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Largest Odd Factors

Pick a number. Call it $$n$$. Write down all the numbers from $$n+1$$ to $$2n$$ (inclusive). For example, if you picked 7, you would write:
$$8,9,10,11,12,13,14$$
Below each number, write down its largest odd factor. Add these factors up. What is the result? Why?

Square Factorials

Source: Woody at Maths Jam
Multiply together the first 100 factorials:
$$1!\times2!\times3!\times...\times100!$$
Find a number, $$n$$, such that dividing this product by $$n!$$ produces a square number.

An Arm and a Leg

If 60% of people have lost an eye, 75% an ear, 80% an arm and 85% a leg, what is the least percentage of people that have lost all four?

Blackboard Sums II

The numbers 1 to 20 are written on a blackboard. Each turn, you may erase two adjacent numbers, $$a$$ and $$b$$ ($$a$$ is to the left of $$b$$) and write the difference $$a-b$$ in their place. You continue until only one number remains.
What is the largest number you can make?
Tags: Numbers

Source: nrich
Find all real solutions to
$$(x^2-7x+11)^{(x^2-11x+30)}=1.$$

Two Tangents

Source: Reddit
Find a line which is tangent to the curve $$y=x^4-4x^3$$ at 2 points.

Blackboard Sums

The numbers 1 to 20 are written on a blackboard. Each turn, you may erase two numbers, $$a$$ and $$b$$ and write the sum $$a+b$$ in their place. You continue until only one number remains.
What is the largest number you can make?
Tags: Numbers

Hat Check

Three logicians, A, B and C, are wearing hats. Each has a strictly positive integer written on it. The number on one of the hats is the sum of the numbers on the other two.
The logicians say:
A: I don't know the number on my hat.
B: The number on my hat is 15.
Which numbers are on hats A and C?

Tags: logic