Found problems: 276
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?
[i]Proposed by Andre Arslan[/i]
Prove that for two non-zero polynomials $ f(x,y),g(x,y)$ with real coefficients the system:
\[ \left\{\begin{array}{c}f(x,y)\equal{}0\\ g(x,y)\equal{}0\end{array}\right.\]
has finitely many solutions in $ \mathbb C^{2}$ if and only if $ f(x,y)$ and $ g(x,y)$ are coprime.
Let $n$ be an arbitrary positive integer.
(a) For every positive integers $a$ and $b$, show that $gcd(n^a + 1, n^b + 1) \le n^{gcd(a,b)} + 1$.
(b) Show that there exist infinitely many composite pairs ($a, b)$, such that each of them is not a multiply of the other number and equality holds in (a).
Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$.
Is it possible to partition the set of integers into three disjoint sets so that for every positive integer $n$ the numbers $n$, $n-50$ and $n+1987$ belong to different sets?
Let $a,m$ be positive integers such that $Ord_m (a)$ is odd and for any integers $x,y$ so that
1.$xy \equiv a \pmod m$
2.$Ord_m(x) \le Ord_m(a)$
3.$Ord_m(y) \le Ord_m(a)$
We have either $Ord_m(x)|Ord_m(a)$ or $Ord_m(y)|Ord_m(a)$.prove that $Ord_m(a)$ contains at most one prime factor.
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
A [i]lattice point[/i] in the plane is a point with integer coordinates. Let $T$ be a triangle in the plane whose vertice are lattice points, but with no other lattice points on its sides. Furthermore, suppose $T$ contains exactly four lattice points in its interior. Prove that these four points lie on a straight line.
Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$.
A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$.
A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$.
Now put a pawn on $(0, 0)$. You can make a ( nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B.
Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.
Let $P(X),Q(X)$ be monic irreducible polynomials with rational coefficients. suppose that $P(X)$ and $Q(X)$ have roots $\alpha$ and $\beta$ respectively, such that $\alpha + \beta $ is rational. Prove that $P(X)^2-Q(X)^2$ has a rational root.
[i]Bogdan Enescu[/i]
For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with
\[\omega(n)\ge\omega(P(n)).\]
Greece (Minos Margaritis - Iasonas Prodromidis)
Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers.
One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins).
In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice.
[i]Nikolai Beluhov[/i]
Assume that $k$ and $n$ are two positive integers. Prove that there exist positive integers $m_1 , \dots , m_k$ such that \[1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).\]
[i]Proposed by Japan[/i]
Let $m,n$ be integers greater than 1,$m \geq n$,$a_1,a_2,\dots,a_n$ are $n$ distinct numbers not exceed $m$,which are relatively primitive.Show that for any real $x$,there exists $i$ for which $||a_ix|| \geq \frac{2}{m(m+1)} ||x||$,where $||x||$ denotes the distance between $x$ and the nearest integer to $x$ .
Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$
[i]Proposed by Carl Schildkraut[/i]
Let $A$ be a set of positive integers satisfying the following properties:
(i) if $m$ and $n$ belong to $A$, then $m+n$ belong to $A$;
(ii) there is no prime number that divides all elements of $A$.
(a) Suppose $n_1$ and $n_2$ are two integers belonging to $A$ such that $n_2-n_1 >1$. Show that you can find two integers $m_1$ and $m_2$ in $A$ such that $0< m_2-m_1 < n_2-n_1$
(b) Hence show that there are two consecutive integers belonging to $A$.
(c) Let $n_0$ and $n_0+1$ be two consecutive integers belonging to $A$. Show that if $n\geq n_0^2$ then $n$ belongs to $A$.
Show that there are infinitely many tuples $(a,b,c,d)$ of natural numbers such that $a^3 + b^4 + c^5 = d^7$.
Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials.
[b]Note: [/b] The [i]degree[/i] of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are [i]proportional[/i] if one of them is the other times a complex constant.
[i]Proposed by Navid Safaie[/i]
The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$.
[hide="comment"]
Hi guys ,
Here is a nice problem:
Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$
Here are some futher question proposed by me :Prove or disprove that :
1) $gcd(n,a_n)=1$
2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$
Thanks kiu si u
[i]Edited by Orl.[/i]
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In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements.
[i]Proposed by Jorge Tipe, Peru[/i]
Bamicin is initially at $(20, 20)$ in a cartesian plane. Every minute, if Bamicin is at point $P$, Bamicin can move to a lattice point exactly $37$ units from $P$. Determine all lattice points Bamicin can visit.
A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$.
Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$.
a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$.
b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.
Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that $f(m) - f(n) = f(m-n)10^n \forall m>n \in \mathbb{N}$. Additionally, gcd$(f(k),f(k+1)) = 1 \forall k \in \mathbb{N}$. Show that if $a,b$ are coprime natural numbers, that is, gcd$(a,b) = 1$ then $f(a),f(b)$ are also coprime.
Let a, b, and c be nonzero integers. Show that there exists an integer k such that
$$gcd\left(a+kb, c\right) = gcd\left(a, b, c\right)$$