Algebra
A glass filled with water has a mass of 1000g. When half of the water is removed, the mass of the glass and remaining water is 700g. What is the mass of the empty glass?
400g (Pascal 2005)
How many positive integer factors of 2020 have more than 3 factors? (As an example, 12 has 6 factors, namely 1,2,3,4,6, and 12.)
7
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 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?
39
Let angle ABC = 24 degree and angle ABD = 20 degree. What is the smallest possible degree measure for angle CBD?
4 degree
Which mathematician from Syracuse discovered the law of buoyancy and approximated the value of pi?
Archimedes
Krystyna has some raisins. She gives one-third of her raisins to Mike. She then eats 4 raisins, after which she gives one-half of her remaining raisins to Anna. If Krystyna then has 16 raisins left, how many raisins did she have to begin?
54 (Pascal 2015)
The least common multiple of a positive integer n and 18 is 180, and the greatest common divisor of n and 45 is 15. What is the sum of the digits of n?
6
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
24
Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7⋅EB. What is the ratio of the area of EFGH to the area of ABCD?
25/32
Which famous unsolved problem in mathematics involves finding the zeroes of the zeta function?
Riemann Hypothesis
If (x-y) / (x+y) = 5, then (2x+3y) / (3x-2y) equals
0 (Cayley 2018)
Find all positive integers n such that n2+1 is divisible by n+1.
1
How many rearrangements of abcd are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either ab or ba.
2
Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?
rt(3)/2
Who solved Fermat's Last Theorem?
Andrew Wiles
For how many positive integers x is the product (x-2)(x-4)(x-6)...(x-2016)(x-2018) less than or equal to 0?
1514 (Fermat 2018)
The integer 636 405 may be written as the product of three 2-digit positive integers. The sum of these three integers is
259 (Pascal 2013)
A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?
1/5
A wire is cut into two pieces, one of length a and the other of length b. The piece of length a is bent to form an equilateral triangle, and the piece of length b is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is a/b?
rt(6)/2
How many perfect scores did China get on the International Mathematics Olympiad in 2022? (hint: in 2022 there were a total of 10 perfect scores over all countries)
6 (yes, that's everyone on their team)
How many pairs (x,y) of non-negative integers with y >= x satisfy the equation 5x2-4xy+2x+y2=624?
7 (Fermat 2015)
If p, q, r, and s are digits, how many of the 14-digit positive integers of the form 88663311pqrs48 are divisible by 792?
50 (Pascal 2019)
How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3?
96
Rhombus ABCD has side length 2 and angle B = 120°. Region R consists of all points inside the rhombus that are closer to vertex B than any of the other three vertices. What is the area of R?
2rt(3)/3
What theorem in combinatorics is named after a popular food item?
The Chicken McNugget Theorem