Let $p\in \mathbb P,p>3$. Calcute: a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $ p\equiv 1 \mod 4$ b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$

Prove that for all $n \in N$, $x^2 + y^2 = z^n$ has solutions with $x,y,z \in N$.

Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day. Given that the low temperatures in December $3$ and January $31$ were $5^\circ \text C$ and $2^\circ \text C$ respectively, find the low temperature in December $25$.

Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.

Let $a$ and $b$ be different positive rational numbers such that the there exist an infinity of positive integers $n$ for which $a^n - b^n$ is an integer. Prove that $a$ and $b$ are also integers.

Let $n$ be a positive integer. Suppose there are exactly $M$ squarefree integers $k$ such that $\left\lfloor\frac nk\right\rfloor$ is odd in the set $\{ 1, 2,\ldots, n\}$. Prove $M$ is odd. An integer is [i]squarefree[/i] if it is not divisible by any square other than $1$.

Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$ Is $N$ even or odd? Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.

Natural number $N$ is given. Let $p_N$ - biggest prime, that $ \leq N$. On every move we replace $N$ by $N-p_N$. We repeat this until we get $0$ or $1$. Prove that exists such number $N$, that we need exactly $1000$ turns to make $0$

[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the positive integer $n$ have at least for positive divisors and $0<d_1<d_2<d_3<d_4$ be its least positive divisors. Find all positive integers $n$ such that: \[ n=d_1^2+d_2^2+d_3^2+d_4^2. \]

Find the smallest positive integer $k$ such that \[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\] for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.) [i]Alex Zhu.[/i] [hide="Clarifications"][list=1][*]${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$. In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$.[/list][/hide]

$M_0 \subset \mathbb{N}$ is a non-empty set with a finite number of elements. Ali produces sets $ M_1,M_2,...,M_n $ in the following order: In step $n$, Ali chooses an element of $M_{n-1} $ like $b_n$ and defines $M_n$ as $$M_n = \left \{ b_nm+1 \vert m\in M_{n-1} \right \}$$ Prove that at some step Ali reaches a set which no element of it divides another element of it.

The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^2$, $8\text{ in}^2$, and $6\text{ in}^2$ respectively is: $\textbf{(A)}\ 576\text{ in}^3 \qquad \textbf{(B)}\ 24\text{ in}^3 \qquad \textbf{(C)}\ 9\text{ in}^3 \qquad \textbf{(D)}\ 104\text{ in}^3 \qquad \textbf{(E)}\ \text{None of these}$

A natural number $N$ is [i]good [/i] if its digits are $1, 2$, or $3$ and all $2$-digit numbers are made up of digits located in consecutive positions of $N$ are distinct numbers. Is there a good number of $10$ digits? Of $11$ digits?

Let $p > 10$ be prime. Prove that there are positive integers $m$ and $n$ with $m + n < p$ exist for which $p$ is a divisor of $5^m7^n-1$.

Prove that when dividing a prime number with $30$, remainder is always not a composite number

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

For each positive integer $n$, let $f(n)$ be a positive integer. Show that if $f(n + 1) > f(f(n))$ for every positive integer n, then $f(x) = x$ for all positive integers $x$.

Solve the following equation in the set of integers $x^5 + 2 = 3 \cdot 101^y$.

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

Let ${p}$ be a prime. $a,b,c\in\mathbb Z,\gcd(a,p)=\gcd(b,p)=\gcd(c,p)=1.$ Prove that: $\exists x_1,x_2,x_3,x_4\in\mathbb Z,| x_1|,|x_2|,|x_3|,|x_4|<\sqrt p,$ satisfying $$ax_1x_2+bx_3x_4\equiv c\pmod p.$$ [i]Proposed by Wang Guangting[/i]

Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$. For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$, the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$, and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$. What is the last digit of the number in the cell $(2021, 2021)$?

Find all triples $(a, b, c)$ of integers that satisfy the equations $ a + b = c$ and $a^2 + b^3 = c^2$