Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m$ and $n$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$

[b]p1. [/b] Evaluate $S$. $$S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}$$ [b]p2. [/b] Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves? [b]p3.[/b] Given that $$(a + b) + (b + c) + (c + a) = 18$$ $$\frac{1}{a + b}+\frac{1}{b + c}+ \frac{1}{c + a}=\frac59,$$ determine $$\frac{c}{a + b}+\frac{a}{b + c}+\frac{b}{c + a}.$$ [b]p4.[/b] Find all primes $p$ such that $2^{p+1} + p^3 - p^2 - p$ is prime. [b]p5.[/b] In right triangle $ABC$ with the right angle at $A$, $AF$ is the median, $AH$ is the altitude, and $AE$ is the angle bisector. If $\angle EAF = 30^o$ , find $\angle BAH$ in degrees. [b]p6.[/b] For which integers $a$ does the equation $(1 - a)(a - x)(x- 1) = ax$ not have two distinct real roots of $x$? [b]p7. [/b]Given that $a^2 + b^2 - ab - b +\frac13 = 0$, solve for all $(a, b)$. [b]p8. [/b] Point $E$ is on side $\overline{AB}$ of the unit square $ABCD$. $F$ is chosen on $\overline{BC}$ so that $AE = BF$, and $G$ is the intersection of $\overline{DE}$ and $\overline{AF}$. As the location of $E$ varies along side $\overline{AB}$, what is the minimum length of $\overline{BG}$? [b]p9.[/b] Sam and Susan are taking turns shooting a basketball. Sam goes first and has probability $P$ of missing any shot, while Susan has probability $P$ of making any shot. What must $P$ be so that Susan has a $50\%$ chance of making the first shot? [b]p10.[/b] Quadrilateral $ABCD$ has $AB = BC = CD = 7$, $AD = 13$, $\angle BCD = 2\angle DAB$, and $\angle ABC = 2\angle CDA$. Find its area. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

Find all pairs $(p,q)$ of prime numbers which $p>q$ and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer.

For two rational numbers $r,s$ we say:$$r\mid s$$whenever there exists $k\in\mathbb{Z}$ such that:$$s=kr$$ ${(a_n)}_{n\in\mathbb{N}}$ is an increasing sequence of pairwise coprime natural numbers and ${(b_n)}_{n\in\mathbb{N}}$ is a sequence of distinct natural numbers. Assume that for all $n\in\mathbb{N}$ we have: $$\sum_{i=1}^{n}\frac{1}{a_i}\mid\sum_{i=1}^{n}\frac{1}{b_i}$$ Prove that [b]for all[/b] $n\in\mathbb{N}$ we have: $a_n=b_n$.

Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$

Given a circle of radius $r= 1995$. Show that around it you can describe exactly $16$ primitive Pythagorean triangles. The primitive Pythagorean triangle is a right-angled triangle, the lengths of the sides of which are expressed by coprime integers.

[b]p1.[/b] Elsie M. is fixing a watch with three gears. Gear $A$ makes a full rotation every $5$ minutes, gear $B$ makes a full rotation every $8$ minutes, and gear $C$ makes a full rotation every $12$ minutes. The gears continue spinning until all three gears are in their original positions at the same time. How many minutes will it take for the gears to stop spinning? [b]p2.[/b] Optimus has to pick $10$ distinct numbers from the set of positive integers $\{2, 3, 4,..., 29, 30\}$. Denote the numbers he picks by $\{a_1, a_2, ...,a_{10}\}$. What is the least possible value of $$d(a_1 ) + d(a_2) + ... + d(a_{10}),$$ where $d(n)$ denotes the number of positive integer divisors of $n$? For example, $d(33) = 4$ since $1$, $3$, $11$, and $33$ divide $33$. [b]p3.[/b] Michael is given a large supply of both $1\times 3$ and $1\times 5$ dominoes and is asked to arrange some of them to form a $6\times 13$ rectangle with one corner square removed. What is the minimum number of $1\times 3$ dominoes that Michael can use? [img]https://cdn.artofproblemsolving.com/attachments/6/6/c6a3ef7325ecee417e37ec9edb5374aceab9fd.png[/img] [b]p4.[/b] Andy, Ben, and Chime are playing a game. The probabilities that each player wins the game are, respectively, the roots $a$, $b$, and $c$ of the polynomial $x^3 - x^2 + \frac{111}{400}x - \frac{9}{400} = 0$ with $a \le b \le c$. If they play the game twice, what is the probability of the same player winning twice? [b]p5.[/b] TongTong is doodling in class and draws a $3 \times 3$ grid. She then decides to color some (that is, at least one) of the squares blue, such that no two $1 \times 1$ squares that share an edge or a corner are both colored blue. In how many ways may TongTong color some of the squares blue? TongTong cannot rotate or reflect the board. [img]https://cdn.artofproblemsolving.com/attachments/6/0/4b4b95a67d51fda0f155657d8295b0791b3034.png[/img] [b]p6.[/b] Given a positive integer $n$, we define $f(n)$ to be the smallest possible value of the expression $$| \square 1 \square 2 ... \square n|,$$ where we may place a $+$ or a $-$ sign in each box. So, for example, $f(3) = 0$, since $| + 1 + 2 - 3| = 0$. What is $f(1) + f(2) + ... + f(2011)$? [b]p7.[/b] The Duke Men's Basketball team plays $11$ home games this season. For each game, the team has a $\frac34$ probability of winning, except for the UNC game, which Duke has a $\frac{9}{10}$ probability of winning. What is the probability that Duke wins an odd number of home games this season? [b]p8.[/b] What is the sum of all integers $n$ such that $n^2 + 2n + 2$ divides $n^3 + 4n^2 + 4n - 14$? [b]p9.[/b] Let $\{a_n\}^N_{n=1}$ be a finite sequence of increasing positive real numbers with $a_1 < 1$ such that $$a_{n+1} = a_n \sqrt{1 - a^2_1}+ a_1\sqrt{1 - a^2_n}$$ and $a_{10} = 1/2$. What is $a_{20}$? [b]p10.[/b] Three congruent circles are placed inside a unit square such that they do not overlap. What is the largest possible radius of one of these circles? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Joaquín and his brother Andrés go to class every day on the $62$ bus. Joaquín always pays for the tickets. Each ticket has a $5$-digit number printed on it. One day, Joaquín observes that the numbers on his tickets - his and his brother's - as well as being consecutive, are such that the sum of the ten digits is precisely $62$. Andrés asks him if the sum of the digits of any of the tickets is $35$ and, knowing the answer, he can directly say the number of each ticket. What were those numbers?

Three natural numbers are such that the product of any two of them is divisible by the sum of those two numbers. Prove that these numbers have a common divisor greater than $1$. [i]Proposed by Evan Chang (squareman), USA[/i]

Juan writes the list of pairs $(n, 3^n)$, with $n=1, 2, 3,...$ on a chalkboard. As he writes the list, he underlines the pairs $(n, 3^n)$ when $n$ and $3^n$ have the same units digit. What is the $2013^{th}$ underlined pair?

The sequence $a_n$ is given as $$a_1=1, a_2=2 \;\;\; \text{and} \;\;\;\; a_{n+2}=a_n(a_{n+1}+1) \quad \forall n\geq 1$$ Prove that $a_{a_n}$ is divisible by $(a_n)^n$ for $n\geq 100$.

For how many integers $ n$ is $ \frac{n}{20\minus{}n}$ the square of an integer? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 10$

A positive integer is called [i]fancy[/i] if it can be expressed in the form $$2^{a_1}+2^{a_2}+ \cdots+ 2^{a_{100}},$$ where $a_1,a_2, \cdots, a_{100}$ are non-negative integers that are not necessarily distinct. Find the smallest positive integer $n$ such that no multiple of $n$ is a [i]fancy[/i] number. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

[b]8.1[/b] A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. Draw another straight line so that after that the rectangle was divided into the largest possible number of parts. [b]8.2[/b] Engineers always tell the truth, but businessmen always lie. F and G are engineers. A declares that, B asserts that, C asserts that, D says that, E insists that, F denies that G is an businessman. C also announces that D is a businessman. If A is a businessman, then how much total businessmen in this company? [b]8.3 [/b]There is a straight road through the field. A tourist stands on the road at a point ?. It can walk along the road at a speed of 6 km/h and across the field at a speed of 3 km/h. Find the locus of the points where the tourist can get there within an hour's walk. [b]8.4 / 7.5 [/b] Let $ [A]$ denote the largest integer not greater than $A$. Solve the equation: $[(5 + 6x)/8] = (15x-7)/5$ . [b]8.5.[/b] In some state, every two cities are connected by a road. Each road is only allowed to move in one direction. Prove that there is a city from which you can travel around everything. state, having visited each city exactly once. [b]8.6[/b] Find all eights of prime numbers such that the sum of the squares of the numbers in the eight is 992 less than their quadruple product. [hide=original wording]Найдите все восьмерки простых чисел такие, что сумма квадратов чисел в восьмерке на 992 меньше, чем их учетверенное произведение.[/hide] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation $$a^3 + 2017a = b^3 -2017b.$$

Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$

Prove that for each positive integer n,the equation $x^{2}+15y^{2}=4^{n}$ has at least $n$ integer solution $(x,y)$

The numbers $1, 2, ..., 2012$ are written on a blackboard, in some order, each of them exactly once. Between every two neighboring numbers the absolute value of their difference is written and the original numbers are deleted. This process is repeated until only a number remains on the board. What is the largest number that can stay on the board?

Let $n$ be a natural number, for which we define $S(n)=\{1+g+g^2+...+g^{n-1}|g\in{\mathbb{N}},g\geq2\}$ $a)$ Prove that: $S(3)\cap S(4)=\varnothing$ $b)$ Determine: $S(3)\cap S(5)$

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

Given set $S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}$.Let $a$ and $n$ natural numbers such that $a+2^k\in S$ for all $k=1,2,3,...,n$.Find the greatest value of $n$.