Found problems: 288
Let $a, b$ be integers such that all the roots of the equation $(x^2+ax+20)(x^2+17x+b) = 0$ are negative integers. What is the smallest possible value of $a + b$ ?
Given a geometric progression whose denominator $q$ is an integer not equal to $0$ or $-1$, prove that the sum of two or more terms in this progression cannot equal any other term in it.
Find integers $a_1,a_2,\cdots,a_{18}$, s.t. $a_1=1,a_2=2,a_{18}=2019$, and for all $3\le k\le 18$, there exists $1\le i<j<k$ with $a_k=a_i+a_j$.
Let n be a positive integer. Prove the following statement:
”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”
Show that an integer $p > 3$ is a prime if and only if for every two nonzero integers $a,b$ exactly one of the numbers
$N_1 = a+b-6ab+\frac{p-1}{6}$ , $N_2 = a+b+6ab+\frac{p-1}{6}$ is a nonzero integer.
Let $x, y$ be distinct positive integers. Show that the number
$$\frac{(x + y)^2}{x^3 + xy^2- x^2y -y^3}$$
is not an integer.
How many different (non similar) triangles are there whose angles have an integer number of degrees?
Let $m \ge 2$ and $n \ge 1$ be integers and $A=(a_{ij})$ a square matrix of order $n$ with integer entries. Prove that for any permutation $\sigma \in S_n$ there is a function $\varepsilon : \{1,2,\ldots,n\} \to \{0,1\}$ such that replacing the entries $a_{\sigma(1)1},$ $a_{\sigma(2)2}, $ $\ldots,$ $a_{\sigma(n)n}$ of $A$ respectively by $$a_{\sigma(1)1}+\varepsilon(1), ~a_{\sigma(2)2}+\varepsilon(2), ~\ldots, ~a_{\sigma(n)n}+\varepsilon(n),$$ the determinant of the matrix $A_{\varepsilon}$ thus obtained is not divisible by $m.$
Consider the integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$.
[b](a)[/b] Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$, where $j$ is not a function of $x$, is $Z(j)=e^{j^{2}/4a} Z(0)$.
[b](b)[/b] Show that
$$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n},$$
where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times\cdots\times3\times 1$.
[b](c)[/b] What is the number of ways to form $n$ pairs from $2n$ distinct objects? Interpret the previous part of the problem in term of this answer.
Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.
The product of three integers is $60$. What is the least possible positive sum of the three integers?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$
Given points $O, A_1, A_2, ..., A_n$ on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers.
(A.Glazyrin)
Is $\frac{19^{92} - 91^{29}}{90}$ an integer?
How many integers $1\le x \le 2021$ make the value of the expression $$\frac{2x^3 - 6x^2 - 3x -20}{5(x - 4)}$$ an integer?
Let \(P(x)=x^{2}+ax+b\) be a quadratic polynomial where \(a\) is real and \(b \neq 2\), is rational. Suppose \(P(0)^{2},P(1)^{2},P(2)^{2}\) are integers, prove that \(a\) and \(b\) are integers.
Let $n \geq 2$ be an integer. Prove that if $$\frac{n^2+4^n+7^n}{n}$$ is an integer, then it is divisible by 11.
Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point $(a, b)$ is one greater than the average of the numbers written on points $ (a+1 , b-1) $ and $ (a-1,b+1)$ . Which numbers could he write on point $(121, 212)$?
Note: The elements of the first quadrant are points where both of the coordinates are non- negative.
Let $n \ge 2$ be an integer and $M= \{1,2,\ldots,n\}.$ For each $k \in \{1,2,\ldots,n-1\}$ we define $$x_k= \frac{1}{n+1} \sum_{\substack{A \subset M \\ |A|=k}} (\min A + \max A).$$
Prove that the numbers $x_k$ are integers and not all of them are divisible by $4.$
[hide=Notations]$|A|$ is the cardinal of $A$
$\min A$ is the smallest element in $A$
$\max A$ is the largest element in $A$[/hide]
Let $a,b,$ be positive integers and $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is never an integer
Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.
Find all three digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! b! c!$
Given positive integers $a, b,$ and $c$ with $a + b + c = 20$.
Determine the number of possible integer values for $\frac{a + b}{c}.$
Is it true that every integer is a sum of finite number of cubes of distinct integers?
Determine all real numbers $x$, such that $x^n+x^{-n}$ is an integer for all integers $n$.
How many integers are there in $(b,2008b]$, where $b$ ($b > 0$) is given.