# Tagged Questions

370 views

### How to find natural solutions of an equation?

When I'm solving problems, I'm often confronted to solving equations, and when I'm solving equations, I'm often confronted to find the natural solutions of these equations. My actual personal ...
58 views

### Sum of square roots of integers

Let $x, y$ be integers and consider the equation $$\sqrt{x}+\sqrt{y}= 8 \sqrt{31}.$$ It is claimed that this implies $\sqrt{x}= a\sqrt{31}$ and $\sqrt{y}=b \sqrt{31}$ for $a,b$ integers. While this ...
181 views

### How many diamonds did they steal?

There are $7$ thieves. They steal diamonds from a diamond merchant and run away into the jungle. Whilst they're running, night falls and they decided to rest in the jungle. When everybody is ...
56 views

### Vieta jumping with non-monic polynomials

I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via ...
56 views

### Troubles with understanding the answer

I don't understand the proof. Where did they get the first line from? 21x11=1+5x46? Fermat's theorem in my view is a^46=_1mod47
64 views

### Olympic problem on irreducible fraction

Prove that the fraction $\frac{21n+4}{14n+3}$ is irreducible for every natural number $n$.
179 views

### Math Olympiads: GCD of terms in a sequence equals GCD of terms in other sequence

Recently, someone asked for a proof of a problem from the Russian Mathematical Olympiad, 1995. Math Olympiads: GCD of terms in a sequence equals GCD of their indices. The problem was to show that if ...
161 views

### 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, ...
80 views

### Contest problem involving primes and factorization

Prove that for any nonnegative integer $n$, the number $$5^{5^{n+1}}+ 5^{5^{n}}+1$$ is not prime. I want only some hints and the method to follow, but I don't need the full solution. Thanks.
13k views

### Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quantity'. The totality ...
35 views

### An unusual inequality

Problem: $(x_i)_{i=1}^n$ is a finite sequence of positive integers. Define $f\big(S\big)=\displaystyle \sum_{i\,\in\, S\,\subseteq\, [n]}x_i$, and suppose $f$ is injective. Prove that: ...
117 views

### Primes $p$ such that $p^2$ divides $x^2 + y^2 + 1$

Call a prime $p$ awesome if there exist positive integers $x$ and $y$ such that $p^2$ divides $x^2+y^2+1$. Observation: $2$ is not awesome, because $x^2+y^2+1\not\equiv 0$ (mod $4$). But $3$ is ...
57 views

29 views

### The product of $i$ consecutive natural numbers is divisible for $i!$ [duplicate]

There is a theorem that I've used it a few times, and never saw a demo of it, and when I tried, I could not, commenting with a teacher, it would not give me much attention and said it would use the ...
192 views

### Smallest positive integer divisible by and having digit sum equal to some 3-digit number.

Let $p,q,r$ be distinct digits among $1,2,4,6,8$, and consider the integer $pqr = 100p + 10q + r$. Let $N$ be the smallest positive integer that is divisible by $pqr$ and has digit sum equal to ...
71 views

### Recommendation for a good book on equation solving theory from the basics

I'm relearning calculus but I often find myself applying algebraic operations to equations mechanically without having a solid understanding of the side-effects of those operations. Such as extra or ...
103 views

### I do not know where to start this challenge, help please

If $x$, $y$ and $z$ are positive numbers such that $1\leq xy+yz+zx\leq3$ which is the set of values ​​of $xyz$? And $x + y + z$?Knowing that $x, y$ and $z$ $\in\mathbb{R^*}$
86 views

### Problem of Ages (Problema das Idades)

English: Somebody help me with this challenge? It's very confusing: Today, both me and my younger brother are between $10$ and $20$ years old. Also, our ages are expressed by prime numbers and the ...
200 views

### The Keys problem

Another challenge: A calculator has two special keys: A key transforms a number x in the number 2x. B key transforms a number x in the number 2x - 1. Is it true that if you start with any positive ...
### Solving for $(x,y): 2+\frac1{x+\frac1{y+\frac15}}=\frac{478}{221}$
Solving for $x,y\in\mathbb{N}$: $$2+\dfrac1{x+\dfrac1{y+\dfrac15}}=\frac{478}{221}$$ This doesn't make any sense; I made $y+\frac15=\frac{5y+1}5$, and so on, but it turns out to be a very ...