Let $N=4^KL$ where $L\equiv\ 7\pmod 8$. Prove that $N$ cannot be written as a sum of 3 squares

You are traveling in a foreign country whose currency consists of five different-looking kinds of coins. You have several of each coin in your pocket. You remember that the coins are worth $1, 2, 5, 10$, and $20$ florins, but you have no idea which coin is which and you don’t speak the local language. You find a vending machine where a single candy can be bought for $1$ florin: you insert any kind of coin, and receive $1$ candy plus any change owed. You can only buy one candy at a time, but you can buy as many as you want, one after the other. What is the least number of candies that you must buy to ensure that you can determine the values of all the coins? Prove that your answer is correct.

Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.

Are there natural number(s) $n$, such that $3^n+1$ has a divisor in the form $24k+20$

Find the sum of all positive integers whose largest proper divisor is $55$. (A proper divisor of $n$ is a divisor that is strictly less than $n$.)

[u]Round 1[/u] [b]p1.[/b] In a chemical lab there are three vials: one that can hold $1$ oz of fluid, another that can hold $2$ oz, and a third that can hold $3$ oz. The first is filled with grape juice, the second with sulfuric acid, and the third with water. There are also $3$ empty vials in the cupboard, also of sizes $1$ oz, $2$ oz, and $3$ oz. In order to save the world with grape-flavored acid, James Bond must make three full bottles, one of each size, filled with a mixture of all three liquids so that each bottle has the same ratio of juice to acid to water. How can he do this, considering he was silly enough not to bring any equipment? [b]p2.[/b] Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the $12$th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p3.[/b] Aquaman has a barrel divided up into six sections, and he has placed a red herring in each. Aquaman can command any fish of his choice to either ‘jump counterclockwise to the next sector’ or ‘jump clockwise to the next sector.’ Using a sequence of exactly $30$ of these commands, can he relocate all the red herrings to one sector? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/0/f/956f64e346bae82dee5cbd1326b0d1789100f3.png[/img] [b]p4.[/b] Is it possible to place $13$ integers around a circle so that the sum of any $3$ adjacent numbers is exactly $13$? [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2[/u] [b]p6.[/b] Eight students participated in a math competition. There were eight problems to solve. Each problem was solved by exactly five people. Show that there are two students who solved all eight problems between them. [b]p7.[/b] There are $3n$ checkers of three different colors: $n$ red, $n$ green and $n$ blue. They were used to randomly fill a board with $3$ rows and $n$ columns so that each square of the board has one checker on it. Prove that it is possible to reshuffle the checkers within each row so that in each column there are checkers of all three colors. Moving checkers to a different row is not allowed. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, ..., 30\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1.$

Given a positive integer $n$, determine how many permutations $\sigma$ of the set $\{1, 2, \ldots , 2022n\}$ have the following property: for each $i \in \{1, 2, \ldots , 2021n + 1\}$, the number $$\sigma(i) + \sigma(i + 1) + \cdots + \sigma(i + n - 1)$$ is a multiple of $n$.

For any positive integers $a,b,c,n$, we define $$D_n(a,b,c)=\mathrm{gcd}(a+b+c,a^2+b^2+c^2,a^n+b^n+c^n).$$ 1. Prove that if $n$ is a positive integer not divisible by $3$, then for any positive integer $k$, there exist three integers $a,b,c$ such that $\mathrm{gcd}(a,b,c)=1$, and $D_n(a,b,c)>k$. 2. For any positive integer $n$ divisible by $3$, find all values of $D_n(a,b,c)$, where $a,b,c$ are three positive integers such that $\mathrm{gcd}(a,b,c)=1$.

Let $n$ be a given integer greater than two, and let $S = \{1, 2,...,n\}$. Suppose the function $f : S^k \to S$ has the property that $f(a) \ne f(b)$ for every pair $a$ and $b$ of elements in $S^k$ with $a$ and $b$ differ in all components. Prove that $f$ is a function of one of its elements.

Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that: $$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$ holds for all $p,q\in\mathbb{P}$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

Let $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$ be the finite field with $13$ elements (with sum and product modulus $13$). Find how many matrix $A$ of size $5$ x $5$ with entries in $\mathbb{F}_{13}$ exist such that $$A^5 = I$$ where $I$ is the identity matrix of order $5$

Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers

Let $n\ge 3$, $d$ be positive integers. For an integer $x$, denote $r(x)$ be the remainder of $x$ when divided by $n$ such that $0\le r(x)\le n-1$. Let $c$ be a positive integer with $1<c<n$ and $\gcd(c,n)=1$, and suppose $a_1, \cdots, a_d$ are positive integers with $a_1+\cdots+a_d\le n-1$. \\ (a) Prove that if $n<2d$, then $\displaystyle\sum_{i=1}^d r(ca_i)\ge n.$ \\ (b) For each $n$, find the smallest $d$ such that $\displaystyle\sum_{i=1}^d r(ca_i)\ge n$ always holds. [i]Proposed by Yeoh Zi Song & Anzo Teh Zhao Yang[/i]

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number.

Let $p$ be the third-smallest prime number greater than $5$ such that: $\bullet$ $2p + 1$ is prime, and $\bullet$ $5^p \not\equiv 1$ (mod $2p + 1$). Find $p$.

Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.

Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$.

For any positive integer n, let $\varphi(n)$ be the number of integers in the set $\{1, 2, \ldots , n\}$ whose greatest common divisor with $n$ is 1. Determine the maximum value of $\frac{n}{\varphi(n)}$ for $n$ in the set $\{2, \ldots, 1000\}$ and all values of $n$ for which this maximum is attained.

Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying $$3a + 5b \equiv 19 \,\,\, (mod \,\,\, n + 1)$$ $$4a + 2b \equiv 25 \,\,\, (mod \,\,\, n + 1)$$ Find $ 2a + 6b$.

Suppose that $m$ and $n$ are relatively prime positive integers with $A = \tfrac mn$, where \[ A = \frac{2+4+6+\dots+2014}{1+3+5+\dots+2013} - \frac{1+3+5+\dots+2013}{2+4+6+\dots+2014}. \] Find $m$. In other words, find the numerator of $A$ when $A$ is written as a fraction in simplest form. [i]Proposed by Evan Chen[/i]

Find all natural numbers whose decimal notation consists of different digits of the same parity and which are perfect squares.