Let $m, n$ integers such that: $(n-1)^3+n^3+(n+1)^3=m^3$ Prove that 4 divides $n$

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

Let $f_n(x) = n + x^2$. Evaluate the product $gcd\{f_{2001}(2002), f_{2001}(2003)\} \times gcd\{f_{2011}(2012), f_{2011}(2013)\} \times gcd\{f_{2021}(2022), f_{2021}(2023)\}$, where $gcd\{x, y\}$ is the greatest common divisor of $x$ and $y$

There are positive integers $x, y$ such that $3x^2 + x = 4y^2 + y$, and $(x - y)$ is equal to (A): $2013$ (B): $2014$ (C): $2015$ (D): $2016$ (E): None of the above.

Prove that every integer can be written as sum of $5$ third powers of integers.

For each positive integer \( n \), list in increasing order all irreducible fractions in the interval \([0, 1]\) that have a positive denominator less than or equal to \( n \): \[ 0 = \frac{p_0}{q_0} < \frac{1}{n} = \frac{p_1}{q_1} < \cdots < \frac{1}{1} = \frac{p_{M(n)}}{q_{M(n)}}. \] Let \( k \) be a positive integer. We define, for each \( n \) such that \( M(n) \geq k - 1 \), \[ f_k(n) = \min \left\{ \sum_{s=0}^{k-1} q_{j+s} : 0 \leq j \leq M(n) - k + 1 \right\}. \] Determine, in function of \( k \), \[ \lim_{n \to \infty} \frac{f_k(n)}{n}. \] For example, if \( n = 4 \), the enumeration is \[ \frac{0}{1} < \frac{1}{4} < \frac{1}{3} < \frac{1}{2} < \frac{2}{3} < \frac{3}{4} < \frac{1}{1}, \] where \( p_0 = 0, p_1 = 1, p_2 = 1, p_3 = 1, p_4 = 2, p_5 = 3, p_6 = 1 \) and \( q_0 = 1, q_1 = 4, q_2 = 3, q_3 = 2, q_4 = 3, q_5 = 4, q_6 = 1 \). In this case, we have \( f_1(4) = 1, f_2(4) = 5, f_3(4) = 8, f_4(4) = 10, f_5(4) = 13, f_6(4) = 17 \), and \( f_7(4) = 18 \).

Find all positive integers $ (x,y)$ such that $ \frac{y^2x}{x\plus{}y}$ is a prime number

Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \] cannot be divided by $5$.

What digits should be placed instead of zeros in the third and fifth places in the number $3000003$ to obtain a number divisible by $13$?

$k$ is a positive integer. The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = k+1$, $a_{n+1} = a_n ^2 - ka_n + k$. Show that $a_m$ and $a_n$ are coprime (for $m \not = n$).

Solve the following system of equations in natural numbers $$\begin{cases} a^4+14ab+1=n^4 \\ b^4+14bc+1=m^4 \\ c^4+14ca+1=k^4 \end{cases}$$

Can the remainder of the division of a prime number $p> 30$ by $30$ be a composite?

Prove that there are infinitely many integers can't be written as $$\frac{p^a-p^b}{p^c-p^d}$$, with a,b,c,d are arbitrary integers and p is an arbitrary prime such that the fraction is an integer too.

Find all pairs of non-negative integers $(m,n)$ satisfying $\frac{n(n+2)}{4}=m^4+m^2-m+1$

For each positive integer $k$, define $a_k$ as the number obtained from adding $k$ zeroes and a $1$ to the right of $2024$, all written in base $10$. Determine wether there's a $k$ such that $a_k$ has at least $2024^{2024}$ distinct prime divisors.

[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$. [b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$. [b]p3.[/b] For a positive integer $n$, there is a sequence $(a_0, a_1, a_2,..., a_n)$ of real values such that $a_0 = 11$ and $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$ for every $k$ with $0 \le k \le n-1$. Find the maximum possible value of $n$. (Be careful that your answer isn’t off by one!) [b]p4.[/b] Persons $A$ and $B$ stand at point $P$ on line $\ell$. Point $Q$ lies at a distance of $10$ from point $P$ in the direction perpendicular to $\ell$. Both persons intially face towards $Q$. Person $A$ walks forward and to the left at an angle of $25^o$ with $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the right, and continues walking. Person $B$ walks forward and to the right at an angle of $55^o$ with line $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the left, and continues walking. Their paths cross at point $R$. Find the distance $PR$. [b]p5.[/b] Compute $$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$ [b]p6.[/b] There is a unique real value $A$ such that for all $x$ with $1 < x < 3$ and $x \ne 2$, $$\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.$$ Compute $A$. [b]p7.[/b] Nine poles of height $1, 2,..., 9$ are placed in a line in random order. A pole is called [i]dominant [/i] if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly $2$ dominant poles. [b]p8.[/b] $\tan (11x) = \tan (34^o)$ and $\tan (19x) = \tan (21^o)$. Compute $\tan (5x)$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be an odd prime and $S = \{1,2,3,\dots, p\}$ Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that $$B\cdot U(U(a)) - a \: \text{ is a multiple of} \: p \: \text{for all} \: a \in S$$ Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.

[i](The following problem is open in the sense that the answer to part (b) is not currently known.)[/i] [list=a] [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a positive real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [*] Let $n$ be a positive integer that is not a perfect square. Find all pairs $(a,b)$ of positive integers for which there exists a real number $r$, such that $$r^a+\sqrt{n} \ \ \text{and} \ \ r^b+\sqrt{n}$$ are both rational numbers. [/list]

For a positive integer $m$ we denote by $\tau (m)$ the number of its positive divisors, and by $\sigma (m)$ their sum. Determine all positive integers $n$ for which $n \sqrt{ \tau (n) }\le \sigma(n)$

Prove that for any integer $k$ ($k \ge 2$) there exists a power of $2$ that among its last $k$ digits, the nines constitute no less than half. For example, for $k = 2$ and $k = 3$ we have the powers $2^{12} = ... 96$ and $2^{53} = ... 992$. [hide=original wording] Probar que para cualquier k entero existe una potencia de 2 que entre sus ultimos k dıgitos, los nueves constituyen no menos de la mitad. [/hide]

Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

What is the least positive integer $m$ such that $m \cdot 2! \cdot 3! \cdot 4! \cdot 5! \cdots 16!$ is a perfect square? $\textbf{(A) }30\qquad\textbf{(B) }30030\qquad\textbf{(C) }70\qquad\textbf{(D) }1430\qquad\textbf{(E) }1001$

a) Determine all 4-tuples $(x_0,x_1,x_2,x_3)$ of pairwise distinct intergers such that each $x_k$ is coprime to $x_{k+1}$(indices reduces modulo 4) and the cyclic sum $\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1}$ is an interger. b)Show that there are infinitely many 5-tuples $(x_0,x_1,x_2,x_3,x_4)$ of pairwise distinct intergers such that each $x_k$ is coprime to $x_{k+1}$(indices reduces modulo 5) and the cyclic sum $\frac{x_0}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+\frac{x_4}{x_0}$ is an interger.

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$: $$n^2 + m^2 = 111...111$$ [b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy? [b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$. [b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].