Geometry
Let d be equal to the distance between two points on a circle of radius 4. If d = 4*sqrt(3), what is the arc length between the two points?
120 degrees
In a bag, there are 12 purple, 7 green, 11 blue and 3 pink marbles. If you select two marbles at random from the bag, without replacement, what is the probability that you will choose first a pink marble and then a purple marble? Express your answer as a common fraction
3/88
An alien tree is growing at a rate of 15 racept per junisk. There are 8 lihoj in 1 racept and 2 sijuy in 3 junisk. How fast is the tree growing in lihoj per sijuy?
180 lihoj per sijuy
What is the remainder when 33^7 is divided by 5?
2
While practicing for MATHCOUNTS, Eli attempted to add the first ten positive integers, accidentally left one out and ended up with a perfect square. Which number did Eli leave out?
6
A triangle with side lengths 15, 14, and 13 has an area of 84. What is the length of the altitude perpendicular to the side of length 14?
12
A pair of fair, standard six-sided dice is rolled. What is the probability that the sum of the numbers rolled is 7 or 10? Express your answer as a common fraction.
1/4
Penelope and George go trick-or-treating together. Penelope gets 152 pieces of candy and George gets 124 pieces of candy. Penelope eats 5 pieces of candy every day, and George eats 4 pieces of candy every day. After how many days will George have the same amount of candy as Penelope?
28
What is the units digit of 26^20 – 12^19?
Given that a mile equals 5280 feet, how many inches are in one-sixth of a mile?
10560
This rectangular prism has integer edge lengths, and the three distinct faces of the prism have areas 35 in^2, 45 in^2 and 63 in^2, what is the volume of the prism?
315
Cayden rolls a fair 20-sided die twice. On the first roll he gets p, and on the second he gets q. The probability that p>q is a/b. What is a+b?
59
A right triangle has legs of length a and b, and a hypotenuse of length c. Another right triangle has legs of lengths of a^2 - b^2 and 2ab, and hypotenuse d. Given that d = 10, a and b are positive integers, and c is not necessarily an integer, what is a+b?
4
How many positive integer divisors of 60 have as factors exactly two of the numbers 2, 3 and 5?
5
How many ways are there to arrange the four integers 1, 2, 3 and 4 in a row so that no two adjacent numbers have a sum of 5?
8
If a square P and an equilateral triangle Q are both inscribed inside the same circle, the ratio between the areas of P and Q, in simplest form, can be expressed as (a√b)/c. What is a+b+c?
20
Peter rolls three six sided dice, labeled A, B, and C. In simplest form, what is the probability that A + B = C?
How many two-digit positive integers are more than twice the sum of their digits? For example, 17 is more than twice the sum of its digits because 17 > 2(1+7)=16.
88
2024 has N positive factors. What is the smallest positive integer A that also has N factors?
120
Amanda is thinking of a positive integer. The sum of the distinct prime factors of her integer is 10 and her integer is less than 100. What is the sum of the possible values of Amanda’s integer?
264
A right circular cone has a base radius of 6 meters and a slant height of 10 meters. This cone is cut parallel to its base to form a smaller cone and a frustum. The volume of the frustum is 1/3 that of the original cone. What is the volume of the frustum?
32 pi
What is the probability of randomly selecting three distinct unit squares on a 3 × 3 grid and getting three unit squares in a row vertically, horizontally or diagonally? Express your answer as a common fraction.
2/21
One-tenth the sum of the positive integers from n to n + 10, inclusive, equals an integer s. What is the least possible value of s?
11
Captain Hook has found a treasure chest filled with over 200 gold coins. If Captain Hook separates the coins into 4 equal piles, there is 1 coin left over. If Captain Hook separates the coins into 7 equal piles, there is 1 coin left over. If Captain Hook separates the coins into 9 equal piles, there are 2 coins left over. What is the least possible number of coins in the treasure chest Jack found?
281 coins
A frog lies in the center tile of 3 rows and columns of tiles (9 tiles total; 3x3). Each minute, the frog jumps randomly up, down, left, or right one tile. The frog always moves after each minute (staying within the 3x3 area), and does not move diagonally. What is the average time (in minutes) that the frog will take to move to a corner tile?