Miscellaneous Algebra Problems & Solutions
In this article, we are going to be solving some various Mathematics Olympiad Problems highly related to the field of Algebra.
First, the questions will be listed. We strongly recommend you to try to solve these questions on your own and then look at the solution. That way, the optimal learning would happen! Also, if some techniques in the solutions are unfamiliar to you, checkout our other articles in both Betamat-EN and Betamat-TR. Some articles about those techniques are yet to be published.
So, let’s start then!
- k is a constant real number. If there are no ( x,y,z) real number triples which satisfy the following system of equations,
What is k?
2. Show that for the positive all x, y, z real numbers, the following inequality holds:
3. The positive integers x, y satisfy the following inequality:
Prove that x³ ≥ 2y
4. Prove that for any positive a, b, c the following inequality holds:
Stop right here to attempt solving these questions. Don’t forget that these are not a piece of cake!
If you’re ready, let’s continue!
- If the equation (1) is multiplied by −2, (2) is multiplied by −1, and added to the equation (3), we would get the following equation:
If k ≠ 5: x could’ve been found and therefore –from the equations given in the question– y, z could’ve been found so there would be (x,y,z) triples satisfying the equations. If k = 5: the last equation would state 0 = −1, creating a contradiction. Therefore, k = 5!
2. If we apply the AM-GM (Arithmetic Mean - Geometric Mean) inequality a few times:
3. x³ ≥ 2y ⇔ x⁹ ≥ 8y³ and x⁵-y³ ≥ 2x ⇔ ( x⁴-2)≥ y³ ⇔ 8x ( x⁴-2)≥ 8y³. If we show the following inequality, the proof will finish.
This inequality also means,
This inequality always holds when x ≥ 0.
4. If this inequality holds, the following inequalities would hold as well.
This inequality is the same with the following inequality
Since a/b, c/b, a/c are all positive numbers, if
the inequality would hold. Let’s observe one of them.
By applying the AM-GM inequality,
Since we can apply the same thing to the others, the proof is concluded.
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