Let $ABC$ be an isosceles triangle in which $AB = AC$. On its sides $BC$ and $AC$ respectively are marked points $P$ and $Q$ so that $PQ\parallel AB$. Let $F$ be the center of the circle circumscribed about the triangle $PQC$, and $E$ the midpoint of the segment $BQ$. Prove that $\angle AEF = 90^o $.

For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard in this way.

On a piece of paper, we write down all positive integers $n$ such that all proper divisors of $n$ are less than $30$. We know that the sum of all numbers on the paper having exactly one proper divisor is $2397$. What is the sum of all numbers on the paper having exactly two proper divisors? We say that $k$ is a proper divisor of the positive integer $n$ if $k | n$ and $1 < k < n$.

Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.

The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n))=f(m)-n.\]

Find all $x$ for which the equation $ x^2 + y^2 + z^2 + 2xyz = 1$ (relative to $z$) has a valid solution for any $y$.

For all nonnegative integer $i$, there are seven cards with $2^i$ written on it. How many ways are there to select the cards so that the numbers add up to $n$?

Determine all triples $(a,b,c)$ of integers that simultaneously satisfy the following relations: \begin{align*} a^2 + a = b + c, \\ b^2 + b = a + c, \\ c^2 + c = a + b. \end{align*}

We consider the quadrilateral $ABCD$, with $AD = CD = CB$ and $AB \parallel CD$. Points $E$ and $F$ belong to the segments $CD$ and $CB$ so that angles $\angle ADE = \angle AEF$. Prove that: a) $4CF \le CB$ , b) if $4CF = CB$, then $AE$ is the bisector of the angle $\angle DAF$.

Let $ABC$ be an acute-angled triangle, $D$ be the foot of the altitude from $A$, the circle with diameter $AD$ intersect $AB$ at $F$ and $AC$ at $E$. Let $P$ be the orthocenter of triangle $AEF$ and $O$ be the circumcenter of $ABC$. Prove that $A, P,$ and $O$ are collinear.

Prove that there are no positive integers $a$ and $b$ such that for all different primes $p$ and $q$ greater than $1000$, the number $ap+bq$ is also prime.

[list=a] [*]Prove that for any real numbers $a$ and $b$ the following inequality holds: \[ \left( a^{2}+1\right) \left( b^{2}+1\right) +50\geq2\left( 2a+1\right)\left( 3b+1\right)\] [*]Find all positive integers $n$ and $p$ such that: \[ \left( n^{2}+1\right) \left( p^{2}+1\right) +45=2\left( 2n+1\right)\left( 3p+1\right) \][/list]

Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that \[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\] Prove that \[5m+12n\le 581.\]

Points $E$ and $F$ lying on sides $BC$ and $AD$ respectively of a parallelogram $ABCD$ are such that $EF=ED=DC$. Let $M$ be the midpoint of $BE$ and $MD$ meet $EF$ at $G$. Prove that $\angle EAC=\angle GBD$.

Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ? [asy] pathpen = linewidth(.7); pointpen = black; pair A=(-1,0), B=-A, C=(0,1); fill(arc(C,1,0,180)--arc(A,1,90,0)--arc(B,1,180,90)--cycle, gray(0.5)); D(CR(D("A",A,SW),1)); D(CR(D("B",B,SE),1)); D(CR(D("C",C,N),1));[/asy] ${ \textbf{(A)}\ 3 - \frac{\pi}{2} \qquad \textbf{(B)}\ \frac{\pi}{2} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \frac{3\pi}{4} \qquad \textbf{(E)}\ 1+\frac{\pi}{2}} $

Prove that, for any integer $a_{1}>1$, there exist an increasing sequence of positive integers $a_{1}, a_{2}, a_{3}, \cdots$ such that \[a_{1}+a_{2}+\cdots+a_{n}\; \vert \; a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\] for all $n \in \mathbb{N}$.

The vertices of a triangle have integer coordinates and one of its sides is of length $\sqrt{n}$, where $n$ is a square-free natural number. Prove that the ratio of the circumradius and the inradius is an irrational number.

Can the cells of a $2002 \times 2002$ table be filled with the numbers from $1$ to $2002^2$ (one per cell) so that for any cell we can find three numbers $a, b, c$ in the same row or column (or the cell itself) with $a = bc$?

Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]D1.[/b] The product of two positive integers is $5$. What is their sum? [b]D2.[/b] Gavin is $4$ feet tall. He walks $5$ feet before falling forward onto a cushion. How many feet is the top of Gavin’s head from his starting point? [b]D3.[/b] How many times must Nathan roll a fair $6$-sided die until he can guarantee that the sum of his rolls is greater than $6$? [b]D4 / Z1.[/b] What percent of the first $20$ positive integers are divisible by $3$? [b]D5.[/b] Let $a$ be a positive integer such that $a^2 + 2a + 1 = 36$. Find $a$. [b]D6 / Z2.[/b] It is said that a sheet of printer paper can only be folded in half $7$ times. A sheet of paper is $8.5$ inches by $11$ inches. What is the ratio of the paper’s area after it has been folded in half $7$ times to its original area? [b]D7 / Z3.[/b] Boba has an integer. They multiply the number by $8$, which results in a two digit integer. Bubbles multiplies the same original number by 9 and gets a three digit integer. What was the original number? [b]D8.[/b] The average number of letters in the first names of students in your class of $24$ is $7$. If your teacher, whose first name is Blair, is also included, what is the new class average? [b]D9 / Z4.[/b] For how many integers $x$ is $9x^2$ greater than $x^4$? [b]D10 / Z5.[/b] How many two digit numbers are the product of two distinct prime numbers ending in the same digit? [b]D11 / Z6.[/b] A triangle’s area is twice its perimeter. Each side length of the triangle is doubled,and the new triangle has area $60$. What is the perimeter of the new triangle? [b]D12 / Z7.[/b] Let $F$ be a point inside regular pentagon $ABCDE$ such that $\vartriangle FDC$ is equilateral. Find $\angle BEF$. [b]D13 / Z8.[/b] Carl, Max, Zach, and Amelia sit in a row with $5$ seats. If Amelia insists on sitting next to the empty seat, how many ways can they be seated? [b]D14 / Z9.[/b] The numbers $1, 2, ..., 29, 30$ are written on a whiteboard. Gumbo circles a bunch of numbers such that for any two numbers he circles, the greatest common divisor of the two numbers is the same as the greatest common divisor of all the numbers he circled. Gabi then does the same. After this, what is the least possible number of uncircled numbers? [b]D15 / Z10.[/b] Via has a bag of veggie straws, which come in three colors: yellow, orange, and green. The bag contains $8$ veggie straws of each color. If she eats $22$ veggie straws without considering their color, what is the probability she eats all of the yellow veggie straws? [b]Z11.[/b] We call a string of letters [i]purple[/i] if it is in the form $CVCCCV$ , where $C$s are placeholders for (not necessarily distinct) consonants and $V$s are placeholders for (not necessarily distinct) vowels. If $n$ is the number of purple strings, what is the remainder when $n$ is divided by $35$? The letter $y$ is counted as a vowel. [b]Z12.[/b] Let $a, b, c$, and d be integers such that $a+b+c+d = 0$ and $(a+b)(c+d)(ab+cd) = 28$. Find $abcd$. [b]Z13.[/b] Griffith is playing cards. A $13$-card hand with Aces of all $4$ suits is known as a godhand. If Griffith and $3$ other players are dealt $13$-card hands from a standard $52$-card deck, then the probability that Griffith is dealt a godhand can be expressed in simplest form as $\frac{a}{b}$. Find $a$. [b]Z14.[/b] For some positive integer $m$, the quadratic $x^2 + 202200x + 2022m$ has two (not necessarily distinct) integer roots. How many possible values of $m$ are there? [b]Z15.[/b] Triangle $ABC$ with altitudes of length $5$, $6$, and $7$ is similar to triangle $DEF$. If $\vartriangle DEF$ has integer side lengths, find the least possible value of its perimeter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The numbers from $1$ to $100$ are placed without repetition in each cell of a \(10 \times 10\) board. An increasing path of length \(k\) on this board is a sequence of cells \(c_1, c_2, \ldots, c_k\) such that, for each \(i = 2, 3, \ldots, k\), the following properties are satisfied: • The cells \(c_i\) and \(c_{i-1}\) share a side or a vertex; • The number in \(c_i\) is greater than the number in \(c_{i-1}\). What is the largest positive integer \(k\) for which we can always find an increasing path of length \(k\), regardless of how the numbers from 1 to 100 are arranged on the board?

Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$.

If $\mathit{B}$ is a point on circle $\mathit{C}$ with center $\mathit{P}$, then the set of all points $\mathit{A}$ in the plane of circle $\mathit{C}$ such that the distance between $\mathit{A}$ and $\mathit{B}$ is less than or equal to the distance between $\mathit{A}$ and any other point on circle $\mathit{C}$ is $\textbf{(A) }\text{the line segment from }P \text{ to }B\qquad$ $\textbf{(B) }\text{the ray beginning at }P \text{ and passing through }B\qquad$ $\textbf{(C) }\text{a ray beginning at }B\qquad$ $\textbf{(D) }\text{a circle whose center is }P\qquad$ $\textbf{(E) }\text{a circle whose center is }B$

We have a $2007 \times 2007$ square table filled with nonnegative integers. For each entry of $0$ in the table, the sum of the elements that are in the same row or column as that entry is at least $2007$. Find the minimum sum of all the elements of such a table.