Let $ABC$ be a triangle with $AB < AC < BC$. On the side $BC$ we consider points $D$ and $E$ such that $BA = BD$ and $CE = CA$. Let $K$ be the circumcenter of triangle $ADE$ and let $F$, $G$ be the points of intersection of the lines $AD$, $KC$ and $AE$, $KB$ respectively. Let $\omega_1$ be the circumcircle of triangle $KDE$, $\omega_2$ the circle with center $F$ and radius $FE$, and $\omega_3$ the circle with center $G$ and radius $GD$. Prove that $\omega_1$, $\omega_2$, and $\omega_3$ pass through the same point and that this point of intersection lies on the line $AK$.

A square of side length $1$ is dissected into two congruent pentagons. Compute the least upper bound of the perimeter of one of these pentagons.

For a positive integer $n$ denote $d(n)$ its greatest odd divisor. Find the value of the sum $$d(1008)+d(1009)+...+d(2015)$$

In acute-angled triangle $ABC$ with $|BC| < |BA|$, point $N$ is the midpoint of $AC$. The circle with diameter $AB$ intersects the bisector of $\angle B$ in two points: $B$ and $X$. Prove that $XN$ is parallel to $BC$. [img]https://cdn.artofproblemsolving.com/attachments/5/1/f0ae8f5df8f2cc1bb80de1ee1807dc845a87b3.png[/img]

Bobbo starts swimming at $2$ feet/s across a $100$ foot wide river with a current of $5$ feet/s. Bobbo doesn’t know that there is a waterfall $175$ feet from where he entered the river. He realizes his predicament midway across the river. What is the minimum speed that Bobbo must increase to make it to the other side of the river safely?

Let $A$ and $B$ be two nonempty finite sets of nonnegative integers. We denote by $\mathcal{F}$ the set of all functions $f:\mathcal{P}(A) \to B$ that satisfy [center]$f(X\cap Y)=\min \{f(X), f(Y)\},$ for all $X,Y \subset A,$[/center] and by $\mathcal{G}$ the set of all functions $g:\mathcal{P}(A) \to B$ that satisfy [center]$g(X\cup Y)=\max \{g(X), g(Y)\},$ for all $X,Y \subset A.$[/center] Prove that $\mathcal F$ and $\mathcal G$ have the same number of elements and find this number.

Let $a, b$, and $c$ be real numbers such that \[\frac{1}{bc-a^2} + \frac{1}{ca-b^2}+\frac{1}{ab-c^2} = 0.\] Prove that \[\frac{a}{(bc-a^2)^2} + \frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2} = 0.\]

[u]Set 6[/u] [b]p16.[/b] Let $x$ and $y$ be non-negative integers. We say point $(x, y)$ is square if $x^2 + y$ is a perfect square. Find the sum of the coordinates of all distinct square points which also satisfy $x^2 + y \le 64$. [b]p17.[/b] Two integers $a$ and $b$ are randomly chosen from the set $\{1, 2, 13, 17, 19, 87, 115, 121\}$, with $a > b$. What is the expected value of the number of factors of $ab$? [b]p18.[/b] Marnie the Magical Cello is jumping on nonnegative integers on number line. She starts at $0$ and jumps following two specific rules. For each jump she can either jump forward by $1$ or jump to the next multiple of $4$ (the next multiple must be strictly greater than the number she is currently on). How many ways are there for her to jump to $2022$? (Two ways are considered distinct only if the sequence of numbers she lands on is different.) PS. You should use hide for answers.

Show that for all integer $k>1$, there are infinitely many natural numbers $n$ such that $k \cdot 2^{2^n} + 1$ is composite.

[b]p1.[/b] A teenager coining home after midnight heard the hall clock striking the hour. At some moment between $15$ and $20$ minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then? [b]p2.[/b] The ratio of two positive integers $a$ and $b$ is $2/7$, and their sum is a four digit number which is a perfect cube. Find all such integer pairs. [b]p3.[/b] Given the integers $1, 2 , ..., n$ , how many distinct numbers are of the form $\sum_{k=1}^n( \pm k) $ , where the sign ($\pm$) may be chosen as desired? Express answer as a function of $n$. For example, if $n = 5$ , then we may form numbers: $ 1 + 2 + 3- 4 + 5 = 7$ $-1 + 2 - 3- 4 + 5 = -1$ $1 + 2 + 3 + 4 + 5 = 15$ , etc. [b]p4.[/b] $\overline{DE}$ is a common external tangent to two intersecting circles with centers at $O$ and $O'$. Prove that the lines $AD$ and $BE$ are perpendicular. [img]https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png[/img] [b]p5.[/b] Find all polynomials $f(x)$ such that $(x-2) f(x+1) - (x+1) f(x) = 0$ for all $x$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A man has part of $ \$4500$ invested at $ 4\%$ and the rest at $ 6\%$. If his annual return on each investment is the same, the average rate of interest which he realizes of the $ \$4500$ is: $ \textbf{(A)}\ 5\% \qquad\textbf{(B)}\ 4.8\% \qquad\textbf{(C)}\ 5.2\% \qquad\textbf{(D)}\ 4.6\% \qquad\textbf{(E)}\ \text{none of these}$

We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points $ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second intersection point (i.e. the one other than $M$) of the intersection points of circles circumscribed around triangles $AKM$ and $MLC$ lies on the diagonal $AC$. (V . N . Dubrovskiy)

Prove that from every set of $200$ integers you can choose a subset of $100$ with the total sum divisible by $100$.

In the Game of Life, each square in an infinite grid of squares is either shaded or blank. Every day, if a square shares an edge with exactly zero or four shaded squares, it becomes blank the next day. If a square shares an edge with exactly two or three shaded squares, it becomes shaded the next day. Otherwise, it does not change. On day $1$, each square is randomly shaded or blank with equal probability. If the probability that a given square is shaded on day 2 is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, find $a + b.$

A large pond contains infinitely many lily pads labelled $1$, $2$, $3$,$ ... $, placed in a line, where for each $k$, lily pad $k + 1$ is one unit to the right of lily pad $k$. A frog starts at lily pad $100$. Each minute, if the frog is at lily pad $n$, it hops to lily pad $n + 1$ with probability $\frac{n-1}{n}$ , and hops all the way back to lily pad $1$ with probability $\frac{1}{n}$. Let $N$ be the position of the frog after $1000$ minutes. What is the expected value of $N$?

The $64$ whole numbers from $1$ through $64$ are written, one per square, on a checkerboard (an $8$ by $8$ array of $64$ squares). The first $8$ numbers are written in order across the first row, the next $8$ across the second row, and so on. After all $64$ numbers are written, the sum of the numbers in the four corners will be $\text{(A)}\ 130 \qquad \text{(B)}\ 131 \qquad \text{(C)}\ 132 \qquad \text{(D)}\ 133 \qquad \text{(E)}\ 134$

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

$$\int\left(\frac{x-1}{x^2+1}\right)^2e^x\,dx$$ [i]Proposed by Connor Gordon[/i]

If A,B are invertible and the set {A^k - B^k | k is a natural number} is finite , then there exists a natural number m such that A^m = B^m.

Let $a$, $b$ be such integers that $gcd(a + n,b + n) > 1$ for every integer $n \geq 1$. Prove that $a = b$.

Prove that for each $ a\in\mathbb N$, there are infinitely many natural $ n$, such that \[ n\mid a^{n \minus{} a \plus{} 1} \minus{} 1. \]

The game of $pebbles$ is played on an infinite board of lattice points $(i,j)$. Initially there is a $pebble$ at $(0,0)$. A move consists of removing a $pebble$ from point $(i,j)$and placing a $pebble$ at each of the points $(i+1,j)$ and $(i,j+1)$ provided both are vacant. Show taht at any stage of the game there is a $pebble$ at some lattice point $(a,b)$ with $0 \leq a+b \leq 3$

If the function $f:[0,1]\to (0.+\infty)$ is increasing and continuous, then for every $a\geq 0$ the following inequality holds: $$\int \limits_0^1 \frac{x^{a+1}}{f(x)}dx \leq \frac{a+1}{a+2} \int \limits_0^1 \frac{x^{a}}{f(x)}dx$$

Prove the following inequality holds if $\{a_n\}$ is a deceasing sequence of positive reals, and $0<\theta<\frac{\pi}{2}$. $$\left|\sum_{n=1}^{2017} a_n \cos n\theta \right| \leq \frac{\pi a_1}{\theta}$$

Let $A,B\in \mathcal{M}_n(\mathbb{R})$. Prove that $\text{rank}\ A+\text{rank}\ B\le n$ if and only if there exists an invertible matrix $X\in \mathcal{M}_n(\mathbb{R})$ such that $AXB=O_n$.