The point $ M$ is chosen inside parallelogram $ ABCD$. Show that $ \angle MAB$ is congruent to $ \angle MCB$, if and only if $ \angle MBA$ and $ \angle MDA$ are congruent.

If $p$ is a prime number and $a>1$ is a natural number , then show that the greatest common divisor of the two numbers $a-1$ and $\frac{a^p-1}{a-1}$ is either $1$ or $p$ .

A regular polygon of 10 sides (a regular decagon) may be inscribed in a circle in the following two distinct ways: Divide the circumference into $10$ equal arcs and (1) join each division point to the next by straight line segments, (2) join each division point to the next but two by straight line segments. (See figures). Prove that the difference in the side lengths of these two decagons is equal to the radius of their circumscribed circle. [img]https://cdn.artofproblemsolving.com/attachments/7/9/41c38d08f4f89e07852942a493df17eaaf7498.png[/img]

Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers. One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins). In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice. [i]Nikolai Beluhov[/i]

(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle $\alpha$. Determine the smallest number of lines for which this is possible. (b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections.

It is known that the quadratic equations $x^2 + 6x + 4a = 0$ and $x^2 + 2bx - 12 = 0$ have a common solution. Prove that then there is a common solution to the quadratic equations $x^2 + 9x + 9a = 0$ and $x^2 + 3bx - 27 = 0$.

A toilet paper roll is a cylinder of radius $8$ and height $6$ with a hole in the shape of a cylinder of radius $2$ and the same height. That is, the bases of the roll are annuli with inner radius $2$ and outer radius $8$. Compute the surface area of the roll.

[b]p1.[/b] Find the number of ordered triples $(a, b, c)$ satisfying $\bullet$ $a, b, c$ are are single-digit positive integers, and $\bullet$ $a \cdot b + c = a + b \cdot c$. [b]p2.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form an increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.) [b]p3.[/b] Suppose that $a + \frac{1}{b} = 2$ and $b +\frac{1}{a} = 3$. If$ \frac{a}{b} + \frac{b}{a}$ can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$. [b]p4.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that $$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B_1B_1B_1...}) + 1$$ Find the sum of all possible values of $10A + B$. [b]p5.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $AC$ and $BC$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE = EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a, b$ positive integers and a squarefree, then find $a + b$. [b]p6.[/b] Five bowling pins $P_1$, $P_2$,..., $P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is a b where a and b are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i -j| = 1$.) [b]p7.[/b] Let triangle $ABC$ have side lengths $AB = 10$, $BC = 24$, and $AC = 26$. Let $I$ be the incenter of $ABC$. If the maximum possible distance between $I$ and a point on the circumcircle of $ABC$ can be expressed as $a +\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$. (The incenter of any triangle $XY Z$ is the intersection of the angle bisectors of $\angle Y XZ$, $\angle XZY$, and $\angle ZY X$.) [b]p8.[/b] How many terms in the expansion of $$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$ have coefficients equal to $1011$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.

Find all positive integers $k$ such that the product of the digits of $k$, in decimal notation, equals \[\frac{25}{8}k-211\]

Seven distinct points are given inside a square with side length $1.$ Together with the square's vertices, they form a set of $11$ points. Consider all triangles with vertices in $M.$ a) Show that at least one of these triangles has an area not exceeding $1\slash 16.$ b) Give an example in which no four of the seven points are on a line and none of the considered triangles has an area of less than $1\slash 16.$

Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

Given an acute triangle $ABC$ with circumcenter $O$. The circumcircle of $BCH$ and a circle with diameter of $AC$ intersect at $P (P \neq C)$. A point $Q$ on segment of $PC$ such that $PB = PQ$. Prove that $\angle ABC = \angle AQP$

In Happy City there are $2014$ citizens called $A_1, A_2, \dots , A_{2014}$. Each of them is either [i]happy[/i] or [i]unhappy[/i] at any moment in time. The mood of any citizen $A$ changes (from being unhappy to being happy or vice versa) if and only if some other happy citizen smiles at $A$. On Monday morning there were $N$ happy citizens in the city. The following happened on Monday during the day: the citizen $A_1$ smiled at citizen $A_2$, then $A_2$ smiled at $A_3$, etc., and, finally, $A_{2013}$ smiled at $A_{2014}$. Nobody smiled at anyone else apart from this. Exactly the same repeated on Tuesday, Wednesday and Thursday. There were exactly $2000$ happy citizens on Thursday evening. Determine the largest possible value of $N$.

Prove that for every integer $n\ge 3$ there exists $N(n)$ with the following property: whenever $P$ is a set of at least $N(n)$ points of the plane such that any three points of $P$ determines a nondegenerate triangle containing at most one point of $P$ in its interior, then $P$ contains the vertices of a convex $n$-gon whose interior does not contain any point of $P$.

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Let $ABCD$ be a isosceles trapezoid. Base $AD$ is $11$ cm long while the other three sides are each $5$ cm long. We draw the line that is perpendicular to $BD$ and contains $C$ and the line that is perpendicular to $AC$ and contains$ B$. We mark the intersection of these two lines with $E$. What is the distance between point $E$ and line $AD$?

There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least $t.$ Find the maximum possible value of $t.$

The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$. Prove that all terms of this sequence are perfect squares.

Label the vertices of a trapezoid $T$ inscribed in the unit circle as $A,B,C,D$ counterclockwise with $AB\parallel CD$. Let $s_1,s_2,$ and $d$ denote the lengths of $AB$, $CD$, and $OE$, where $E$ is the intersection of the diagonals of $T$, and $O$ is the center of the circle. Determine the least upper bound of $\frac{s_1-s_2}d$ over all $T$ for which $d\ne0$, and describe all cases, if any, in which equality is attained.

In a tennis ball factory there are $4$ machines $m_1 , m_2 , m_3 , m_4$, which produce, respectively, $10\%$, $20\%$, $30\%$ and $40\%$ of the balls that come out of the factory. The machine $m_1$ introduces defects in $1\%$ of the balls it manufactures, the machine $m_2$ in $2\%$, $m_3$ in $4\%$ and $m_4$ in $15\%$. Of the balls manufactured In one day, one is chosen at random and it turns out to be defective. What is the probability that Has this ball been made by the machine $ m_3$ ?

[hide=Note][i]The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.[/i][/hide] For each integer $n\ge2$, consider a regular polygon with $2n$ sides, all of length $1$. Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length $1$. Determine the limit inferior and the limit superior of the sequence defined by $$\frac1{n^2}\log_2C(n).$$

$S$ is a subset of Natural numbers which has infinite members. $$S’=\left\{x^y+y^x: \, x,y\in S, \, x\neq y\right\}$$ Prove the set of prime divisors of $S’$ has also infinite members [i]Proposed by Yahya Motevassel[/i]

Find all polynomial $P(x)$ with degree $\leq n$and non negative coefficients such that $$P(x)P(\frac{1}{x})\leq P(1)^2$$ for all positive $x$. Here $n$ is a natuaral number

Consider the set of all 6-digit numbers consisting of only three digits, $a,b,c$ where $a,b,c$ are distinct. Suppose the sum of all these numbers is $593999406$. What is the largest remainder when the three digit number $abc$ is divided by $100$?