find the number of distinct pairs of integers (x,y) such that 0 < x < y and
√(1984) = √(x) + √(y).
100
Use operations on 8 8 8 to = 6. For example (9+9)/√(9) = 6
Example answer: 8-√(√(8+8)), (√(8-8/8))!
200
Solve this system of equations:
Answer: x= -3, y=½, z=2, this is basic algebra please don’t get this wrong.
200
4 congruent circles, each of which is tangent externally to 2 of the other circles, are circumscribed by a square of area 16. A small circle is then placed in the center so that it is tangent to each of the four circles. Determine the radius of the smallest circle.
Answer: √(2)-1, draw a diagonal through the square. The length of it is 4√(2). Wecan look at only half of the diagonal(from a corner to the center) which has length 2√(2). The length from the corner to the center of one of the congruent circles is √(2). This can be seen if you draw a line through the center to the edge of the square(it’s a radius). Using the pythagorean theorem we see that the length is √(2). 2√(2)-√(2)-1 = √(2)-1 which is our answer.
200
Select numbers a and b between 0 and 1 independently and at random, and let c be their sum. Let A, B and C be the results when a, b and c, respectively, are rounded to the nearest integer. What is the probability that A + B= C?
200
Compute the largest prime factor of 3(3(3(3(3(3(3(3(3(3(3+1)+1)+1)+1)+1)+1)+1)+1)+1)+1)+1
Answer: 73 Let
be the sequence with
and
. We wish to find the largest prime factor of
.
Expanding the first few terms, we get
The sequence appears to be
. Indeed, we prove by induction that
This is just a geometric series, so
Therefore
Since
is bigger than all the other factors, and is a prime number, the largest prime factor of
is
.
200
What is the highest possible award given in the field of mathematics?
Fields Medal
300
Dylan has a 100×100 square, and wants to cut it into pieces of area at least 1. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?
Answer: 9999, Since each piece has area at least 1 and the original square has area 10000, Dylan can end up with at most 10000 pieces. There is initially 1 piece, so the number of pieces can increase by at most 9999. Each cut increases the number of pieces by at least 1, so Dylan can make at most 9999 cuts. Notice that this is achievable if Dylan makes 9999 vertical cuts spaced at increments of 1/100 units.
300
Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that ∠AMD = ∠CMD. What is the degree measure of ∠AMD?
300
The numbers 1-16 are randomly placed into the 16 squares of a 4x4 grid. Each square gets one number, and each of the numbers is used once. What is the probability that the numbers 1, 2, 3, and 4 are in the same line.
Answer: 1/182, there are 10 “lines” in a 4x4 grid. There are 4! ways to arrange 1, 2, 3, 4. For the other 12 numbers, there are 12! ways to arrange them. There are 16! ways to arrange the numbers into boxes in general. (10*4!*12!)/16! gives 1/182.
300
For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marshmallow pieces. He defines the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number of pieces. One day, Mihir notices that his breakfast cereal has exactly 90 oat pieces and 9 marshmallow pieces, and exclaims, “This is such an unlucky bowl!” How many marshmallow pieces does Mihir need to add to his bowl to double its luckiness?
Answer: 11
Let x be the number of marshmallows to add. We are given that 2 · 9/99 = (9 + x)/(99 + x). Rearranging this gives 2(99 + x) = 11(9 + x). Thus 9x = 99 and x = 11.
300
In what dimension(number of dimensions) is the volume(area in 2 dimensions) of a “unit-sphere(circle in 2 dimensions, sphere in 3)” the largest.
Answer: 5, there is an equation 
400
Find c if a, b, and c are positive integers which satisfy c=(a+bi)^3 - 107i, where i^2 = -1
400
On square ABCD, point E lies on side AD and point F lies on side BC, so that BE = EF = FD = 30. Find the area of the square ABCD.
400
How many positive integers less than 10,000 have at most two different digits?
Answer: 927, casework https://artofproblemsolving.com/wiki/index.php/2004_AIME_II_Problems/Problem_4
400
Find all natural numbers x such that the product of their digits (in decimal notation) is equal to x^2 - 10x - 22.
400
Who dis ppls?
Answer: Issac Newton, Euclid, Archimedes
500
If a ≥ b > 1, what is the largest possible value of log_a(a/b) + log_b(b/a)?
500
For breakfast, Milan is eating a piece of toast shaped like an equilateral triangle. On the piece of toast rests a single sesame seed that is one inch away from one side, two inches away from another side, and four inches away from the third side. He places a circular piece of cheese on top of the toast that is tangent to each side of the triangle. What is the area of this piece of cheese?
Answer: 49π/9
Suppose the toast has side length s. If we draw the three line segments from the sesame seed to the three vertices of the triangle, we partition the triangle into three smaller triangles, with areas s/2 , s, and 2s, so the entire piece of toast has area 7s/2 . Suppose the cheese has radius r. We similarly see that the toast has area 3rs/2 . Equating these, we see that r = 7/3 , so the area of the cheese is π(7/3)^2 = 49π/9
500
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?
500
Meghana writes two (not necessarily distinct) primes q and r in base 10 next to each other on a blackboard, resulting in the concatenation of q and r (for example, if q = 13 and r = 5, the number on the blackboard is now 135). She notices that three more than the resulting number is the square of a prime p. Find all possible values of p.
Answer: 5
Trying p = 2, we see that p 2 − 3 = 1 is not the concatenation of two primes, so p must be odd. Then p 2 − 3 is even. Since r is prime and determines the units digit of the concatenation of q and r, r must be 2. Then p 2 will have units digit 5, which means that p will have units digit 5. Since p is prime, we find that p can only be 5, and in this case, p 2 − 3 = 22 allows us to set q = r = 2 to satisfy the problem statement. So there is a valid solution when p = 5, and this is the only possibility.
500
What does the Tau function represent in number theory and define what it means for a function to be multiplicative.?
Answer: The tau function (generally called the divisor function) outputs the number of divisors of a number. A function is multiplicative if f(ab)=f(a)f(b) when gcd(a,b)=1, f(1)=1, and the function lies over the positive integers.