Algabra IMO

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51   
Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.

175   
Prove there exist infinite number of positive integer $a$ such that for any positive integer $n$, $n^4 + a$ is not a prime number.

2239   
Prove that $\frac{5^{125}-1}{5^{25}-1}$ is composite.

2325   
Find the maximal value of $m^2+n^2$ if $m$ and $n$ are integers between $1$ and $1981$ satisfying $(n^2-mn-m^2)^2=1$.

2678   
Show that $x^n + 5x^{n-1} + 3 = 0$ cannot be factorized into two non-constant polynomials with integer coefficients.

2695   
Let sequence $\{a_n\}$ satisfy $a_0=0, a_1=1$, and $a_n = 2a_{n-1}+a_{n-2}$. Show that $2^k\mid n$ if and only if $2^k\mid a_n$.

2698   
Let $\{a_n\}$ be a sequence defined as $a_n=\lfloor{n\sqrt{2}}\rfloor$ where $\lfloor{x}\rfloor$ indicates the largest integer not exceeding $x$. Show that this sequence has infinitely many square numbers.

2699   

Let sequence $g(n)$ satisfy $g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$ where $n\ge 1$. Show that if $n$ is a prime greater than 5, then $n\mid g(n)[g(n)+1]$.


3176   
If a sequence $\{a_n\}$ satisfies $a_1=1$ and $a_{n+1}=\frac{1}{16}\big(1+4a_n+\sqrt{1+24a_n}\big)$, find the general term of $a_n$.

3389   
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$. Proposed by Dorlir Ahmeti, Albania

3406   
Find all functions $f:\mathbb Z\rightarrow \mathbb Z$ such that, for all integers $a,b,c$ that satisfy $a+b+c=0$, the following equality holds: \[f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\] (Here $\mathbb{Z}$ denotes the set of integers.) Proposed by Liam Baker, South Africa

3632   
In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m − 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?

3844   
For each integer $a_0 >$ 1, define the sequence $a_0, a_1, a_2, \cdots$ by: $$ a_{n+1} = \left\{ \begin{array}{ll} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}\\ a_n + 3 & \text{otherwise} \end{array} \right. $$ For all $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.

3845   
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f (f(x)f(y)) + f(x + y) = f(xy)$$

3849   
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is 1. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0$, $a_1$, $\cdots$, an such that, for each $(x, y)$ in $S$, we have: $$a_0x^n + a_1x^{n-1}y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1$$

4096   

Find all numbers $n \ge 3$ for which there exists real numbers $a_1, a_2, ..., a_{n+2}$ satisfying $a_{n+1} = a_1, a_{n+2} = a_2$ and\[a_{i}a_{i+1} + 1 = a_{i+2}\]for $i = 1, 2, ..., n.$


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