ALGEBRA

GEOMETRY

PROBABILITY

NUMBER THEORY

LOGIC

100

f(f(64)) when f(x) = √x + 1

What is 4?

100

What is the area of a triangle with a base length of 4 and a height of 6?

What is 12?

100

How many ways can a person choose from three different shirts and two different socks?

What is 6?

100

How many positive integers solution pairs (x, y) exist for x + y = 10?

What is 9?

100

Below are various 4x4 arrangements chips. Each chip has a blue side and a black side. You are allowed to manipulate the chips by choosing a row or column and flipping every chip in that row or column so that all blue chips become black and vice versa. In the 5 configurations below, which one can never be completely blue no matter how you manipulate it?

What is D?

200

The only real solution of x for x(x^{2} − 2x + 100) = 200

What is 2?

200

What is the area of triangle with (0,8), (4,0), and (4,10)

What is 20?

200

How many ways are there to sort six cards A, B, C, D, E, and F from one to six?

What is 720?

200

How many ways can positive integers x and 47 − x^{2} both be prime numbers?

What is 3?

200

There are 5 people, each with a letter of the alphabet as their name. They either wear a blue shirt or red shirt. People with blue shirts always tell the truth, and people with red shirts always lie. List out everyone who is wearing blue.

A: I have a blue shirt.

B: A does not have a blue shirt.

C: If B is not wearing a blue shirt, then I am not wearing a blue shirt.

D: I am wearing the same color shirt as E.

E: I am wearing a different color shirt than D.

What are B, C, and E?

300

What is the value of a_{100} given that a_{1} =1 and a_{n}=(a_{n-1)}/(√(a^{2}_{n-1+1}))?

What is 1/10?

300

What is the area of an octagon with side length 2?

What is 8+8√2?

300

What is the probability that at least one side of three tossed standard six-sided dice is a three?

What is 91/216?

300

What is the positive integer value of x such that 6^{x} has exactly 100 factors?

What is 9?

300

Which of the following statements is guaranteed to always be true about IM3+ students and their IM3+ classmates?

A) There is an odd number of students who have an odd number of classmates.

B) There is an even number of students who have an even number of classmates.

C) There is an odd number of students who have an even number of classmates.

D) There is an even number of students who have an odd number of classmates.

E) None of the above is guaranteed to be true.

What is D?

400

The solution of x for the equation log_{4}x = 3 + log_{64}x?

What is 512?

400

A triangle with side lengths 6, 8, and 10 is inscribed in a circle. What is the area of the circle?

What is 25π?

400

What is the probability that two positive integers randomly chosen from 1 to 10 (including 1 and 10) are relatively prime?

What is 31/45?

400

What is the units digit of the number: 2^{100} + 3^{100}?

What is 7?

400

A and B are playing a game with two non-empty piles of marbles. On each player’s turn, they will choose a pile to take marbles from. They can choose to take as many marbles as they want(including all the marbles in the pile). The winner is the one who takes the last marble/marbles. A always plays first. Which of the following statements is true?

A: A always wins if the total number of marbles is odd.

B: A always wins if the total number of marbles is even.

C: B always wins if the total number of marbles is odd.

D: B always wins if the total number of marbles is even.

E: A wins if and only if both piles have the same amount of marbles.

What is A?

500

x^{2} + y^{2} for the following system of equations:

x ≠ y

(x-3)^{2}=y+9

(y-3)^{2}=x+9

What is 35?

500

What is the maximum amount of intersection points within 19 lines?

What is 171?

500

How many ways can a frog hop up a 10-step staircase, if it can only leap 2 or 3 steps each jump?

What is 7?

500

What is the number of positive integer solution pairs (x, y) to 1 = 3/x + 4/y?

What is 4?

500

There are 10 stacks of coins, each with 10 coins in them. 9 of the stacks have legitimate coins, each weighing 1 gram. The last stack is full of counterweigh coins, which weigh 1.1 grams each. You have a digital scale you can weight the coins with. In the worst case scenario, what is the minimum number of times you have to weigh the coins to guarantee that you’ll be able to find the stack with the counterfeit coins?

What is 1?

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