1
One year a manufacturer announced a 10% price increase, and the cost of their product went up by $40. The next year the manufacturer announced a 15% price increase. Find the additional number of dollars the cost went up the second year.
66
A rectangular piece of paper measures 32 cm by 24 cm. Start by cutting a 2-cm-wide strip off the top side of the piece of paper leaving a 30 cm by 24 cm rectangle. Then rotate the paper 90◦ and cut another 2-cm-wide strip off the new top side of the piece of paper. Continue rotating the paper by 90◦ and cutting another 2-cm-wide strip off the top side of the paper. Continue this until the entire paper has been cut into 2-cm-wide strips. Finally, line up all the strips end-to-end to form one long 2-cm-wide strip. Find the length in centimeters of this strip.
384
Yan needs to grow 20% taller before she will be allowed to ride the roller coaster. Her younger brother, Sile, is three quarters as tall as Yan. Find the percentage taller that Sile must grow before he will be allowed to ride the roller coaster.
60
The integers m and n are each greater than 4 and satisfy the equation
(4m + 5)(4n + 5) − (3m + 2)(3n + 2) = 2023.
Find m + n.
35
Find the positive even integer n for which
1^2 + 2^2 + 3^2 + · · · + n^2 = 4(1 + 3 + 5 + · · · + 3(n − 1)
24
Jaylen holds 2 coins, and Hailey holds 3 coins. They perform four exchanges. On each exchange, each of Jaylen and Hailey randomly selects one of the coins that they currently hold, and they exchange those coins. The probability that each ends up with the same coins that they started with is m/n , where m and n are relatively prime positive integers. Find m + n
239
Find the least positive integer with the property that if its digits are reversed and then 450 is added to this reversal, the sum is the original number. For example, 621 is not the answer because it is not true that 621 = 126 + 450
1501
Jasmin selects a real number between 5 and 11, and Trenton selects a real number between 3 and 10. Given that the numbers are selected randomly and independently, the probability that Jasmin’s number and Trenton’s number differ by at most 2 is m/n , where m and n are relatively prime positive integers. Find m + n.
41
For real numbers a, b, and c, the roots of the polynomial x^5 − 10x^4 + ax^3 + bx^2 + cx − 320 are real numbers that form an arithmetic progression. Find a + b + c.
49
Four indistinguishable red blocks, four indistinguishable white blocks, and four indistinguishable blue blocks are randomly placed into three boxes with four blocks in each box. The probability that at least two of the boxes receive identical collections of blocks is m/n, where m and n are relatively prime positive integers. Find m + n
95
Find the number of sequences of 10 letters where all the letters are either A or B, the first letter is A, the last letter is B, and the sequence contains no three consecutive letters reading ABA. For example, count AAABBABBAB and ABBBBBBBAB but not AABBAABABB or AAAABBBBBA.
86
Each face of a cube is painted solid white or solid black, with the colors chosen independently and at random. The probability that the cube contains at least one vertex such that the three faces of the cube sharing that vertex are all painted the same color is m/n , where m and n are relatively prime positive integers. Find m + n.
55