Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

[b]p1.[/b] Suppose Mitchell has a fair die. He is about to roll it six times. The probability that he rolls $1$, $2$, $3$, $4$, $5$, and then $6$ in that order is $p$. The probability that he rolls $2$, $2$, $4$, $4$, $6$, and then $6$ in that order is $q$. What is $p - q$? [b]p2.[/b] What is the smallest positive integer $x$ such that $x \equiv 2017$ (mod $2016$) and $x \equiv 2016$ (mod $2017$) ? [b]p3.[/b] The vertices of triangle $ABC$ lie on a circle with center $O$. Suppose the measure of angle $ACB$ is $45^o$. If $|AB| = 10$, then what is the distance between $O$ and the line $AB$? [b]p4.[/b] A “word“ is a sequence of letters such as $YALE$ and $AELY$. How many distinct $3$-letter words can be made from the letters in $BOOLABOOLA$ where each letter is used no more times than the number of times it appears in $BOOLABOOLA$? [b]p5.[/b] How many distinct complex roots does the polynomial $p(x) = x^{12} - x^8 - x^4 + 1$ have? [b]p6.[/b] Notice that $1 = \frac12 + \frac13 + \frac16$ , that is, $1$ can be expressed as the sum of the three fractions $\frac12 $, $\frac13$ , and $\frac16$ , where each fraction is in the form $\frac{1}{n}$, with each $n$ different. Give a $6$-tuple of distinct positive integers $(a, b, c, d, e, f)$ where $a < b < c < d < e < f$ such that $\frac{1}{a} +\frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} = 1$ and explain how you arrived at your $6$-tuple. Multiple answers will be accepted. [b]p7.[/b] You have a Monopoly board, an $11 \times 11$ square grid with the $9 \times 9$ internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by rolling two standard $6$-sided dice. Let $S$ be the set of squares on the board such that if you are initially on a square in $S$, no matter what you roll with the dice, you will always either land on Go (move forward enough squares such that you end up on Go) or you pass Go (you move forward enough squares such that you step on Go during your move and then you advance past Go). You randomly and uniformly select one square in $S$ as your starting position. What is the probability that you land on Go? [b]p8.[/b] Using $L$-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a $3$-by-$2017$ rectangle without any gaps? [b]p9.[/b] Does there exist a pair of positive integers $(x, y)$, where $x < y$, such that $x^2 + y^2 = 1009^3$? If so, give a pair $(x, y)$ and explain how you found that pair. If not, explain why. [b]p10.[/b] Triangle $ABC$ has inradius $8$ and circumradius $20$. Let $M$ be the midpoint of side $BC$, and let $N$ be the midpoint of arc $BC$ on the circumcircle not containing $A$. Let $s_A$ denote the length of segment $MN$, and define $s_B$ and $s_C$ similarly with respect to sides $CA$ and $AB$. Evaluate the product $s_As_Bs_C$. [b]p11.[/b] Julia and Dan want to divide up $256$ dollars in the following way: in the first round, Julia will offer Dan some amount of money, and Dan can choose to accept or reject the offer. If Dan accepts, the game is over. Otherwise, if Dan rejects, half of the money disappears. In the second round, Dan can offer Julia part of the remaining money. Julia can then choose to accept or reject the offer. This process goes on until an offer is accepted or until $4$ rejections have been made; once $4$ rejections are made, all of the money will disappear, and the bargaining process ends. If Julia or Dan is indifferent between accepting and rejecting an offer, they will accept the offer. Given that Julia and Dan are both rational and both have the goal of maximizing the amount of money they get, how much will Julia offer Dan in the first round? [b]p12.[/b] A perfect partition of a positive integer $N$ is an unordered set of numbers (where numbers can be repeated) that sum to $N$ with the property that there is a unique way to express each positive integer less than $N$ as a sum of elements of the set. Repetitions of elements of the set are considered identical for the purpose of uniqueness. For example, the only perfect partitions of $3$ are $\{1, 1, 1\}$ and $\{1, 2\}$. $\{1, 1, 3, 4\}$ is NOT a perfect partition of $9$ because the sum $4$ can be achieved in two different ways: $4$ and $1 + 3$. How many integers $1 \le N \le 40$ each have exactly one perfect partition? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$|5x^2-\tfrac25|\le|x-8|$ if and only if $x$ is in the interval $[a, b]$. There are relatively prime positive integers $m$ and $n$ so that $b -a =\tfrac{m}{n}$ . Find $m + n$.

Let $n$ and $k$ be two positive integers such that $n>k$. Prove that the equation $x^n+y^n=z^k$ has a solution in positive integers if and only if the equation $x^n+y^n=z^{n-k}$ has a solution in positive integers.

Determine if there exist $101$ positive integers (not necessarily distinct) such that their product is equal to the sum of all their pairwise LCM.

Find all natural integers $m, n$ such that $m, 2+m, 2^n+m, 2+2^n+m$ are all prime numbers

A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called [i]quadratic[/i] if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$. $\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$. $\text{(ii)}$ Find the smallest positive integer $n$ for which there exists a quadratic sequence with $a_0=0$ and $a_n=2021$.

An integer consists of 7 different digits, and is a multiple of each of its digits. What digits are in this nubmer?

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions: $i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$ $ii) f$ takes all integer values

[b]8.1[/b] In the parallelogram $ABCD$ , the diagonal $AC$ is greater than the diagonal $BD$. The point $M$ on the diagonal $AC$ is such that around the quadrilateral $BCDM$ one can circumscribe a circle. Prove that $BD$ is the common tangent of the circles circumscribed around the triangles $ABM$ and $ADM$. [img]https://cdn.artofproblemsolving.com/attachments/b/3/9f77ff1f2198c201e5c270ec5b091a9da4d0bc.png[/img] [b]8.2 [/b] $A$ is an odd integer, $x$ and $y$ are roots of equation $t^2+At-1=0$. Prove that $x^4 + y^4$ and $x^5+ y^5$ are coprime integer numbers. [b]8.3[/b] A regular triangle is reflected symmetrically relative to one of its sides. The new triangle is again reflected symmetrically about one of its sides. This is repeated several times. It turned out that the resulting triangle coincides with the original one. Prove that an even number of reflections were made. [b]8.4 /7.6[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles. [b]8.5 [/b] All two-digit numbers that do not end in zero are written one after another so that each subsequent number begins with that the same digit with which the previous number ends. Prove that you can do this and find the sum of the largest and smallest of all multi-digit numbers that can be obtained in this way. [url=https://artofproblemsolving.com/community/c6h3390996p32049528]8,6*[/url] (asterisk problems in separate posts) PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$. Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively prime to $i$.

For every pair of reals $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$: \begin{align*} x_n & = (1 - a) x_{n - 1} + a y_{n - 1}, \\ y_n & = (1 - b) x_{n - 1} + b y_{n - 1}. \end{align*} The [i]supermean[/i] of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying $\left (p - \textstyle\frac{1}{2} \right)^2 + \left (q - \textstyle\frac{1}{2} \right)^2 \le \left(\textstyle\frac{1}{10}\right)^2$, the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\textstyle\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$. [i]Proposed by Lewis Chen[/i]

Solve the equation $x^{2}+y^{2}=10xy$ for integers $x$ and $y$

Find the largest positive integer $k$ such that $\phi ( \sigma ( 2^k)) = 2^k$. ($\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$). As a hint, you are given that $641|2^{32}+1$.

Let k be a positive integer. Find the number of non-negative integers n less than or equal to $10^k$ satisfying the following conditions: (i) n is divisible by 3; (ii) Each decimal digit of n is one of the digits 2,0,1 or 7.

Let $n>1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor$ such that the following equation has integer solutions with $a_m>0:$ $$\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}$$ [i]Proposed by Navid Safaei[/i]

Suppose $p,q$ are distinct primes and $S$ is a subset of $\{1,2,\dots ,p-1\}$. Let $N(S)$ denote the number of solutions to the equation $$\sum_{i=1}^{q}x_i\equiv 0\mod p$$ where $x_i\in S$, $i=1,2,\dots ,q$. Prove that $N(S)$ is a multiple of $q$.

Find all pairs of positive integers $(x,n) $ that are solutions of the equation $3 \cdot 2^x +4 =n^2$.

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible? [img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img] [b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$. [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy? [b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img] [b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove: (a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$ (b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$

$g$ and $n$ are natural numbers such that $gcd(g^2-g,n)=1$ and $A=\{g^i|i \in \mathbb N\}$ and $B=\{x\equiv (n)|x\in A\}$(by $x\equiv (n)$ we mean a number from the set $\{0,1,...,n-1\}$ which is congruent with $x$ modulo $n$). if for $0\le i\le g-1$ $a_i=|[\frac{ni}{g},\frac{n(i+1)}{g})\cap B|$ prove that $g-1|\sum_{i=0}^{g-1}ia_i$.( the symbol $|$ $|$ means the number of elements of the set)($\frac{100}{6}$ points) the exam time was 4 hours

Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers? [i]Proposed by F. Petrov[/i]