Algebra
Suppose n% of m is 4 and m% of mn is 6. What is n?
800/3
The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eight graders taking a foreign language class. How many eight graders take only a math class and not a foreign language class?
39
What is the value of (1 · 1! + 2 · 2! + 3 · 3! + 4 · 4!) / [ (2 × 0 × 2 × 1) + (2^3 + 0^3 + 2^3 + 1^3 )]?
7
The faces of each of two fair dice are numbered 1, 2, 3, 5, 7, and 8. When the two dice are tossed, what is the probability that their sum will be an even number?
5/9
A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?
30
A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased?
40
Steven has 6 coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats (fatty). In order to make the coupons last, she decides that she will redeem one every 10 days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the 6 dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Steven redeem her first coupon?
Wednesday
A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome.) Let N be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of N?
2
On a beach 50 people are wearing sunglasses and 35 people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is 2/5. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?
7/25
Triangles ABC and ADC are isosceles with AB=BC and AD=DC. Point D is inside triangle ABC, angle ABC measures 40 degrees, and angle ADC measures 140 degrees. What is the degree measure of angle BAD?
Rayna drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?
110
Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?
5760
Call a positive integer troubling if its only prime divisors are 2 and 5. Two troubling numbers have a sum of 135, 000. Find the number of positive integer divisors of their product.
88
A box contains five cards, numbered 1, 2, 3, 4, and 5. Three cards are selected randomly without replacement from the box. What is the probability that 4 is the largest value selected?
3/10
What is the area of the triangle formed by the lines y=5, y=1+x, and y=1-x?
16
Let x be a real number such that x^4 − 7x^3 + 10x^2 + 7x + 1 = 0. Find the sum of all possible values of x^3 − 1/x^3 .
(A) 111 (B) 112 (C) 113 (D) 114 (E) 115
(B) 112
In a tournament there are six teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?
24
How many positive factors does 23,232 have?
42
An integer between 1000 and 9999, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?
56/225
Suppose the interior angle measure of a regular m-gon is exactly 24 degrees greater than the interior angle measure of a regular n-gon. What is the sum of all possible values of n?
42
Let x, y be positive real numbers such that log_2(x/y) + xy = 69 and log_2(x^2 * y) + log_2(x^2 * y^4 )/2 = 18. Compute x^2 + 8y^2 .
(A) 2020 (B) 2032 (C) 2048 (D) 2064 (E) 2080
(D) 2064
Steven has 24 apples. In how many ways can she share them with Jason and Rayna so that each of the three people has at least two apples?
190
For positive integers 𝑛, let the 𝑛th triangular number be the sum of the first 𝑛 positive integers. For how many integers 𝑛 between 1 and 100, inclusive, does the 𝑛th triangular number have the same last digit as the product of the first 𝑛 triangular numbers?
(A) 11 (B) 12 (C) 20 (D) 21 (E) 22
(E) 22
Fifteen people are split into teams Alpha and Beta for a game so that Team Alpha has 2 players, Team Beta has 13, and teams are chosen at random. For any game, the probability that Team Alpha wins is the same. Juwang, one of the 15 players, is on the winning team23of the time. For any game in which Juwang is on the winning team, the probability that Juwang is on Team Alpha may be expressed as mn for relatively prime positive integers m, n. Find m+n.
3/55
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?
(10-7sqrt(2))/3