Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

Inner Equality

Age 16 to 18 Short
Challenge Level

Suppose that we are told that four numbers $a, b, c, d$ lie between $-5$ and $5$. Suppose also that the numbers are constrained so that
$$5< a+b < 10 \quad\mbox{ and }\quad -10< c+d < -5$$

Given this information, what can you deduce about these inequalities?

$$ ?? < a+ b- c - d < ?? $$ $$ ?? < a- c < ?? $$ $$ ?? < a - c + d - b < ?? $$ $$ ?? < abcd < ?? $$ $$ ?? < \frac{|a|+|c|}{2}-\sqrt{|ac|} < ??$$

Did you know ... ?

There are many useful general inequalities in mathematics, such as the AM-GM, Cauchy-Schwarz and Jensen's inequalities. These general inequalities are powerful tools which greatly simplify a wide variety of problems in mathematics, in applications from integration to probability via linear algebra.

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Shades of Fermat's Last Theorem

Exhaustion

Code to Zero

Inner Equality

Age 16 to 18 ShortChallenge Level

Age 16 to 18 Short
Challenge Level