Find all real solutions $x$ to the equation $$x =\sqrt{x -\frac{1}{x}} +\sqrt{1 -\frac{1}{x}}$$

Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions: $a)$ every natural number occurs in this sequence and exactly once; $b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$ ?

Let $a, b$ and $c$ be pairwise relatively prime natural numbers. Find all possible values of $\frac{(a + b)(b + c)(c + a)}{abc}$ if known what it is integer.

Does there exist a sequence of positive integers $a_1,a_2,...$ such that every positive integer occurs exactly once and that the number $\tau (na_{n+1}^n+(n+1)a_n^{n+1})$ is divisible by $n$ for all positive integer. Here $\tau (n)$ denotes the number of positive divisor of $n$.

We number the columns of an $n\times n$-board from $1$ to $n$. In each cell, we place a number. This is done in such a way that each row precisely contains the numbers $1$ to $n$ (in some order), and also each column contains the numbers $1$ to $n$ (in some order). Next, each cell that contains a number greater than the cell's column number, is coloured grey. In the figure below you can see an example for the case $n = 3$. [asy] unitsize(0.6 cm); int i; fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray(0.8)); fill(shift((1,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); fill(shift((0,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); for (i = 0; i <= 3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$1$", (0.5,3.5)); label("$2$", (1.5,3.5)); label("$3$", (2.5,3.5)); label("$3$", (0.5,2.5)); label("$1$", (1.5,2.5)); label("$2$", (2.5,2.5)); label("$1$", (0.5,1.5)); label("$2$", (1.5,1.5)); label("$3$", (2.5,1.5)); label("$2$", (0.5,0.5)); label("$3$", (1.5,0.5)); label("$1$", (2.5,0.5)); [/asy] (a) Suppose that $n = 5$. Can the numbers be placed in such a way that each row contains the same number of grey cells? (b) Suppose that $n = 10$. Can the numbers be placed in such a way that each row contains the same number of grey cells?

Alice wrote a sequence of $n > 2$ nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those n numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the same as in Alice’s sequence. What are the possible values of $n{}$? [i]Alexey Zaslavsky[/i]

Given convex quadrilateral $ABCD$. We construct the similar triangles $APB, BQC, CRD, DSA$ outside $ABCD$ so that $\angle PAB = \angle QBC = \angle RCD = \angle SDA, \angle PBA = \angle QCB = \angle RDC = \angle SAD$. Prove that if $PQRS$ is a parallelogram, so is $ABCD$.

How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.)

A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line.

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent. [i]Proposed by Robin Park[/i]

Find digits $x, y, z$ such that the equality \[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\] holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.

Show that for any quadrilateral inscribed in a circle of radius 1, the length of the shortest side is less than or equal to $\sqrt{2}$.

Billy's basketball team scored the following points over the course of the first 11 games of the season: \[42, 47, 53, 53, 58, 58, 58, 61, 64, 65, 73\] If his team scores 40 in the 12th game, which of the following statistics will show an increase? $ \textbf{(A) } \text{range} \qquad \textbf{(B) } \text{median} \qquad \textbf{(C) } \text{mean} \qquad \textbf{(D) } \text{mode} \qquad \textbf{(E) } \text{mid-range} $

Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened.

$\frac{15^{30}}{45^{15}}=$ $ \textbf{(A)}\ \left(\frac{1}{3}\right)^{15} \qquad\textbf{(B)}\ \left(\frac{1}{3}\right)^2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 3^{15} \qquad\textbf{(E)}\ 5^{15}$

In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.

For an integer $n \geq 5,$ two players play the following game on a regular $n$-gon. Initially, three consecutive vertices are chosen, and one counter is placed on each. A move consists of one player sliding one counter along any number of edges to another vertex of the $n$-gon without jumping over another counter. A move is legal if the area of the triangle formed by the counters is strictly greater after the move than before. The players take turns to make legal moves, and if a player cannot make a legal move, that player loses. For which values of $n$ does the player making the first move have a winning strategy?

Derek the Dolphin and Kevin the Frog are playing a game where they take turns taking coins from a stack of $N$ coins, except with one rule: The number of coins someone takes each turn must be a power of $6$. The person who cannot take any more coins loses. If Derek goes first, how many integers $N$ from $1$ to $6^{2021}$ inclusive will guarantee him a win? (Example: If $N = 37$, then a possible sequence of turns is: Derek takes one coin, Kevin takes $36$ coins, and Kevin wins.)

In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$? [asy] pair X = (0, 0); pair W = (0, 4); pair Y = (8, 0); pair Z = (8, 4); label("$X$", X, dir(180)); label("$W$", W, dir(180)); label("$Y$", Y, dir(0)); label("$Z$", Z, dir(0)); draw(W--X--Y--Z--cycle); dot(X); dot(Y); dot(W); dot(Z); pair M = (2, 0); pair A = (8, 3); label("$A$", A, dir(0)); dot(M); dot(A); draw(W--M--A--cycle); markscalefactor = 0.05; draw(rightanglemark(W, M, A)); label("$M$", M, dir(-90)); [/asy] $ \textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad $

Joey and his five brothers are ages $3,5,7,9,11,$ and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the 5-year-old stayed home. How old is Joey? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 13 $

In the following figure, it is desired to go from point \( A \) to point \( B \) by walking only along the lines of the figure up and to the right. How many different paths can we take? [img]https://cdn.artofproblemsolving.com/attachments/7/c/dce2e0bcb69c9e2014474ab3699b4ef0470497.png[/img]

Let $A$ be a point on a circle with center $O$ and $B$ be the midpoint of $[OA]$. Let $C$ and $D$ be points on the circle such that they lie on the same side of the line $OA$ and $\widehat{CBO} = \widehat{DBA}$. Show that the reflection of the midpoint of $[CD]$ over $B$ lies on the circle.

In an acute-angled triangle $ABC$, $AM$ is a median, $AL$ is a bisector and $AH$ is an altitude ($H$ lies between $L$ and $B$). It is known that $ML=LH=HB$. Find the ratios of the sidelengths of $ABC$.

The equation $(x+a) (x+b) = 9$ has a root $a+b$. Prove that $ab\le 1$.

Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation $b$ comes exactly after $a$