Problem of the Week
Welcome to the Problem of the Week (POTW). Every Sunday, a new POTW will be posted below along with the answer to the previous week's POTW. If you have any questions or would like to submit answers regarding the POTW, please email infinitysquared1113@gmail.com and we will get back to you as soon as possible! Good luck and happy problem solving!
Problem of the Week
POTW #48 (5/13-5/19):
Difficulty: Medium
If Celeste can guess Alex's zodiac sign in one try (there are 12 in total), Alex promises to give her a stuffed animal. Assuming Alex tells the truth and Celeste guesses randomly, what is the expected number of stuffed animal Celeste obtains?
Problems Archive
Featuring previous Problems of the Week (with answers)!
POTW #1 (6/11/23-6/17/23):
Difficulty: Medium
Melissa draws a tangerine in the shape of a circle with the radius of 4 inches in the center of a 10inch x 20inch paper. Assuming the tangerine is a perfect circle, find the smallest distance between the circumference of the circle and a corner of the page.
ANSWER: 5 √5 - 4 inches
POTW #2 (6/18/23-6/24/23):
Difficulty: Easy
At the orchard, Jessica ate 1/3 of her tangerines and then gave 2 tangerines to Jesse. The next day she ate 1/5 of her remaining tangerines and then gave 1 tangerine to Jesse. On the third day she ate her last 7 tangerines. How many tangerines did Jessica have at the beginning?
ANSWER: 18 tangerines
POTW #3 (6/25/23-7/1/23):
Difficulty: Easy
Nathan goes to the grocery store and buys 5 tangerines and 5 limes that cost a total of $4.65. Assuming all tangerines are the same price and all limes are the same price, what is the cost of 11 tangerine and 11 limes?
ANSWER: $10.23
POTW #4 (7/2/23-7/8/23):
Difficulty: Hard
Eight distinct tangerines are distributed amongst Allison, Jennifer, Allen, and Nathan. If Allison must receive at least two tangerines, and Jennifer must get at least one, how many different arrangements are possible?
ANSWER: 172032 ways
POTW #5 (7/9/23-7/15/23):
Difficulty: Medium
The Farmers Market sells apples by the bushel, and each bushel of apples weighs 42 lbs. A bushel of green apples contains 84 apples. If a bushel of red apples contains twice as many apples as a bushel of green apples, how many red apples are in 1 lb of apples?
ANSWER: 4 apples
POTW #6 (7/16/23-7/22/23):
Difficulty: Hard
There were originally a red and green apple on a table. The green apple started off 20 cm taller than the red apple. They were both eaten starting at the same time by Jonathan and at a constant rate. You may assume they were eaten perfectly from top to bottom. Five minutes after Jonathan started they were both the same height. The green apple was finished 6 minutes after Jonathan started, and the red apple after 10 minutes. What was the ratio of the original height of the red apple to the original height of the green apple?
ANSWER: 3
POTW #7 (7/23/23-7/29/23):
Difficulty: Easy
Nine chairs in a row are to be occupied by six students and Professors Alpha, Beta, and Gamma. These three professors arrive before the six students and decide to choose their chairs so that each professor will be between two students. In how many ways can Professors Alpha, Beta, and Gamma choose their chairs?
ANSWER: 60 ways
POTW #8 (7/30/23-8/5/23):
Difficulty: Hard
There is a train traveling from City A to City B at some constant speed. One hour after leaving City A, the train breaks down and now travels at 75% of its normal speed, causing it to arrive at City B 1 hour and 40 minutes later than it would have if it never broke down. If instead, the train had broken down 60 miles further from City A, then it would only be late by 1 hour and 25 minutes. What is the distance between the two cities?
ANSWER: 480 miles
POTW #9 (8/6/23-8/12/23):
Difficulty: Easy
What is the area of a hexagon with side length of 4?
ANSWER: 24√3
POTW #10 (8/13/23-8/19/23):
Difficulty: Medium
Apollo, Ares, and Athena are counting the number of animals in the zoo. Apollo said, “There are at least 77 animals,” Ares replied, “I think there are at most 46 animals.” Athena then said, “Actually, there are at most 68 animals.” Let a be the # of animals at the zoo. What is the smallest value of a (as a whole number) such that the statements made by all three of them are false?
ANSWER: 69
POTW #11 (8/20/23-8/26/23):
Difficulty: Easy
Two, Three, Four, Five, Six, Seven, and Eight are eight friends who are trying to find a common favorite number. Two doesn't like any number that is a multiple of 2, Three doesn't like any number that is a multiple of 3, Four doesn't like any number that is a multiple of 4, and so on so that Eight doesn't like any number that is a multiple of 8. In addition, all of them dislike any negative numbers or zero. If this is the case, what is the least number they can denote their common favorite number?
ANSWER: 1
POTW #12 (8/27/23-9/2/23):
Difficulty: Easy
In how many ways can the integer 12,345 be expressed as the sum of two primes?
ANSWER: 1
12,345
POTW #13 (9/3/23-9/9/23):
Difficulty: Medium
In Parva Ava, all words are of length 1-3 letters and only consist of the letters A, P, R, and V. How many total words can be formed? (letters can repeat)
ANSWER: 84 words
P
A
V
R
POTW #14 (9/17/23-9/23/23):
Difficulty: Easy
In how many ways can 15 be written as the sum of an increasing sequence of two or more consecutive positive integers?
ANSWER: 3 ways
POTW #15 (9/24/23-9/30/23):
Difficulty: Hard
For how many pairs of consecutive integers in the set
{1000, 1001, 1002,..., 2000}
is no carrying required when the two integers are added?
ANSWER: 156
POTW #16 (10/1/23-10/7/23):
Difficulty: Medium
Solve this Sudoku:
POTW #17 (10/8/23-10/14/23):
Difficulty: Medium
How many ways are there to choose two of the POTW's (including this one) to be displayed on our website's front page?
ANSWER: 136 ways
POTW #18 (10/15/23-10/21/23):
Difficulty: Medium
Find the simplified value of this sum:
ANSWER: 1/3
POTW #19 (10/22/23-10/28/23):
Difficulty: Medium
How many ways are there to arrange the letters in "Infinity" such that the arrangement ends with "ty"?
ANSWER: 180 ways
POTW #20 (10/30-11/5/23):
Difficulty: Hard
Let N be the # of subsets of S where S={1,2,3,..11} such that there are no two consecutive integers in the subset. Find the remainder when N is divided by 100.
ANSWER: 33
POTW #21 (11/6/23-11/12/23):
Difficulty: Medium
What is the volume of a regular tetrahedron with side length 2?
ANSWER:
POTW #22 (11/13/23-11/19/23):
Difficulty: Hard
On a digital clock, times are displayed using the digits 0-9 with hours running from 1-12 and minutes running from 00 to 59. the hours do not display leading zeros and the minutes do display leading zeros. how many minutes of a 24-hour day are there such that no digit is repeated in the displayed time?
ANSWER: 896 minutes
POTW #23 (11/20/23-11/26/23):
Difficulty: Easy
If Thanksgiving occurs on the fourth Thursday of November, and Thanksgiving Day 2023 occurs on November 23rd, 2022, what day November X (where X is the date) will Thanksgiving Day 2024 occur?
ANSWER: 28
POTW #24 (11/27/23-12/3/23):
Difficulty: Easy
What is the sum of the roots of
?
ANSWER: 4
POTW #25 (12/4/23-12/10/23):
Difficulty: Hard
If N is the number of ways one can arrange eight 0's and ten 1's so that every 0 is followed by a 1, what is the value of N?
ANSWER: 45
POTW #26 (12/11/23-12/17/23):
Difficulty: Medium
There are 6 boxes, each a cube with an integer side length 1≤k≤6, such that for each box with side length k, the box directly above it cannot have a side length larger than k+1. how many ways are there to arrange the 6 boxes?
ANSWER: 32
POTW #27 (12/18/23-12/24/23):
Difficulty: Easy
What is the product of the GCD and LCM of the numbers 24 and 36?
ANSWER: 864
POTW #28 (12/25/23-12/31/23):
Difficulty: Easy
Edward conducts a survey on people's fruit preferences. 63% of them said that they liked apples, 62% of them said they liked tangerines, and 25% of them said that they liked neither. What percent of them liked both?
ANSWER: 50%
POTW #29 (1/1/24-1/7/24):
Difficulty: Medium
If New Years Day 2024 occurs on a Monday, what day of the week did New Years Day 2000 occur on?
ANSWER: Saturday
POTW #30 (1/8/24-1/14/24):
Difficulty: Medium
For what positive integer n is n!+5 a perfect cube?
ANSWER: 5
POTW #31 (1/15/24-1/21/24):
Difficulty: Hard
Let a(n)=2^n+4^n. What is the remainder a(10) is divided by 3?
ANSWER: 2
POTW #32 (1/22/24-1/28/24):
Difficulty: Hard
Let N = 888,888 × 9,999,999. Compute the sum of the digits of N.
ANSWER: 63
POTW #33 (1/29/24-2/4/24):
Difficulty: Hard
What is the maximum number of intersection points formed by 5 lines and 2 circles?
ANSWER: 32
POTW #34 (2/5/24-2/11/24):
Difficulty: Medium
In the figure, the area of the smallest square is 5 cm^2. What is the area of the largest square?
ANSWER: 20 cm^2
POTW #35 (2/12/24-2/18/24):
Difficulty: Easy
How many terms are in the arithmetic sequence 1, 4, ..., 214?
ANSWER: 72
POTW #36 (2/19/24-2/25/24):
Difficulty: Medium
Abby wants to listen to her 5 hour long playlist consisting of only 2, 3, or 4 minute songs. 46 of the songs on her playlist are 4 minutes long and 14 of the songs are 3 minutes long. How many songs are 2 minutes long?
ANSWER: 37
POTW #37 (2/26/24-3/3/24):
Difficulty: Hard
There are 15 chairs lined up in a row. Elizabeth, Fred, and Gerald want to sit on these chairs. However, Elizabeth can only sit next to Fred, but not Gerald. How many ways are there to arrange these three people?
ANSWER: 338 ways
POTW #38 (3/4/24-3/10/24):
Difficulty: Easy
What is the sum of the positive factors of 48?
ANSWER: 124
POTW #39 (3/11/24-3/17/24):
Difficulty: Hard
Define a pair of integers to be relatively prime if their greatest common factor (GCD) is 1.
How many positive integers less than or equal to 50 are relatively prime to 50?
ANSWER: 20
POTW #40 (3/18/24-3/24/24):
Difficulty: Hard
There are 15 chairs lined up in a row. Elizabeth, Fred, and Gerald want to sit on these chairs. However, Elizabeth can only sit next to Fred, but not Gerald. How many ways are there to arrange these three people?
ANSWER: 338 ways
POTW #41 (3/25/24-3/31/24):
Difficulty: Hard
An acidic solution is a mixture of two solutions of the same substance with different concentrations. The first solution contains 10% of the substance by mass, and the second solution contains 20% of the substance by mass. If we want to make a 100 gram solution that is 12% concentrated by mass, then how much of the 10% solution is required (in grams)?
ANSWER: 80 grams
POTW #42 (4/1-4/7):
Difficulty: Easy
Loother and Lony are shuffling ten cards labeled 1 through 10. If Loother randomly picks up a card, what is the probability that the card shows an even number?
ANSWER: 1/2
POTW #43 (4/8-4/14):
Difficulty: Medium
A ladybug is crawling from the origin (0,0) to the point (4,4). Each second, the lady bug moves one step in either the positive x-direction or positive y-direction. How many paths to (4,4) can it make?
ANSWER: 70 paths
POTW #44 (4/15-4/21):
Difficulty: Hard
Find the number of positive integers less than 2024 whose digit sum is 5.
ANSWER: 39
POTW #45 (4/22-4/28):
Difficulty: Hard
Find the sum of the whole numbers n<100, such that ?
ANSWER: 0
How many ways are there to arrange the letters of the word MATHCOUNTS, where the first letter of each arrangement must be a T?
POTW #46 (4/29/24-5/5/24):
Difficulty: Medium
How many ways are there to arrange the letters of the word MATHCOUNTS, where the first letter of each arrangement must be a T?
ANSWER: 362880 ways
POTW #47 (5/6/24-5/12/24):
Difficulty: Easy
Greg has a box with one of each ball numbered 1 - 30. What is the probability that Jeremy can draw an even number from the box?
ANSWER: 1/2