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A parallelogram has coordinates (0, 0), (144, 0), (377, 89) and (233, 89).

Determine the number of lattice points strictly within the interior of the parallelogram.

Prove the following identity for a 2×2 matrix A and B

(I+A)(I-BA)^-1(I+B)=(I+B)(I-AB)^-1(I+A)

In the rally point scoring method now widely adopted in volleyball, either team can score a point by winning the rally after a serve, regardless of who serves the ball. The first team to earn 25 points wins the set, except that a team must win by at least two points — that is, if the score is 24-24, then the final score must be 26-24, 27-25, 28-26, etc.

Assume two evenly matched teams are playing each other.

To at least four significant figures, what is the expected number of rallies (total points scored) per set?

Suppose a type of glass is such that, for any incoming light: 70 percent of light shining from one side is transmitted through to the other side; 20 percent of the light is reflected (off of the outer surface) back in the direction from which it came; the remaining 10 percent is absorbed in the glass.

How much of an original light source will be transmitted through three panes of glass? It is assumed that the panes are parallel and at a small distance from each other.

Ignore any loss of light above or below the panes (which is the same as assuming the panes extend infinitely in all four directions). Express your answer as a ratio of integers.

Find all nonnegative integer solutions to the following equation

64 + 3^x5^y = 17^z

I am getting a new sign for my shop, and instead of showing the shop number as numeric digits, it will be spelled out in capital letters, one word per digit. For example, if my shop number were 103, it would be painted as ONE ZERO THREE.

The sign painter is a little eccentric, and instead of charging by the hour, he charges by the brush stroke, which can be any shape and can touch, but must not cover, an already painted area (except where changing direction). He paints capital letters in a simple style (no serifs) and does not use two strokes where a single stroke will do. For example, E, F, G, H, I, N, O, R, S, T, U, V, W, X, Z would require 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, and 1 strokes, respectively.

My shop number, which I had spelled out for the painter when he arrived, is a prime number, consisting of three different digits. He charges $1 per stroke, regardless of the length of the stroke. By coincidence, the cost of my number in dollars was equal to the sum of the three digits in the shop number and was a prime.

What is my shop number?

S is a square with no repeat digits.
C is a cube with no repeat digits.
F, a pandigital fifth power, is a scramble of the concatenation of the digits of S and C.
F does have repeat digits, but there are no digits which appear in F more than twice.

None of S, C, or F can end in zero.

F is the smallest pandigital fifth power that meets these conditions.
There are several (S,C) pairs that work with this F.

Find F and list all of the (S,C) pairs (there are less than 10)

Suppose that, in a distant galaxy, there is a solar system in which, instead of being spheres, the planets are right circular cones (with heights equal to the diameters of their bases). Suppose one of these planets has the same total volume and mass as our Earth, but a uniform density.

What would be the gravitational acceleration on a person standing in the center of the circular base, and what would be the gravitational acceleration on a person standing at the apex?

Assume the Earth is a perfect sphere with a radius of 6,370 km and an average density of 5,518 kg/m^3. Use a value of 6.673×10-11 N m^2/ kg^2 for G. Express your answers correct to three significant figures.

Frank has a tablecloth covered with 1 cm diameter black polka dots. The polka dots form a square grid with 2 cm between neighboring centers.

He takes a flat 4 cm by 4 cm piece of clear plastic and paints four 1 cm diameter circles on it in the same pattern as on the tablecloth. After laying the tablecloth on a flat surface, he randomly tosses the piece of plastic onto it.

To three significant figures, what is the probability that none of the polka dots on the plastic will overlap any portion of any of the polka dots on the tablecloth?

How many ways are there to choose integers a, b, and c such that 1 ≤ a < b < c ≤ 2024 and a + b + c = 2027?

Find the missing digits

29!=8a41b6199373970c95454361de00000

Each team has three players ranked 1 through 3, where the player ranked 1 is the strongest player of the three, and rank 3 is the weakest. In the tournament, each player from one team is paired against one player from the opponent in a head-to-head match. If a player’s ranking is A and his opponent’s ranking is B, then the first player’s probability of winning the match is B ÷ (A + B).

The Nets win the coin toss, which gives them the privilege of determining the pairings.

What is the Nets best pairing strategy, and using that strategy, what is their probability of winning the tournament by winning a majority of the matches?

A circle is inscribed in a quadrilateral ABCD, where M and N are midpoints of AC and BD, respectively. If AB=5, CD=9, and AC²+BD²+4MN²=218. Find the value of BC*DA.

Find the angle at the top corner of the teardrop curve defined by the equation √(x²+y²)=e^y-1

The point (10,26) is a focus of a non-degenerate ellipse tangent to the positive x and y axes. The locus of the center of the ellipse lies along a line. Find the equation of this line.

Last spring equinox, brothers Doug and Ryan Euclid were standing at the equator looking out to sea. Doug was standing on the shore, while Ryan was standing above him on top of a cliff. Their eyes were 2m and 23m above sea level, respectively. When Doug saw the sun set, he called out; 26 seconds later Ryan observed the sun setting.

Doug and Ryan were observant and knew the time it takes the Earth to rotate once. The brothers estimated the diameter of the Earth.

What was their estimate to the nearest 100km?

Determine the number of non-negative integers x with x < 1010 that have a digit sum of 21.

Find the smaller of two consecutive positive 30-digit integers such that the difference of their cubes is a perfect square.

That is: (j+1)³- j³ = k², where j and k are positive integers.

Given that there are exactly four primes that divide the number

257⁴ + 32³ - 8193² - 640²

find the largest of the four primes.

There are two rectangles 1 × 2 and 3 × 4 with parallel sides centered at a common point O. Points F and G are selected on the boundary of the inner and outer rectangle, respectively. What is the maximum possible area of triangle FOG?

Let z be a complex number with 2≤|z|≤4. When all possible values of z+1/z are graphed on the complex plane, they form a region R. Compute the area of R.

Let ⌈x⌉ denote the smallest integer greater than or equal to x. The sequence (a_i) is defined as follows: a₁=1, and for all i≥1,

a_i+1=min(7⌈(a_i+1)/7⌉, 19⌈(a_i+1)/19⌉)

Compute a₁₀₀.

John is shopping for a used car and has found one being offered by Alice for $12,000, based on $2,000 down and a $10,000 balance to be paid to Alice over 36 months at 6% (annual) simple interest paid monthly on the unpaid balance (John’s monthly payments would be $10,000/36 + unpaid balance x 0.06/12).

However, if he pays cash, he can get a reduction of D dollars so the cash payment would be $12,000 - D.

Although Alice will finance the car at 6%, she believes she can better invest her money at 10% elsewhere.

If the 36 monthly payments are discounted to the present at a rate of 10%, how big a reduction in the cash price (to the nearest dollar) can Alice afford to give John so that buying the car for a single cash payment and the net present value of financing over 36 months (plus $2,000 added to account for the down payment), are equivalent propositions?

Assume all months have 30 days, and a year is 360 days.

A positive integer is called "maybe prime" if all of its digits are primes and the number is not divisible by 2 or 3. Find the number of positive integers less than 10000 that are "maybe prime".

Three spheres S₁, S₂, and S₃ are pairwise externally tangent and are tangent to a plane P at points A, B, and C, respectively. If AB=13, BC=14, and CA=15, compute the product of the radii of the three spheres.

In the interior of an equilateral triangle ABC a point P is chosen such that PA² = PB² + PC². Find the measure of angle BPC.

In a certain British soccer pool, the objective is to pick games that end in a tie. The ticket buyer picks 8 games from a list of 45 or more. For each of these games,

if the teams tie, the player gets 3 points, if the visiting team wins, they get 2 points, and if the home team wins, they get 1.5 points. The entry with the highest point total wins.

Assume that for each game, the probability of the home team’s winning is 0.5, the probability of the visiting team’s winning is 0.4, and the probability of a tie is 0.1.

Determine the probability that the total points for an entry will be 22 or higher.