Algebra
Suppose 15% of x equals 20% of y. Compute the percentage of x that y is.
What is 75?
A circular table has 60 chairs around it. There are N people seated at this table in such a way that the next person seated must sit next to someone. Evaluate the smallest possible value for N.
What is 20?
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, compute the area of the hexagon.
What is 6?
Find the smallest two-digit positive integer that is a divisor of 201020112012.
What is 12?
According to legend, the ancient Greek Pythagoreans murdered Hippasus for proving that this number was irrational.
What is √2?
Compute the sum of all distinct real values of x such that
||| ... ||x| + x| ... |+x|+x| = 1
Where there are 2025 x's in the equation.
What is -2024/2025?
There are 15 stones placed in a line. Evaluate the number of ways you can mark 5 of these stones so that there are an odd number of stones between any two of the stones you marked.
What is 77?
Let ABC be an isosceles triangle with AB = AC. Let D and E be the midpoints of segments AB and AC, respectively. Suppose that there exists a point F on ray DE outside of ABC such that triangle BFA is similar to triangle ABC. Compute AB/BC.
What is √2?
When Larry turned 16 years old, his parents gave him a cake with n candles, where n has exactly 16 different positive integer divisors. Compute the smallest possible value of n.
What is 120?
This mathematician/philosopher is known for his famous statement "cogito, ergo sum" and is considered a pioneer of analytic geometry.
Who is René Descartes?
Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, compute the minimum number of pills he must take to make his weight 2015 pounds.
What is 13?
In Middle-Earth, nine cities form a 3 by 3 grid. The top left city is the capital of Gondor and the bottom right city is the capital of Mordor. Compute the number of ways that the remaining cities be divided among the two nations such that all cities in a country can be reached from its capital via the grid-lines without passing through a city of the other country.
What is 30?
Let ABCD be an isosceles trapezoid with bases AB = 92 and CD = 19. Suppose AD = BC = x and a circle with center on AB is tangent to segments AD and BC. If m is the smallest possible value of x, compute m2.
What is 1679?
For each nonnegative integer n we define An = 23n + 36n+2 + 56n+2. Find the greatest common divisor of the numbers A1, A2, ... A2024.
What is 7?
This famous French mathematician died young due to a duel during the French Revolution of 1830. He laid down the foundations of group theory and studied algebraic solutions to polynomial equations. Provide his last name (bonus points if you get his first name.)
Who is Évariste Galois?
Compute the remainder when 2202 + 202 is divided by 2101 + 251 + 1.
What is 201?
Let T = {1, 2, 3, . . . , 14, 15}. Say that a subset S of T is handy if the sum of all the elements of S is a multiple of 5. For example, the empty set is handy (because its sum is 0) and T itself is handy (because its sum is 120). Compute the number of handy subsets of T.
What is 6560?
In triangle ABC, the external angle bisector of <BAC intersects line BC at D. E is a point on ray AC such that <BDE = 2<ADB. If AB = 10, AC = 12, and CE = 33, compute DB/DE.
What is 2/3?
A set S of distinct positive integers has the following property: for every integer x in S the arithmetic mean of the set of values obtained by deleting x from S is an integer. Given that 1 belongs to S and that 2002 is the largest element of S, what is the greatest number of elements that S can have?
What is 30?
This internationally revered, 20th century logician wrote a proof of God's existence, met his wife at a nightclub, and later developed a phobia of being poisoned and starved to death. His work justified mathematicians in assuming the axiom of choice in their proofs.
Who is Kurt Gödel?
Compute the largest prime divisor of 1001110116. Give your answer in base 10.
What is 181?
A 5x5 table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly once in every 2x2 subtable.The sum of all numbers of a regular table is called the total sum of the table. With any four numbers, one constructs all possible regular tables, computes their total sums, and counts the number of distinct outcomes. Determine the maximum possible count.
What is 60?
Let ABC be a triangle with AB = 6, AC=7, circumcircle w, and incenter I. Let X be a point on w such that AX = CX and segments BX and AC do not intersect. Let Y be a point on w such that AY = BY and segments CY and AB do not intersect. If I lies on line XY, compute the length of BC.
What is 13/3?
Consider a string of n 7's, 777...77, into which + signs are inserted to produce an arithmetic expression. For example, 7 + 77 + 777 + 7 + 7 = 875 could be obtained from eight 7's in this way. Compute the number of distinct values of n for which it is possible to insert + signs so that the resulting expression has value 7000.
What is 108?
This 13th century Chinese mathematician proved the generalized Chinese Remainder Theorem in his work "Mathematical Treatise in Nine Sections" and introduced the use of the zero symbol in written Chinese mathematics. He then became a corrupt government official.
Who is Qin Jiushao (秦九韶)?