Found problems: 276
Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials)
such that
[list]
[*] for any polynomials $p, q \in \mathbb{Z}[x]$, $\theta(p + q) = \theta(p) + \theta(q)$;
[*] for any polynomial $p \in \mathbb{Z}[x]$, $p$ has an integer root if and only if $\theta(p)$ does.
[/list]
[i]Carl Schildkraut[/i]
Suppose that $S\subseteq \mathbb Z$ has the following property: if $a,b\in S$, then $a+b\in S$. Further, we know that $S$ has at least one negative element and one positive element. Is the following statement true?
There exists an integer $d$ such that for every $x\in \mathbb Z$, $x\in S$ if and only if $d|x$.
[i]proposed by Mahyar Sefidgaran[/i]
Let $P$ be the set of all primes. Given any positive integer $n$, define $$\displaystyle f(n) = \max_{p \in P}v_p(n)$$ Prove that for any positive integer $k\ge 2$, there exists infinitely many positive integers $m$ such that \[ f(m+1) = f(m+2) = \cdots = f(m+k) \]
[i]Proposed by Ivan Chan Guan Yu[/i]
Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.)
[i]David Yang.[/i]
Let $\mathbb Z_{\ge 0}$ be the set of non-negative integers, and let $f:\mathbb Z_{\ge 0}\times \mathbb Z_{\ge 0} \to \mathbb Z_{\ge 0}$ be a bijection such that whenever $f(x_1,y_1) > f(x_2, y_2)$, we have $f(x_1+1, y_1) > f(x_2 + 1, y_2)$ and $f(x_1, y_1+1) > f(x_2, y_2+1)$.
Let $N$ be the number of pairs of integers $(x,y)$ with $0\le x,y<100$, such that $f(x,y)$ is odd. Find the smallest and largest possible values of $N$.
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]
Find the number of pairs $(m,n)$ of positive integers with $1\le m<n\le 30$ such that there exists a real number $x$ satisfying $$\sin(mx)+\sin(nx)=2.$$
Let $ABC$ be a scalene triangle and let $P$ and $Q$ be two distinct points in its interior. Suppose that the angle bisectors of $\angle PAQ,\,\angle PBQ,$ and $\angle PCQ$ are the altitudes of triangle $ABC$. Prove that the midpoint of $\overline{PQ}$ lies on the Euler line of $ABC$.
(The Euler line is the line through the circumcenter and orthocenter of a triangle.)
[i]Proposed by Holden Mui[/i]
Let $\mathbb{N}$ be the set of positive integers. Across all function $f:\mathbb{N}\to\mathbb{N}$ such that $$mn+1\text{ divides } f(m)f(n)+1$$ for all positive integers $m$ and $n$, determine all possible values of $f(101).$
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and
$(i) f(xy) + f(x)f(y) = f(x) + f(y)$
$(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $
for all $x,y \in \mathbb{Z}$, simultaneously.
$(a)$ Find the set of all possible values of the function $f$.
$(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find the set of all integers $n$ such that $f(n) \ne 0$.
Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits.
(Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)
$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$. Show that $a_1 = a_3$.
The set of positive integers is partitioned into $n$ disjoint infinite arithmetic progressions $S_1, S_2, \ldots, S_n$ with common differences $d_1, d_2, \ldots, d_n$, respectively. Prove that there exists exactly one index $1\leq i \leq n$ such that\[ \frac{1}{d_i}\prod_{j=1}^n d_j \in S_i.\]
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)\plus{}xQ(x^5)\plus{}x^2R(x^5)\equal{}(x^4\plus{}x^3\plus{}x^2\plus{}x\plus{}1)S(x),\] prove that $ x\minus{}1$ is a factor of $ P(x)$.
Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that
\[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.
Determine all positive integers $N$ for which there exists a strictly increasing sequence of positive integers $s_0<s_1<s_2<\cdots$ satisfying the following properties:
[list=disc]
[*]the sequence $s_1-s_0$, $s_2-s_1$, $s_3-s_2$, $\ldots$ is periodic; and
[*]$s_{s_n}-s_{s_{n-1}}\leq N<s_{1+s_n}-s_{s_{n-1}}$ for all positive integers $n$
[/list]
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{R}$ such that
\[
f(x)^2-f(y)^2=f(x+y)\cdot f(x-y),
\]
for all $x,y\in \mathbb{Q}$.
[i]Proposed by Portugal.[/i]
Let $n, \in \mathbb {N^*} , n\ge 3$
a) Prove that the polynomial $f (x)=\frac {X^{2^n-1}-1}{X-1}-X^n $ has a divisor of form $X^p +1$ where $p\in\mathbb {N^*} $
b) Show that for $n=7$ the polynomial $f (X) $ has three divisors with integer coefficients .
Let $x_0, x_1, \dots , x_{n_0-1}$ be integers, and let $d_1, d_2, \dots, d_k$ be positive integers with $n_0 = d_1 > d_2 > \cdots > d_k$ and $\gcd (d_1, d_2, \dots , d_k) = 1$. For every integer $n \ge n_0$, define
\[
x_n = \left\lfloor{\frac{x_{n-d_1} + x_{n-d_2} + \cdots + x_{n-d_k}}{k}}\right\rfloor.
\]
Show that the sequence $\{x_n\}$ is eventually constant.
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 $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that
$$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$
is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial.
[i]Proposed by Vlad Matei, Romania[/i]
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$
for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.
[i]Ankan Bhattacharya[/i]
For a finite nonempty set $A$ of positive integers, $A =\{a_1, a_2,\dots , a_n\}$, we say the calamitous complement of A is the set of all positive integers $k$ for which there do not exist nonnegative integers $w_1, w_2, \dots , w_n$ with $k = a_1w_1 + a_2w_2 +\dots + a_nw_n.$ The calamitous complement of $A$ is denoted $cc(A)$. For example, $cc(\{5, 6, 9\}) = \{1, 2, 3, 4, 7, 8, 13\}$.
Find all pairs of positive integers $a, b$ with $1 < a < b$ for which there exists a set $G$ satisfying all of the following properties:
1. $G$ is a set of at most three positive integers,
2. $cc(\{a, b\})$ and $cc(G)$ are both finite sets, and
3. $cc(G) = cc(\{a, b\})\cup \{m\}$ for some $m$ not in $cc(\{a, b\})$.
Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas.
[list](i) Prove that all the discs that he cuts have integral areas.
(ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.[/list]