Found problems: 288
The sum of several integers (not necessarily distinct) equals $1492$. Decide whether the sum of their seventh powers can equal (a) $1996$; (b) $1998$.
Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$
Also, $m,n$ are relatively prime positive integers.
Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them $45$ degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.
Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.
Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).
Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.
For positive integers $a$ and $b$, if the expression $\frac{a^2+b^2}{(a-b)^2}$ is an integer, prove that the expression $\frac{a^3+b^3}{(a-b)^3}$ is an integer as well.
Show that for $n > 1$ we can find a polynomial $P(a, b, c)$ with integer coefficients such that
$$P(x^{n},x^{n+1},x+x^{n+2})=x.$$
For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.
A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings.
Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]
$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$.
Find the positive real numbers $x,y$ for which $\frac{x+y}{2},\sqrt{xy},\frac{2xy}{x+y},\sqrt{\frac{x^2 +y^2}{2}}$ are integers whose sum is $66$.
What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots?
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$.
Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.
Let ${A}$ be a finite collection of squares in the coordinate plane such that the vertices of all squares that belong to ${A}$ are ${(m, n), (m + 1, n), (m, n + 1)}$, and ${(m + 1, n + 1)}$ for some integers ${m}$ and ${n}$. Show that there exists a subcollection ${B}$ of ${A}$ such that ${B}$ contains at least ${25 \% }$ of the squares in ${A}$, but no two of the squares in ${B}$ have a common vertex.
If the ratio $$\frac{17m+43n}{m-n}$$ is an integer where $m$ and $n$ positive integers, let's call $(m,n)$ is a special pair. How many numbers can be selected from $1,2,..., 2021$, any two of which do not form a special pair?
Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$
is integer for every integer $d\ge 0$
Find the smallest real number $x$ for which exist two non-congruent triangles, whose sides have integer lengths and the numerical value of the area of each triangle is $x$.
Solve the equation $x^2+y^4+1=6^z$ in the set of integers.
Determine the integral part of $A$, where $A =\frac{1}{672}+\frac{1}{673}+... +\frac{1}{2014}$
Let $f (n)$ be a function satisfying the following three conditions for all positive integers $n$:
(a) $f (n)$ is a positive integer,
(b) $f (n + 1) > f (n)$,
(c) $f ( f (n)) = 3n$.
Find $f (2001)$.
Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.
The number $50$ is written as a sum of several positive integers (not necessarily distinct) whose product is divisible by $100.$ What is the largest possible value of this product?
Prove that for every real number $N$ the equation
$$ x_{1}^{2}+x_{2}^{2} +x_{3}^{2} +x_{4}^{2} = x_1 x_2 x_3 +x_1 x_2 x_4 + x_1 x_3 x_4 +x_2 x_3 x_4$$
has an integer solution $(x_1 , x_2 , x_3 , x_4)$ for which $x_1, x_2 , x_3 $ and $x_4$ are all larger than $N.$