Interesting Fact About 12-th Power

Show that every -th power of an integer can be written as a sum of three third power of integers.

Solution : Note that for an arbitrary integer . One can also choose .

Inequality Involving Integers

Let be an integer and be pairwise distinct integers. Prove that

.

Solution : We can see that since are distinct, the minimal sum for happens when or . In both cases, the value for is and . We see that .

Source : Societatea de Stiinte Matematice din Romania

Elementary Counting

There are type of breads at ZumaZuma store. students went to the store and each one buys different breads. Each type of breads is bought by students, and each pair of bread types is chosen by students.

Prove that

i.)

ii.) .

Exponent Condition for Prime Numbers

Suppose that are two integers such that is a prime number. Show that is a power of

Source : European Mathematical Society.

Inequalities from Art of Problem Solving Forum

1.) Suppose that are real numbers that satisfy

show that

i.)

ii.)

iii.)

iv.)

v.)

vi.)

2.) Suppose that are real numbers that satisfy and , show that

i.)

ii.)

iii.)

iv.)

v.)

Source : Art of Problem Solving, post by user sqing.

Unexpected Inequality

Show that for ,

.

Source : Mathematics Magazine.

Inequality from American Mathematical Monthly

Let be positive numbers. Show that

.

Solution : Standard QM-AM-HM seems enough to solve this problem, so

which can be manipulated and concluded to be equivalent with the statement.

Source : American Mathematical Monthly Problem 12083.

Trigonometric Identity Using Complex Number.

Prove that .

My Take on Mathematical Reflection Problem

This is problem U610 from Mathematical Reflection, Edition 6 2022.

If is a positive integer, evaluate

.

Solution : Write

,

we want to evaluate the value of , while on the other hand

.

Take , we thus have , and

.

Corollary : If is a positive integer, then we have

Convex Function to Integral Inequality

Suppose that is a convex function. Show that for all integer .