Found problems: 288
Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$.
(i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices.
(ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.
Show that $1 + 1/2 + 1/3 + ... + 1/n$ is not an integer for $n > 1$.
Find all integer solutions to the equation
$$x^3+2y^3+4z^3+8xyz=0$$
Find all integers $m$ such that $\frac{2m^2+7m-9}{m^2+m+1}$ is integer
Let $p, q$ be two distinct primes. Prove that there are positive integers $a, b$ such that the arithmetic mean of all positive divisors of the number $n = p^aq^b$ is an integer.
The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.
Kobar and Borah are playing on a whiteboard with the following rules: They start with two distinct positive integers on the board. On each step, beginning with Kobar, each player takes turns changing the numbers on the board, either from $P$ and $Q$ to $2P-Q$ and $2Q-P$, or from $P$ and $Q$ to $5P-4Q$ and $5Q-4P$. The game ends if a player writes an integer that is not positive. That player is declared to lose, and the opponent is declared the winner.
At the beginning of the game, the two numbers on the board are $2024$ and $A$. If it is known that Kobar does not lose on his first move, determine the largest possible value of $A$ so that Borah can win this game.
Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where
a) $N = 19$,
b) $N = 20$ ?
Mikhail Malkin
Find all positive integers that can be written in the following way $\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}$ .
Also, $a,b, c$ are positive integers that are pairwise relatively prime.
Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5 $.
In how many ways can the number $2000$ be written as a sum of three positive, not necessarily different integers? (Sums like $1 + 2 + 3$ and $3 + 1 + 2$ etc. are the same.)
In a triangle $ \vartriangle ABC $ with sides integer numbers, it is known that the radius of the circumcircle circumscribed to $ \vartriangle ABC $ measures $ \dfrac {65} {8} $ centimeters and the area is $84$ cm². Determine the lengths of the sides of the triangle.
Specify at least one right triangle $ABC$ with integer sides, inside which you can specify a point $M$ such that the lengths of the segments $MA, MB, MC$ are integers. Are there many such triangles, none of which are are similar?
Determine all nonnegative integers $x, y$ satisfying the equation
\begin{align*}
\sqrt{xy}=\sqrt{x+y}+\sqrt{x}+\sqrt{y}.
\end{align*}
The function is given $f(x) = \frac{2x^3 -6x^2 + 13x + 10}{2x^2 - 9x}$.
Determine all positive integers $x$ for which $f(x)$ is an integer
Let $a_1, ... ,a_5$ be real numbers such that at least $N$ of the sums $a_i+a_j$ ($i < j$) are integers. Find the greatest value of $N$ for which it is possible that not all of the sums $a_i+a_j$ are integers.
Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.
Four odd positive intgers $a, b, c, d (a\leq b \leq c\leq d)$ are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as $1$. Find all $(a, b, c, d)$ that satisfies this.
Sequence of real numbers $a_1, a_2, . . . , a_{2000}$ is such that for any natural number $n$, $1\le n \le 2000$, the equality $$a^3_1+ a^3_2+... + a^3_n = (a_1 + a_2 +...+ a_n)^2.$$ Prove that all terms of this sequence are integers.
Let $C$ be the circle with center $O(0,0)$ and radius $1$, and $A(1,0), B(0,1)$ be points on the circle. Distinct points $A_1,A_2, ....,A_{n-1}$ on $C$ divide the smaller arc $AB$ into $n$ equal parts ($n \ge 2$). If $P_i$ is the orthogonal projection of $A_i$ on $OA$ ($i =1, ... ,n-1$), find all values of $n$ such that $P_1A^{2p}_1 +P_2A^{2p}_2 +...+P_{n-1}A^{2p}_{n-1}$ is an integer for every positive integer $p$.
Suppose that $p,q$ are prime numbers such that $\sqrt{p^2 +7pq+q^2}+\sqrt{p^2 +14pq+q^2}$ is an integer.
Show that $p = q$.
Determine all positive integers $n$ such that $\frac{2^n}{n^2}$ is an integer.
All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is:
(a) an integer;
(b) the square of an integer.
(A Kanel)
Let $a_1,a_2,\dots,a_{22}$ be positive integers with sum $59$.
Prove the inequality
\[\frac{a_1}{a_1+1}+\frac{a_2}{a_2+1}+\dots+\frac{a_{22}}{a_{22}+1}<16.\]
Find the smallest positive integer $n$ with the following property: if we write down all positive integers from $1$ to $10^n$ and add together the reciprocals of every non-zero digit written down, we obtain an integer.