Found problems: 700
Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?
Prove that for any positive integers $p,q$ such that $\sqrt{11}>\frac{p}{q}$, the following inequality holds:
\[\sqrt{11}-\frac{p}{q}>\frac{1}{2pq}.\]
Given an arbitrary prime $p>2011$. Prove that there exist positive integers $a, b, c$ not all divisible by $p$ such that for all positive integers $n$ that $p\mid n^4- 2n^2+ 9$, we have $p\mid 24an^2 + 5bn + 2011c$.
Let $p$ be some prime number.
a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$.
b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.
[b]8.1[/b] On the median drawn from the vertex of the triangle to the base, point $A$ is taken. The sum of the distances from $A$ to the sides of the triangle is equal to $s$. Find the distances from $A$ to the sides if the lengths of the sides are equal to $x$ and $y$.
[b]8.2[/b] Fraction $0, abc...$ is composed according to the following rule: $a$ and $c$ are arbitrary digits, and each next digit is equal to the remainder of the sum of the previous two digits when divided by $10$. Prove that this fraction is purely periodic.
[b]8.3[/b] Two convex polygons with $m$ and $n$ sides are drawn on the plane ($m>n$). What is the greatest possible number of parts, they can break the plane?
[b]8.4 [/b]The sum of three integers that are perfect squares is divisible by $9$. Prove that among them, there are two numbers whose difference is divisible by $9$.
[b]8.5 / 9.5[/b] Given $k+2$ integers. Prove that among them there are two integers such that either their sum or their difference is divisible by $2k$.
[b]8.6[/b] A right angle rotates around its vertex. Find the locus of the midpoints of the segments connecting the intersection points sides of an angle and a given circle.
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].
The last digit of the number $x^2 +xy+y^2$ is zero (where $x$ and $y$ are positive integers). Prove that two last digits of this numbers are zeros.
[b]7.[/b] Prove that for any odd prime number $p$, the polynomial
$2(1+x^{ \frac{p+1}{2} }+(1-x)^{\frac {p+1}{2}})$
is congruent mod $p$ to the square of a polynomial with integer coefficients. [b](N. 21)[/b]
*This problem was proposed by P. Erdõs in the American Mathematical Monthly 53 (1946), p. 594
Prove that $n^3-n-3$ is not a perfect square for any integer $n$.
[i]Calvin Deng.[/i]
Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.
$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
Let $C_k=\frac{1}{k+1}\binom{2k}{k}$ denote the $k^{\text{th}}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set
\[\left\{\sum_{k=1}^{p-1}C_kn^k\,\middle\vert\, n\in\{1,2,\ldots,p-1\}\right\}\]
are divisible by $p$.
[i]Tristan Shin[/i]
Let ${u_n}$ be the Fibonacci sequence, i.e., $u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}$ for $n>1$. Prove that there exist infinitely many prime numbers $p$ that divide $u_{p-1}$.
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.
Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.
Let $p=4k+1$ be a prime and let $|x| \leq \frac{p-1}{2}$ such that $\binom{2k}{k}\equiv x \pmod p$. Show that $|x| \leq 2\sqrt{p}$.
Let p(x) be the polynomial $x^3 + 14x^2 - 2x + 1$. Let $p^n(x)$ denote $p(p^(n-1)(x))$. Show that there is an integer N such that $p^N(x) - x$ is divisible by 101 for all integers x.
Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.
Call a positive integer $k$ $\textit{pretty}$ if for every positive integer $a$, there exists an integer $n$ such that $n^2+n+k$ is divisible by $2^a$ but not $2^{a+1}$. Find the remainder when the $2021$st pretty number is divided by $1000$.
[i]Proposed by i3435[/i]
Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other.
[i]Linus Hamilton.[/i]
For prime number $q$ the polynomial $P(x)$ with integer coefficients is said to be factorable if there exist non-constant polynomials $f_q,g_q$ with integer coefficients such that all of the coefficients of the polynomial $Q(x)=P(x)-f_q(x)g_q(x)$ are dividable by $q$ ; and we write:
$$P(x)\equiv f_q(x)g_q(x)\pmod{q}$$
For example the polynomials $2x^3+2,x^2+1,x^3+1$ can be factored modulo $2,3,p$ in the following way:
$$\left\{\begin{array}{lll}
X^2+1\equiv (x+1)(-x+1)\pmod{2}\\
2x^3+2\equiv (2x-1)^3\pmod{3}\\
X^3+1\equiv (x+1)(x^2-x+1)
\end{array}\right.$$
Also the polynomial $x^2-2$ is not factorable modulo $p=8k\pm 3$.
a) Find all prime numbers $p$ such that the polynomial $P(x)$ is factorable modulo $p$:
$$P(x)=x^4-2x^3+3x^2-2x-5$$
b) Does there exist irreducible polynomial $P(x)$ in $\mathbb{Z}[x]$ with integer coefficients such that for each prime number $p$ , it is factorable modulo $p$?
Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list]
Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.
For any positive integer $n$, define the subset $S_n$ of natural numbers as follow
$$ S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$
Find all positive integers $n$ such that there exists an element of $S_n$ which [u]doesn't belong[/u] to any of the sets $S_1, S_2,\dots,S_{n-1}$.
[i]Proposed by Yahya Motevassel[/i]
Find all non-negative integers $ m$ and $ n$ that satisfy the equation:
\[ 107^{56}(m^2\minus{}1)\minus{}2m\plus{}5\equal{}3\binom{113^{114}}{n}\]
(If $ n$ and $ r$ are non-negative integers satisfying $ r\le n$, then $ \binom{n}{r}\equal{}\frac{n}{r!(n\minus{}r)!}$ and $ \binom{n}{r}\equal{}0$ if $ r>n$.)