Found problems: 85335
Consider all ordered pairs $(a, b)$ of positive integers such that $\frac{a^2 + b^2 + 2}{ab}$ is an integer and $a\le b$. We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus $P_1 = (1, 1), P_2 = (1, 3)$, and so on. If $P_{2020} = (x, y),$ then compute the remainder when $x + y$ is divided by $2017$.
[i]Proposed by Ashwin Sah[/i]
Find the locus of the affix $M$, of the complex number $z$, so that it is aligned with the affixes of $i$ and $iz$ .
Write down $1, 2, 3, ... , 2015$ in a row on a whiteboard. Every minute, select a pair of adjacent numbers at random, erase them, and insert their sum where you selected the numbers. (For instance, selecting $3$ and $4$ from $1, 2, 3, 4, 5$ would result in $1, 2, 7, 5$.) Repeat this process until you have two numbers remaining. What is the probability that the smaller number is less than or equal to $2015$?
Ayse knows the weights of nine balls with different colors are $1,2,\cdots, 9$ grams, but she doesn't know the weight of a specific ball. But Baris knows the weight of each ball. Baris wants to prove his knowledge to Ayse. There is a double pan balance which shows the heavier pan and the difference of the two pans. At least how many weighs are required for proof of Ali's knowledge?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 6
$
Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that $f(n)=f\left(\frac{n}{p}\right)-f(p)$. Given that $f(2001)=1$, what is the value of $f(2002)$?
There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked $X$?
An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example is shown on the picture for $ n \equal{} 5$. Determine the value of $ f(2005)$.
Let $\left(a_k\right)_{k=1}^\infty$ be a sequence of real numbers such that \[a_{k-1}+a_{k+1}\ge2a_k\] for all $k>1.$ For $n\ge1$ denote \[A_n=\frac1n\left(a_1+\cdots+a_n\right).\] Show that also the inequality \[A_{n-1}+A_{n+1}\ge2A_n\] holds for every $n>1.$
The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$.
Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.
There is a set of $ n$ coins with distinct integer weights $ w_1, w_2, \ldots , w_n$. It is known that if any coin with weight $ w_k$, where $ 1 \leq k \leq n$, is removed from the set, the remaining coins can be split into two groups of the same weight. (The number of coins in the two groups can be different.) Find all $ n$ for which such a set of coins exists.
Let $k$ be a positive integer. At the European Chess Cup every pair of players played a game in which somebody won (there were no draws). For any $k$ players there was a player against whom they all lost, and the number of players was the least possible for such $k$. Is it possible that at the Closing Ceremony all the participants were seated at the round table in such a way that every participant was seated next to both a person he won against and a person he lost against.
[i]Proposed by Matija Bucić.[/i]
Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let $x, y, z$ and $u$ denote the side lengths of the quadrilateral spanned by these four points. Prove that $25 \leq x^2+y^2+z^2+u^2 \leq 50$.
Find all the finite sequences with $ n$ consecutive natural numbers $ a_1, a_2,..., a_n$, with $ n\geq3$ such that $ a_1\plus{} a_2\plus{}...\plus{} a_n\equal{}2009$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$
$f(f(x)f(y)+y)=f(x)y+f(y-x+1)$
For all $x,y \in \mathbb{R}$
Let there be $3399$ numbers arbitrarily chosen among the first $6798$ integers $1, 2, \ldots , 6798$ in such a way that none of them divides another. Prove that there are exactly $1982$ numbers in $\{1, 2, \ldots, 6798\}$ that must end up being chosen.
In $\triangle ABC$ with $AC>AB$, let $D$ be the foot of the altitude from $A$ to side $\overline{BC}$, and let $M$ be the midpoint of side $\overline{AC}$. Let lines $AB$ and $DM$ intersect at a point $E$. If $AC=8$, $AE=5$, and $EM=6$, find the square of the area of $\triangle ABC$.
[i]Proposed by [b]DeToasty3[/b][/i]
Let $g_1$ and $g_2$ be some lines, which intersect in point $A$. A circle $k_1$ is tangent to $g_1$ at point $A$ and intersects $g_2$ for a second time in $C$. A circle $k_2$ is tangent to $g_2$ at point $A$ and intersects $g_1$ for a second time in $D$. The circles $k_1$ and $k_2$ intersect for a second time in point $B$. Prove that, if $\frac{AC}{AD}=\sqrt{2}$, then $\frac{BC}{BD}=2$.
$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.
Given a right triangular prism $A_1B_1C_1 - ABC$ with $\angle BAC = \frac{\pi}{2}$ and $AB = AC = AA_1$, let $G$, $E$ be the midpoints of $A_1B_1$, $CC_1$ respectively, and $D$, $F$ be variable points lying on segments $AC$, $AB$ (not including endpoints) respectively. If $GD \bot EF$, the range of the length of $DF$ is
${ \textbf{(A)}\ [\frac{1}{\sqrt{5}}, 1)\qquad\textbf{(B)}\ [\frac{1}{5}, 2)\qquad\textbf{(C)}\ [1, \sqrt{2})\qquad\textbf{(D)}} [\frac{1}{\sqrt{2}}, \sqrt{2})\qquad $
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
Study the real function $f(x) = \left(1 +\frac{1}{x}\right)^x$ defined for $ x \in R - \{-1, 0\}$ . Graphic representation.
Find all non-negative integer solutions of the equation
\[2^x + 3^y = z^2 .\]
On the plane are given $n > 2$ points, no three of which are collinear. Prove that among all closed polygonal lines passing through these points, any one with the minimum length is non-selfintersecting.
Given a point $p$ and a line segment $l$, let $d(p,l)$ be the distance between them. Let $A$, $B$, and $C$ be points in the plane such that $AB=6$, $BC=8$, $AC=10$. What is the area of the region in the $(x,y)$-plane formed by the ordered pairs $(x,y)$ such that there exists a point $P$ inside triangle $ABC$ with $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$