In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\angle AMN=\angle NMO$. Determine the measure of $\angle A B N$ in degrees. [asy] defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90); draw(K--O--A--M--N--B--A^^Circle(origin,1)); label("$A$", A, dir(O--A)); label("$K$", K, dir(O--K)); label("$B$", B, dir(O--B)); label("$N$", N, E); label("$M$", M, S); label("$O$", O, SE);[/asy]

A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were $ 100$ tourists on the ferry boat, and that on each successive trip, the number of tourists was $ 1$ fewer than on the previous trip. How many tourists did the ferry take to the island that day? $ \textbf{(A)}\ 585\qquad \textbf{(B)}\ 594\qquad \textbf{(C)}\ 672\qquad \textbf{(D)}\ 679\qquad \textbf{(E)}\ 694$

For real positive numbers $a, b, c$, the equality $abc = 1$ holds. Prove that $$\frac{a^{2014}}{1 + 2 bc}+\frac{b^{2014}}{1 + 2ac}+\frac{c^{2014}}{1 + 2ab} \ge \frac{3}{ab+bc+ca}.$$

Let $ S(n) $ be the sum of the squares of the positive integers less than and coprime to $ n $. For example, $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $, but $ S(4) = 1^2 + 3^2 $. Let $ p = 2^7 - 1 = 127 $ and $ q = 2^5 - 1 = 31 $ be primes. The quantity $ S(pq) $ can be written in the form $$ \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) $$ where $ a $, $ b $, and $ c $ are positive integers, with $ b $ and $ c $ coprime and $ b < c $. Find $ a $.

We have $4$ sticks. It is known, that for every $3$ sticks we can build a triangle with the same area. Is it true, that sticks have the same length?

Let $f(x)$ be a polynomial with rational coefficients, and let $\alpha$ be a real number. If \[\alpha^3-2019\alpha=(f(\alpha))^3-2019f(\alpha)=2021,\] prove that $(f^n(\alpha))^3-2019f^n(\alpha)=2021$ for any positive integer $n$. (Here, we define $f^n(x)=\underbrace{f(f(f\cdots f}_{n\text{ times}}(x)\cdots ))$.)

[b]p1.[/b] If $a \diamond b = ab - a + b$, find $(3 \diamond 4) \diamond 5$ [b]p2.[/b] If $5$ chickens lay $5$ eggs in $5$ days, how many chickens are needed to lay $10$ eggs in $10$ days? [b]p3.[/b] As Alissa left her house to go to work one hour away, she noticed that her odometer read $16261$ miles. This number is a "special" number for Alissa because it is a palindrome and it contains exactly $1$ prime digit. When she got home that evening, it had changed to the next greatest "special" number. What was Alissa's average speed, in miles per hour, during her two hour trip? [b]p4.[/b] How many $1$ in by $3$ in by $8$ in blocks can be placed in a $4$ in by $4$ in by $9$ in box? [b]p5.[/b] Apple loves eating bananas, but she prefers unripe ones. There are $12$ bananas in each bunch sold. Given any bunch, if there is a $\frac13$ probability that there are $4$ ripe bananas, a $\frac16$ probability that there are $6$ ripe bananas, and a $\frac12$ probability that there are $10$ ripe bananas, what is the expected number of unripe bananas in $12$ bunches of bananas? [b]p6.[/b] The sum of the digits of a $3$-digit number $n$ is equal to the same number without the hundreds digit. What is the tens digit of $n$? [b]p7.[/b] How many ordered pairs of positive integers $(a, b)$ satisfy $a \le 20$, $b \le 20$, $ab > 15$? [b]p8.[/b] Let $z(n)$ represent the number of trailing zeroes of $n!$. What is $z(z(6!))?$ (Note: $n! = n\cdot (n-1) \cdot\cdot\cdot 2 \cdot 1$) [b]p9.[/b] On the Cartesian plane, points $A = (-1, 3)$, $B = (1, 8)$, and $C = (0, 10)$ are marked. $\vartriangle ABC$ is reflected over the line $y = 2x + 3$ to obtain $\vartriangle A'B'C'$. The sum of the $x$-coordinates of the vertices of $\vartriangle A'B'C'$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Compute $a + b$. [b]p10.[/b] How many ways can Bill pick three distinct points from the figure so that the points form a non-degenerate triangle? [img]https://cdn.artofproblemsolving.com/attachments/6/a/8b06f70d474a071b75556823f70a2535317944.png[/img] [b]p11.[/b] Say piece $A$ is attacking piece $B$ if the piece $B$ is on a square that piece $A$ can move to. How many ways are there to place a king and a rook on an $8\times 8$ chessboard such that the rook isn't attacking the king, and the king isn't attacking the rook? Consider rotations of the board to be indistinguishable. (Note: rooks move horizontally or vertically by any number of squares, while kings move $1$ square adjacent horizontally, vertically, or diagonally). [b]p12.[/b] Let the remainder when $P(x) = x^{2020} - x^{2017} - 1$ is divided by $S(x) = x^3 - 7$ be the polynomial $R(x) = ax^2 + bx + c$ for integers $a$, $b$, $c$. Find the remainder when $R(1)$ is divided by $1000$. [b]p13.[/b] Let $S(x) = \left \lfloor \frac{2020}{x} \right\rfloor + \left \lfloor \frac{2020}{x + 1} \right\rfloor$. Find the number of distinct values $S(x)$ achieves for integers $x$ in the interval $[1, 2020]$. [b]p14.[/b] Triangle $\vartriangle ABC$ is inscribed in a circle with center $O$ and has sides $AB = 24$, $BC = 25$, $CA = 26$. Let $M$ be the midpoint of $\overline{AB}$. Points $K$ and $L$ are chosen on sides $\overline{BC}$ and $\overline{CA}$, respectively such that $BK < KC$ and $CL < LA$. Given that $OM = OL = OK$, the area of triangle $\vartriangle MLK$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p15.[/b] Euler's totient function, $\phi (n)$, is defined as the number of positive integers less than $n$ that are relatively prime to $n$. Let $S(n)$ be the set of composite divisors of $n$. Evaluate $$\sum^{50}_{k=1}\left( k - \sum_{d\in S(k)} \phi (d) \right)$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $J$ , $E$, $R$, and $Y$ be four positive integers chosen independently and uniformly at random from the set of factors of $1428$. What is the probability that $JERRY = 1428$? Express your answer in the form $\frac{a}{b\cdot 2^n}$ where $n$ is a nonnegative integer, $a $and $b$ are odd, and gcd $(a,b) = 1$.

Prove or disprove the following statements: (a) There exists a monotone function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$. (b) There exists a continuously differentiable function $f : [0, 1] \rightarrow [0, 1]$ such that for each $y \in [0, 1]$ the equation $f(x) = y$ has uncountably many solutions $x$.

If $x$, $y$, $z$ are real numbers satisfying \begin{align*} (x + 1)(y + 1)(z + 1) & = 3 \\ (x + 2)(y + 2)(z + 2) & = -2 \\ (x + 3)(y + 3)(z + 3) & = -1, \end{align*} find the value of $$ (x + 20)(y + 20)(z + 20). $$

Let $r$ be a positive integer. Show that if a graph $G$ has no cycles of length at most $2r$, then it has at most $|V|^{2016}$ cycles of length exactly $2016r$, where $|V|$ denotes the number of vertices in the graph $G$.

Find the sum of the coefficients of the polynomial $(63x-61)^4$.

If all vertices of a triangle on the square grid are grid points, then the triangle is called a [i]lattice[/i] triangle. What is the area of the lattice triangle with (one) of the smallest area, if one grid has area $1$ square unit?

Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.

Let $ABCD$ be a convex quadrilateral, $I_1$ and $I_2$ be the incenters of triangles $ABC$ and $DBC$ respectively. The line $I_1I_2$ intersects the lines $AB$ and $DC$ at points $E$ and $F$ respectively. Given that $AB$ and $CD$ intersect in $P$, and $PE=PF$, prove that the points $A$, $B$, $C$, $D$ lie on a circle.

A positive integer $K$ is given. Define the sequence $(a_n)$ by $a_1 = 1$ and $a_n$ is the $n$-th positive integer greater than $a_{n-1}$ which is congruent to $n$ modulo $K$. [b](a)[/b] Find an explicit formula for $a_n$. [b](b)[/b] What is the result if $K = 2$?

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\theta$ be the acute angle formed by that side and the median. What is $\sin{\theta}$? $ \textbf{(A)}\ \frac{3}{10}\qquad\textbf{(B)}\ \frac{1}{3}\qquad\textbf{(C)}\ \frac{9}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{9}{10} $

Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.

$ABC$ is a triangle with $\angle{B}>45^{\circ}$ , $\angle{C}>45^{\circ}$. We draw the isosceles triangles $CAM,BAN$ on the sides $AC,AB$ and outside the triangle, respectively, such that $\angle{CAM}=\angle{BAN}=90^{\circ}$. And we draw isosceles triangle $BPC$ on the side $BC$ and inside the triangle such that $\angle{BPC}=90^{\circ}$. Prove that $\Delta{MPN}$ is an isosceles triangle, too, and $\angle{MPN}=90^{\circ}$.

Let $ABC$ be a triangle with incenter $I,$ and let $D,E,F$ be the midpoints of sides $BC, CA, AB$, respectively. Lines $BI$ and $DE$ meet at $P $ and lines $CI$ and $DF$ meet at $Q$. Line $PQ$ meets sides $AB$ and $AC$ at $T$ and $S$, respectively. Prove that $AS = AT$

At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$, respectively, which number thought student who told publicly number $6$

Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.

Let $a,b$ and $c$ be real numbers such that \begin{align*} a^2+ab+b^2&=9 \\ b^2+bc+c^2&=52 \\ c^2+ca+a^2&=49. \end{align*} Compute the value of $\dfrac{49b^2+39bc+9c^2}{a^2}$.

Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations: $$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$ Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$ doesn't have limit at $ \infty . $ [i]Romeo Ilie[/i]