Algebra / NT
What is the remainder of the sum of the first 2024 natural numbers when divided by the sum of the first 7 natural numbers?
8
[AMC12 Q4] The acute angles of a right triangle are a and b, where a>b and both a and b are prime numbers. What is the least possible value of b?
A Chinese word is formed from a radical and a "body" section. Suppose there are 30 such radicals and 30 such bodies. However, for each radical, the CCP has censored exactly three words containing it. How many uncensored words exist?
Knights always tell the truth, and Knaves always lie.
A: B is a knave
B: A is a knave
C: A and B are either both knights or both Knaves.
Determine if each person is a Knight, Knave, indeterminable (multiple solutions), or paradoxical (impossible).
C is a Knave. A and B are indeterminable.
Saturn and Jupiter play a game. Saturn goes first, and on either player's turn, they may reduce the "shared number" by 1 or 2. The first person to reduce it to 0 or below wins. The shared number starts as some integer N, where 50 < N < 100. For how many N does Saturn win?
32
What is the area of a regular octagon with side length 1?
2+2sqrt(2)
Knights always tell the truth and Knaves always lie. Spies may do either.
Ashitaka says: "Celeste is a knave."
Nova says: "Ashitaka is a knight."
Celeste says: "I am the spy."
Can the spy be determined? If so, who is the spy? If not, is it because the puzzle is indeterminable (multiple possible solutions), or paradoxical (no possible solution)?
[AMC12 Q7] While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?
Let the maximal distance on a solid be the furthest possible distance between two objects on the surface of a solid. For instance, the maximal distance of a sphere is twice its radius. What is the maximal distance of a regular octahedron with side length 1?
[AMC12 Q15] Let k=2008^2 + 2^2008. What is the units digit of k^2+2^k?
[Note: You can only answer once, and I won't reveal if you're right or wrong until everyone answers.]
An infinity long list of statements is numbered in order, giving each statement a natural number, such that statement N states: "Each statement whose number exceeds N is false."
E.g.
1. Statements 2, 3, 4... are false
2. Statements 3, 4, 5... are false
etc.
Is statement 1 true, false, indeterminable (could be either), or paradoxical (can't be either)?
Paradoxical.
Determine all solutions to:
2^(2x) + 7^(2x) + 1 = 14^x+7^x+2^x
Where -1<x<1.
[AMC12 Q9] A three-quarter sector of a circle of radius 4 inches together with its interior can be rolled up to form the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
3sqrt(7)pi
Three points are selected uniformly randomly on a circle. What is the probability that the triangle formed by these points contains the center within its area?
1/4. The three dimensional version of this problem was on the PUTNAM, a very difficult test.
[AIME PROBLEM!!! Wrong answer / Dont answer = -400 points]
Alice knows that 3 red cards and 3 black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is m/n where m and n are relatively prime positive integers. Find m+n.
51 (m = 41, n = 10)
The following transformations belong to set S:
Transformation R (Rotation of 90 degrees clockwise)
Transformation L (Rotation of 90 degrees counterclockwise)
Transformation H (Reflection about the x axis)
Transformation V (Reflection about the y axis)
How many sequences of 2024 transformations belonging to set S will send the line x=1 to the line x=-1?
2^4046 (4^2023)
Determine the maximum possible value of the expression 4xyz / (8x^3+8y^3+z^3) for positive values x, y, z.
Six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the region inside the hexagon but outside all of the semicircles?
3sqrt(3)-pi
Cosmo is going on a vacation to a three planet system. However, they want to visit not only each planet, but each possible order of the planets. For instance, if there were two planets, they might go:
1 -> 2 -> 1
as this trip contains both the order 1 -> 2 and 2 -> 1.
What is the shortest possible trip for three planets? (The above example has a length of 3).
9. The general case of this problem is known as the superpermutation problem.
free 500 points :3
:3
The following game is played within a sphere with radius R:
Neil Armstrong and Buzz Aldrin place unit spheres anywhere within the large sphere, such that they stay completely within the sphere (consider spheres tangent to the outer sphere as outside the sphere), and no two unit spheres intersect. Neil goes first. Given perfect play, there exists some n such that if R > n, Neil wins. What is n?