This includes weeks 13-31 of 2022

There are chickens and rabbits in the same cage. There are a total of 35 heads and 94 legs. How many chickens and rabbits are there? (this problem came from ancient Chinese math book)

Assume x chickens and y rabbits in the cage, then we can have

x + y = 35 (1)

2x + 4y = 94 (2)

Solve the above equation by (2) - (1)*2, we got

2y = 24 => y = 12

Therefore, x = 23. So we have 23 chickens and 12 rabbits in the cage!

Jack and his three friends went trick-or-treating together. If Jack got 11 pieces of candy, and each of his friends got three times as much candy as he did, how many pieces of candy did Jack and his friends get in all?

If Jack’s friends got three times as much candy as Jack, and Jack got 11 pieces of candy, that means that all three of Jack’s friends got 33 pieces of candy. So in order to find the sum, we can do this calculation.

33*3 + 11 = 110

Since 3 of Jack’s friends each got 33 pieces of candy, we multiplied 33 by 3. Therefore, Jack and his friends got 110 pieces of candy in total.

More than 200 students signed up to participate in the Science Olympiad competition. If teams of 3 could be made with no students left out, and teams of 4 could also be made with no students left out, what is the fewest number of students who could have signed up for the Geography Bowl?

We need an integer divisible by both 3 and 4, and greater than 200. If an integer is divisible by both 3 and 4, it is divisible by 12. 200 divided by 12 is 16 remainder 4. Therefore, the next multiple of 12 up from 200 is 204. Therefore, the fewest amount of students who could have signed up is 204.

Goldbach’s conjecture states that every even number greater than two can be expressed as the sum of two prime numbers. For example, 2022 = 191 + 1831. How many ordered pairs of prime numbers have a sum of 60?

The ordered pairs who's components are prime numbers with sum of 60 are: (7,53), (13,47), (17,43), (19,41), (23,37), and (29,31). We see 6 ordered pairs, but we must also account for the reverse of each of these ordered pairs. Therefore, there are 6 x 2 = 12 ordered pairs.

Fyodor and his three sons, Ivan, Dmitri and Alyosha, are standing exactly on the corners of a rectangular room. Fyodor is 3 meters from Dmitri and 5 meters from Ivan. What is the minimum possible distance that Fyodor could be from Alyosha?

With a rectangle, two vertices may be the endpoints of a short side, of a long side, or of a diagonal. Given two such lengths, the shorter must be a side, while the longer may be a side or a diagonal. If the 5 meters is a diagonal, then we have a 3-4-5 right triangle and the unknown length is the second side, which must be 4 meters. If the 5 meter is the longer side, the unknown length is the diagonal, even longer than the 5 meters. Thus, the minimum possible distance in question is 4 meters.

A ship anchored in a port has a ladder which hangs over the side. The length of the ladder is 200cm, the distance between each rung in 20cm and the bottom rung touches the water. The tide rises at a rate of 10cm an hour. When will the water reach the fifth rung?

Never. The tide raises both the water and the boat so the water will never reach the fifth rung.

While driving, Carl notices that his odometer reads 25,952 miles, which happens to be a palindrome. He thought this was pretty rare, but 2.5 hours later, his odometer reads as the next palindrome number of miles. What was Carl’s average speed during those 2.5 hours?

The first palindrome is at 25952. Any 5-digit palindrome starting with 25 must end in 52 and we can have an arbitrary digit in the middle, but getting the next palindrome in the 25 thousands would require the next digit after 9 for the middle digit, but there is no such digit. Thus, we must go to the 26 thousands, and the palindrome must end in 62; the least middle digit is 0. The next palindrome after 25952 is 26062, with a separation of 110. Therefore, the travel is 11- miles in 2.5 hours, for an average speed of (110mi/2.5h) = 44 mi/h

A machine randomly generates one of the nine numbers 1, 2, 3, ... , 9 with equal likelihood. What is the probability that when Tsuni uses this machine to generate four numbers their product is divisible by 14? Express your answer as a common fraction.

To have the product of the 4 digits be divisible by 14, the digit 7 must be chosen at least once, and one or more of the digits 2,4,6, and 8 must be chosen at least once. This is the complement of never choosing 7, which has probability (8/9)^4 = 4096/6561 or never choosing an even digit, which has the probability (5/9)^4 = 625/6561. however, these two cases overlap (choosing 1, 3, 5 or 9 every time), which has probability (4/9)^4 = 256/6561. Thus, the answer is 1 – ( 4096+625-256/6561) = 2096/6561

A positive integer divisor of 11! Is chosen at random. What is the probability that this divisor is prime? Express your answer as a common fraction?

11! = 11 x (2 x 5) x 3^2 x 2^3 x 7 x (2 x 3) x 5 x 2^2 x 3 x 2 = 2^8 x 3^4 x 5^2 x 7 x 11, which has (8 + 1)(4 + 1)(2 + 1) (1 + 1)(1 + 1) = 9 x 5 x 3 x 2 x 2 = 27 x 20 = 540 divisors. The only prime divisors are 2, 3, 5, 7, and 11 – thus, 5 of them. Therefore, the probability of a randomly chosen divisor of 11! Being prime is 5/540 or 1/108.

The sum of the integer n and eighteen is equal to the product of four and five. What is the value of n?

N + 18 = 4 x 5 = 20, so n = 20 - 18 = 2

There are 49 dogs signed up for a dog show. There are 36 more small dogs than large dogs. How many small dogs have signed up to compete

To figure out how many small dogs are competing, you have to subtract 36 from 49 and then divide that answer, 13 by 2, to get 6.5 dogs, or the number of big dogs competing. But you’re not done yet! You then have to add 6.5 to 36 to get the number of small dogs competing, which is 42.5. Of course, it’s not actually possible for half a dog to compete in a dog show, but for the sake of this math problem let’s assume that it is.

Phil wrote down a six-digit number on the board, and then Caleb erased two of the digits. What remains on the board is 234__6__. Phil remembers that the original number is divisible by 2, 3 and 5. What is the greatest possible value of Phil’s original number?

The number being divisible by both 2 and 5 means the ones digit must be 0. To be divisible by 3, the sum of the digits must be divisible by 3: 2 + 3 + 4 + h + 6 + 0 = 15 + h, so the hundreds digit must be divisible by 3, thus 0, 3, 6 or 9. The largest of these, 9, yields the greatest possible overall value for the original number, 234960.

Four equilateral triangles are arranged around a square which has an area of 12. What’s the shaded area?

Draw a couple of diagonal lines. These divide the four triangles into four pairs of right triangles. Four such right triangles can be arranged around the inside of the center square and you’re left with a tilted square in red. What’s its area? Well, clearly a side of the red square is equal to the difference between the long and short legs of a right triangle. But also clearly, half the diagonal of the original central square is also the difference between the long and short legs of a right triangle! That means the area of the red square is half the area of the original central square, meaning four right triangles also are half that area, or all eight right triangles are equal in area to the central square: 12.

A haiku is a poem with three lines: the first line contains 5 syllables, the second line 7 syllables, and the last line 5 syllables. If each word in each list shown is used at most once, how many different haiku can be made with these words?

Line 1: either ‘2’ or ‘3’ first, so 2 orderings; 4 choices for the ‘2’; 3 choices for the ‘3’; thus, 2 x 4 x 3 = 24 distinct lines.

Line 2: ‘3’ in any of 3 positions, so 3 orderings; 3 x 2 choices for the ‘2’s; 2 choices for the ‘3’; thus 3 x 3 x 2 x 2 = 36 distinct lines.

Line 3: either ‘2’ or ‘3’ first, so 2 orderings; 1 choice left for the ‘2’; 1 choice left for the ‘3’; thus, 2 x 1 x 1 = 2 distinct lines.

Putting the options together for all three lines: 24 x 36 x 2 = 1728 haiku.

Replace the question mark with the appropriate number

The answer is 6. All of the rows add up to 15. 6 is also the only number not represented out of numbers 1 through 9.

In a regular icosagon (20-sided polygon), all the diagonals are drawn. If a diagonal is selected at random, what is the probability of selecting a diagonal that is neither the shortest possible length nor the longest possible length? Express your answer as a common fraction.

For a regular *n*-gon, there are always *n(n-*3)/2 diagonals, so for *n* = 20, there are 170 diagonals. When *n* is even, the longest diagonals go straight across passing through the center to the opposite vertex, and there are 1- such distinct pairings of vertices. The shortest diagonals are obtained by taking any vertex; that vertex is adjacent to two other vertices, and the diagonal joining those two other vertices has the shortest possible length – there are 20 such vertices and 20 such diagonals. There fore, of the 170 diagonals, we are rejecting 10 + 20 = 30 and thus keeping 140. The probability is, therefore, 140/170 = 14/17.

If you go to the movies and you’re paying, is it cheaper to take one friend to the movies twice or two friends to the movies at the same time?

It’s cheaper to take two friends at the same time. In this case, you would only be buying three tickets, whereas if you take the same friend twice, you are purchasing four tickets.

Cut up the Greek letter Pi below into five pieces as shown. Then re-arrange the five pieces to make a square. Is there more than one way?

Only one solution has been found so far, can you find another?

In a bag full of small balls, 1/4 of these balls are green, 1/8 are blue, 1/12 are yellow and the remaining 26 white. How many balls are blue?

Let us first find the fraction of green, blue and yellow balls

1/4 + 1/8 + 1/12 = 6 / 24 + 3 / 24 + 2 / 24 = 11 / 24. Make sure you have a common denominator and add the numerators.

The fraction of white balls is given by

24 / 24 - 11 / 24 = 13 / 24

Since 13 / 24 corresponds to 26 balls. If x is the total number of balls then

(13 / 24) of x = 26 balls

x = 26 × (24 / 13) = 48 , total number of balls

The fraction of blue balls is 1 / 8 of x. The number of blue balls is given by

(1 / 8) of 48 = 6 balls.

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