# Tagged Questions

Use this tag when you want to determine the thinking that is needed to solve a certain type of problem, as opposed to looking for a specific answer to a question.

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### How to solve the problem that determines the age of Diophantus?

How to solve the problem that determines the age of Diophantus?"God gave him to be a boy for the sixth part of his life, and adding one to it twelfth part covered her cheeks fluff, He gave him the ...
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### How to solve the sequence: $87, 89, 95, 107, ?, 157$

This question appeared in a competitive exam. The question is: Q. Find the unknown term in $87,89,95,107,?,157$ 1)127 $\ \ \ \ \ \ \ \$ 2)122 3)139 $\ \ \ \ \ \ \ \$ ...
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### An example of a problem which is difficult but is made easier when a diagram is drawn

I am writing a blog post related to problem solving and one of the main techniques used in problem solving is drawing a diagram. Essentially, I want to illustrate that some hard problems (for example, ...
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### “Here's a cool problem”: a collection of short questions with clever solutions

This game will be familiar to many mathematicians, and it is always good fun to play. I am looking to find a list of good questions with short, when-you-see-it solutions. The kind of question one ...
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### How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?

How do I integrate this guy? I've been stuck on this for hours.. $$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
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### Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers

For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
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### Is it possible to construct a sequence that ends in 1000000000?

Starting from the number $1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it or rearranging its digits (not allowing ...
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### Sum of the sum of the sum of the first $n$ natural numbers

I have here another problem of mine, which I couldn't manage to solve. Given that: $$x_n = 1 + 2 + \dots + n \\ y_n = x_1 + x_2 + \dots + x_n \\ z_n = y_1 + y_2 + \dots + y_n$$ Find $z_{20}$...
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### Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me....
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### Calculating Non-Integer Exponent

I just wanted to directly calculate the value of the number $2^{3.1}$ as I was wondering how a computer would do it. I've done some higher mathematics, but I'm very unsure of what I would do to solve ...
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### Math Olympiads: GCD of terms in a sequence equals GCD of their indices.

The sequence $a_1 ,a_2 ,a_3 ,...$ of positive integers satisfies $\text{gcd}(a_i ,a_j ) = \text{gcd} (i, j)$ for $i \neq j$. Prove that $a_i = i$ for all $i$. Source: Russian Mathematical Olympiad, ...
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### “Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it's acceptable. EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and ...
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### How to solve an overdetermined system of point mappings via rotation and translation

I have a set of points in one coordinate system $P_1, \ldots, P_n$ and their corresponding points in another coordinate system $Q_1, \ldots , Q_n$. All points are in $\mathbb{R}^3$. I'm looking for a ...
### $W$ white balls, $B$ black balls, adding $K$ of the resultant color each iteration
The problem is stated as follows. We have a box with $W$ white balls and $B$ black ones. Repeat $N$ times: each iteration a ball is taken out (uniformly), and put back along with $K$ (constant) more ...