Algebra
The ratio of w to x is 4:3, the ratio of y to z is 3:2, and the ratio of z to x is 1:6. What is the ratio of w to y?
16 to 3
The area of a rectangle with integer side lengths is 32 cm^2. What is the least possible perimeter of the rectangle?
24
Dak has a quarter, a dime, a nickel, and a penny. How many different amounts can be obtained by using one or more of the coins in Dak's collection?
15
An abundant number is a number for which the sum of its positive proper factors is greater than the number itself. For example, because the sum of the positive proper factors of 24 is 36, it follows that 24 is an abundant number. What is the least abundant number greater than 24?
30
Name the three math competitions that Spartan Vanguard helps to host.
Aleph, Vanguard Math Tournament (VMT) and Mathzilla
The sum of two natural numbers is 17,402 . One of the two numbers is divisible by 10. If the units digit of that number is erased, the other number is obtained.
What is the difference of these two numbers?
14238
A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?
26
Ms. Pauling’s chemistry class has 5 lab benches, each of which seats 2 students. If 6 students enter her otherwise empty classroom, and each student picks a random available open seat, what is the probability that at least one of the lab benches is completely empty?
13/21
Find the positive base a in which the equation 4a*12a = 103a is valid.
5
Found naturally in pinecones and sunflowers, the spiral arrangement of structures and seeds follow this geometric sequence.
The Fibonacci Sequence
What is the sum of all real numbers x for which |x2-12x+34| = 2?
18
For positive integers n and m, each exterior angle of a regular n-sided polygon is 45 degrees larger than each exterior angle of a regular m-sided polygon. One example is n = 4 and m = 8 because the measures of each exterior angle of a square and a regular octagon are 90 degrees and 45 degrees, respectively. What is the greatest of all possible values of m?
56
A point (x,y) is randomly picked from inside the rectangle with vertices (0,0), (4,0), (4,1), and (0,1). What is the probability that x<y?
1/8
A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x+32 are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?
24
This German mathematician is credited with discovering the formula for summing n consecutive numbers, supposedly when he was in elementary school and his teacher gave the class the busy work of summing the first 100 natural numbers.
Carl Friedrich Gauss
What is the sum of all real numbers x for which the median of the numbers 4,6,8,17, and x is equal to the mean of those five numbers?
-5
Triangle ABC has AB=2*AC. Let D and E be on lines AB and BC respectively, such that angle BAE = angle ACD. Let F be the intersection of segments AE and CD, and suppose that triangle CFE is equilateral. How many degrees is angle ACB?
90
I have tiled my square bathroom wall with congruent square tiles. All the tiles are red, except those along the two diagonals, which are all blue (i.e. the corners are blue and all the tiles along the diagonals between each pair of opposite corners are blue). If I used 121 blue tiles, how many red ones did I use?
3600
If a, b and c are all positive integers greater than 1 for which a + ab + abc = 415, what is the value of c?
40
This famous equation, ei*pi+1 = 0, was called "the most beautiful equation in mathematics" by many, including Richard Feynman.
Euler's Identity
In the complex plane, let A be the set of solutions to z3-8 = 0and let B be the set of solutions to z3 - 8z2 - 8z + 64 = 0. What is the greatest distance between a point of A and a point of B?
2√21
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle (the circle around the triangle). What is the radius, in inches, of the circle?
(3*sqrt3)/pi
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number 1, then Todd must say the next two numbers (2 and 3), then Tucker must say the next three numbers (4,5,6), then Tadd must say the next four numbers (7,8,9,10), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number 10,000 is reached. What is the 2025th number said by Tadd?
5985
Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?
An Ancient Greek polymath who used shadows and observations of the sun's rays to calculate the Earth's circumference with remarkable accuracy.
Eratosthenes