Let $ n$ be a positive integer. Show that \[ \left(\sqrt{2} \plus{} 1 \right)^n \equal{} \sqrt{m} \plus{} \sqrt{m\minus{}1}\] for some positive integer $ m.$

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.

given a 4-digit number $(abcd)_{10}$ such that both$(abcd)_{10}$and$(dcba)_{10}$ are multiples of $7$,having the same remainder modulo $37$.find $a,b,c,d$.

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

We define [i]similar numbers[/i] as positive integers that have exactly the same digits (but possibly in another order). For example, $1241$, $2114$ and $4211$ are similar numbers, but $1424$ is not similar to the other three. Determine whether there exist three similar numbers, each with $300$ digits (all digits being non-zero), such that the sum of two of them equals the third. If the answer is yes, provide an example; if not, justify why it is impossible.

Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.

Find the smallest positive integer $n$ such that the number $2^n + 2^8 + 2^{11}$ is a perfect square. (A): $8$, (B): $9$, (C): $11$, (D): $12$, (E) None of the above.

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

Find the sum of the greatest common factor and the least common multiple of $12$ and $18$.

Let $p$ be a prime. Prove that any complete graph with $1000p$ vertices, whose edges are labelled with integers, has a cycle whose sum of labels is divisible by $p$.

Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$). Then Arnaldo tells the number of divisors of $ab$. Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.

[u]Round 5[/u] [b]p13.[/b] The expression $\sqrt2 \times \sqrt[3]{3} \times \sqrt[6]{6}$ can be expressed as a single radical in the form $\sqrt[n]{m}$, where $m$ and $n$ are integers, and $n$ is as small as possible. What is the value of $m + n$? [b]p14.[/b] Bertie Bott also produces Bertie Bott’s Every Flavor Pez. Each package contains $6$ peppermint-, $2$ kumquat-, $3$ chili pepper-, and $5$ garlic-flavored candies in a random order. Harold opens a package and slips it into his Dumbledore-shaped Pez dispenser. What is the probability that of the first four candies, at least three are garlic-flavored? [b]p15.[/b] Quadrilateral $ABCD$ with $AB = BC = 1$ and $CD = DA = 2$ is circumscribed around and inscribed in two different circles. What is the area of the region between these circles? [u] Round 6[/u] [b]p16.[/b] Find all values of x that satisfy $\sqrt[3]{x^7} + \sqrt[3]{x^4} = \sqrt[3]{x}$. [b]p17.[/b] An octagon has vertices at $(2, 1)$, $(1, 2)$, $(-1, 2)$, $(-2, 1)$, $(-2, -1)$, $(-1, -2)$, $(1, -2)$, and $(2, -1)$. What is the minimum total area that must be cut off of the octagon so that the remaining figure is a regular octagon? [b]p18.[/b] Ron writes a $4$ digit number with no zeros. He tells Ronny that when he sums up all the two-digit numbers that are made by taking 2 consecutive digits of the number, he gets 99. He also reveals that his number is divisible by 8. What is the smallest possible number Ron could have written? [u]Round 7[/u] [b]p19.[/b] In a certain summer school, 30 kids enjoy geometry, 40 kids enjoy number theory, and 50 kids enjoy algebra. Also, the number of kids who only enjoy geometry is equal to the number of kids who only enjoy number theory and also equal to the number of kids who only enjoy algebra. What is the difference between the maximum and minimum possible numbers of kids who only enjoy geometry and algebra? [b]p20.[/b] A mouse is trying to run from the origin to a piece of cheese, located at $(4, 6)$, by traveling the shortest path possible along the lattice grid. However, on a lattice point within the region $\{0 \le x \le 4, 0 \le y \le 6$, $(x, y) \ne (0, 0),(4, 6)\}$ lies a rock through which the mouse cannot travel. The number of paths from which the mouse can choose depends on where the rock is placed. What is the difference between the maximum possible number of paths and the minimum possible number of paths available to the mouse? [b]p21.[/b] The nine points $(x, y)$ with $x, y \in \{-1, 0, 1\}$ are connected with horizontal and vertical segments to their nearest neighbors. Vikas starts at $(0, 1)$ and must travel to $(1, 0)$, $(0, -1)$, and $(-1, 0)$ in any order before returning to $(0, 1)$. However, he cannot travel to the origin $4$ times. If he wishes to travel the least distance possible throughout his journey, then how many possible paths can he take? [u]Round 8[/u] [b]p22.[/b] Let $g(x) = x^3 - x^2- 5x + 2$. If a, b, and c are the roots of g(x), then find the value of $((a + b)(b + c)(c + a))^3$. [b]p23.[/b] A regular octahedron composed of equilateral triangles of side length $1$ is contained within a larger tetrahedron such that the four faces of the tetrahedron coincide with four of the octahedron’s faces, none of which share an edge. What is the ratio of the volume of the octahedron to the volume of the tetrahedron? [b]p24.[/b] You are the lone soul at the south-west corner of a square within Elysium. Every turn, you have a $\frac13$ chance of remaining at your corner and a $\frac13$ chance of moving to each of the two closest corners. What is the probability that after four turns, you will have visited every corner at least once? PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of ordered triples of positive integers $(a,b,c),$ where $1 \leq a,b,c \leq 10,$ with the property that $\gcd(a,b), \gcd(a,c),$ and $\gcd(b,c)$ are all pairwise relatively prime. [i]Proposed by Kyle Lee[/i]

Find all numbers $x \in \mathbb Z$ for which the number \[x^4 + x^3 + x^2 + x + 1\] is a perfect square.

Let $a_0, a_1, \ldots, a_{10}$ be integers such that, for each $i \in \{0,1,\ldots,2047\}$, there exists a subset $S \subseteq \{0,1,\ldots,10\}$ with \[ \sum_{j \in S} a_j \equiv i \pmod{2048}. \] Show that for each $i \in \{0,1,\ldots,10\}$, there is exactly one $j \in \{0,1,\ldots,10\}$ such that $a_j$ is divisible by $2^i$ but not by $2^{i+1}$. Note: $\sum_{j \in S} a_j$ is the summation notation, for instance, $\sum_{j \in \{2,5\}} a_j = a_2 + a_5$, while for the empty set $\varnothing$, one defines $\sum_{j \in \varnothing} a_j = 0$.

Determine all positive integers in the form $a<b<c<d$ with the property that each of them divides the sum of the other three.

Find all functions $f : \{1, 2, ... , n,...\} \to Z$ with the following properties (i) if $a, b$ are positive integers and $a | b$, then $f(a) \ge f(b)$; (ii) if $a, b$ are positive integers then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.

Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N} $ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ . (Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)

Let $p$ be a prime and let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with integer coefficients such that $0 < a, b, c \le p$. Suppose $f(x)$ is divisible by $p$ whenever $x$ is a positive integer. Find all possible values of $a + b + c$.

Let $\pi(n)$ denote the number of primes less than or equal to $n$. A subset of $S=\{1,2,\ldots, n\}$ is called [i]primitive[/i] if there are no two elements in it with one of them dividing the other. Prove that for $n\geq 5$ and $1\leq k\leq \pi(n)/2,$ the number of primitive subsets of $S$ with $k+1$ elements is greater or equal to the number of primitive subsets of $S$ with $k$ elements. [i]Proposed by Cs. Sándor, Budapest[/i]

Determine all integer solutions to the equation $x^3 + y^3 + 2015 = 0$.

[b]p1.[/b] Let $a(1), a(2), ..., a(n),...$ be an increasing sequence of positive integers satisfying $a(a(n)) = 3n$ for every positive integer $n$. Compute $a(2019)$. [b]p2.[/b] Consider the function $f(12x - 7) = 18x^3 - 5x + 1$. Then, $f(x)$ can be expressed as $f(x) = ax^3 + bx^2 + cx + d$, for some real numbers $a, b, c$ and $d$. Find the value of $(a + c)(b + d)$. [b]p3.[/b] Let $a, b$ be real numbers such that $\sqrt{5 + 2\sqrt6} = \sqrt{a} +\sqrt{b}$. Find the largest value of the quantity $$X = \dfrac{1}{a +\dfrac{1}{b+ \dfrac{1}{a+...}}}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that a)$\sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0$ (mod $2017^2$ ) b)$\sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )$ (mod $2017^2$ )

Let $r$ and $s$ be real numbers in the interval $[1, \infty)$ such that for all positive integers $a$ and $b$ with $a \mid b \implies \left\lfloor ar \right\rfloor$ divides $\left\lfloor bs \right\rfloor$. a) Prove that $\frac{s}{r}$ is a natural number. b) Show that both $r$ and $s$ are natural numbers. Here, $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x$.

$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.