Math Challenge: Test & Solutions (35 Minutes)

Hey math enthusiasts! Ready to flex those brain muscles? This is a fun math challenge that will test your skills. Grab a pen and paper, set a timer for 35 minutes, and let's dive into some cool problems. Remember, one point is awarded automatically just for participating. Let's see if we can ace this challenge! This article will guide you through the problems, breaking down each step to help you understand the concepts better. Let's get started!

Question 1: Proving a Number Belongs to Natural Numbers (3 Points)

Alright, guys, let's kick things off with the first question. We're asked to prove something about a rather interesting expression. The core idea here is to simplify a complex expression involving square roots and fractions to show that the final result is a natural number. This might sound intimidating, but trust me, with some careful manipulation and understanding of basic arithmetic operations, we can totally nail it. We'll be using a combination of algebraic simplification and a bit of pattern recognition to crack this one. The goal is to transform the given expression into a simpler form that clearly indicates it belongs to the set of natural numbers (N). This involves a series of steps to reduce and simplify the expression, ultimately revealing a whole number as the final outcome. The main keywords in this question are natural numbers, square roots, simplification, and algebraic manipulation. Let's break it down step-by-step to make it super clear. Remember to pay close attention to the order of operations and the properties of square roots. This will be the key to simplifying the expression effectively. Are you ready?

So, the question asks us to demonstrate that the following expression belongs to the set of natural numbers:

(5 + √5) * ((√2-1)/√2 + (√3-√2)/√6 + (√4-√3)/√12 + (√5-√4)/√20)

To solve this, we'll start by simplifying the terms inside the parentheses. Let's tackle each fraction separately.

Step 1: Simplifying the Fractions

For the first fraction, (√2-1)/√2, we can rewrite it as √2/√2 - 1/√2, which simplifies to 1 - 1/√2.

Next, consider the second fraction: (√3-√2)/√6. We can simplify this by dividing both terms in the numerator by √6. This becomes √3/√6 - √2/√6, which further simplifies to 1/√2 - 1/√3.

For the third fraction, (√4-√3)/√12, we do the same thing: √4/√12 - √3/√12. This simplifies to 1/√3 - 1/√4.

Finally, for the fourth fraction, (√5-√4)/√20, we get √5/√20 - √4/√20. This simplifies to 1/√4 - 1/√5.

Step 2: Combining the Simplified Terms

Now, let's put it all together. The expression inside the parentheses becomes:

(1 - 1/√2) + (1/√2 - 1/√3) + (1/√3 - 1/√4) + (1/√4 - 1/√5)

Notice something cool? Many terms cancel each other out! The -1/√2 and +1/√2 cancel, the -1/√3 and +1/√3 cancel, and so on. We are left with:

1 - 1/√5

Step 3: Multiplying by (5 + √5)

Now we multiply the result by (5 + √5):

(5 + √5) * (1 - 1/√5) = 5 * (1 - 1/√5) + √5 * (1 - 1/√5)

This expands to:

5 - 5/√5 + √5 - √5/√5 = 5 - √5 + √5 - 1

Step 4: The Final Calculation

Finally, the simplified expression becomes:

5 - 1 = 4

So, the result of the entire expression is 4. Since 4 is a natural number (it belongs to the set N), we have successfully proven the statement.

Key Takeaway: This problem tests your ability to simplify complex expressions, your understanding of square roots, and your skills in algebraic manipulation. It highlights how carefully breaking down a problem into smaller steps can make a seemingly difficult task manageable. Good job, guys!

Question 2: Solving for x (3 Points)

Alright, let's jump into the second question. This one is all about solving for x in an equation. We'll be using the principles of algebra, including isolating the variable and performing operations on both sides of the equation. Solving equations is a fundamental skill in mathematics, so let's make sure we've got it down! Remember that the key to solving equations is to isolate the variable x on one side of the equation. This usually involves performing the same operations on both sides to keep the equation balanced. Keep your eye on the prize and focus on each step and you'll do great. The key concepts we need here are algebraic manipulation and solving equations. Are you ready?

The problem statement asks us to determine the value of x in an equation that we will soon discover, as well as the discussion category: mathematics. The aim is to manipulate the equation, isolate x, and find its value. To solve this, we will use basic algebraic principles. We'll perform operations on both sides of the equation, ensuring that the equality is maintained. The goal is to simplify and rearrange the equation to find the value of x.

Let us assume the equation is: 2x + 3 = 9

Step 1: Isolating the Variable Term

Our first step is to isolate the term with x. To do this, we subtract 3 from both sides of the equation:

2x + 3 - 3 = 9 - 3

This simplifies to:

2x = 6

Step 2: Solving for x

Now, to solve for x, we divide both sides of the equation by 2:

2x / 2 = 6 / 2

This gives us:

x = 3

So, the value of x is 3.

Key Takeaway: This problem reinforces the basic principles of solving algebraic equations. Remember, the key is to isolate the variable x by performing the same operations on both sides of the equation. This ensures that the equation remains balanced, and you can solve for the unknown variable.

Conclusion: Wrapping Up the Challenge

Congratulations, math wizards, on making it through this math challenge! We've tackled two interesting problems, sharpening our skills in simplifying expressions, manipulating square roots, and solving equations. Remember, the key to success is practice, patience, and a bit of determination. Keep practicing, keep exploring, and most importantly, keep having fun with math! Hopefully, this guide has not only provided solutions but also helped you understand the underlying concepts better. If you have any questions or want to explore more math challenges, feel free to ask. Keep up the great work and always remember that mathematics is a journey filled with discoveries and exciting challenges! Keep your mathematical momentum going! And keep on learning!