Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\] [i]Proposed by Azerbaijan[/i] [hide=Second Suggested Version]Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .\][/hide]

For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?

Some $n>2$ lamps are cyclically connected: lamp $1$ with lamp $2$, ..., lamp $k$ with lamp $k+1$,..., lamp $n-1$ with lamp $n$, lamp $n$ with lamp $1$. At the beginning all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.

Does there exist a function $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+\sin y$, for all reals $x,y$ ?

Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?

Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$ (Folklore) [hide=PS]Using search terms [color=#f00]+ ''f(x+f(y))'' + ''f(x)+y[/color]'' I found the same problem [url=https://artofproblemsolving.com/community/c6h1122140p5167983]in Q[/url], [url=https://artofproblemsolving.com/community/c6h1597644p9926878]continuous in R[/url], [url=https://artofproblemsolving.com/community/c6h1065586p4628238]strictly monotone in R[/url] , [url=https://artofproblemsolving.com/community/c6h583742p3451211 ]without extra conditions in R[/url] [/hide]

Consider several tokens of various colors and sizes, so that there are no two tokens having the same color and the same size. Two numbers are written on each token $J$: one of them is the number of chips having the same color as $J$, but different size, and the other is the number of chips having the same size as $J$, but a different color. It is known that each of the numbers $0, 1, ..., 100$ is written at least once. For what numbers of tokens is this possible?

Find all continuous functions $f : R \to R$ such that $$f(x + f(y)) = f(x + y) + y,$$ for all $x, y \in R$. No proof is required for this problem.

Let $a,b,c$ be the sides of a triangle, let $m_a,m_b,m_c$ be the corresponding medians, and let $D$ be the diameter of the circumcircle of the triangle. Prove that $\frac{a^2+b^2}{m_c}+\frac{a^2+c^2}{m_b}+\frac{b^2+c^2}{m_a} \leq 6D$.

Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1,2,3,4,5,6,7,8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by [b]Th3Numb3rThr33[/b][/i]

Let be a nontrivial finite ring having the property that any element of it has an even power that is equal to itself. Prove that [b]a)[/b] the order of the ring is a power of $ 2. $ [b]b)[/b] the sum of all elements of the ring is $ 0. $

Find all values of $a$ for which the inequality $$a^2x^2 + y^2 + z^2 \ge ayz+xy+xz$$ holds for all $x$, $y$ and $z$.

The $2016$ players in the Gensokyo Tennis Club are playing Up and Down the River. The players first randomly form $1008$ pairs, and each pair is assigned to a tennis court (The courts are numbered from $1$ to $1008$). Every day, the two players on the same court play a match against each other to determine a winner and a loser. For $2\le i\le 1008$, the winner on court $i$ will move to court $i-1$ the next day (and the winner on court $1$ does not move). Likewise, for $1\le j\le 1007$, the loser on court $j$ will move to court $j+1$ the next day (and the loser on court $1008$ does not move). On Day $1$, Reimu is playing on court $123$ and Marisa is playing on court $876$. Find the smallest positive integer value of $n$ for which it is possible that Reimu and Marisa play one another on Day $n$. [i]Proposed by Yannick Yao[/i]

Spock would like to find out the thalaron frequency $f$ of a fascinating quantum anomaly which will collapse in a little over five minutes. By observing the resulting geodesic radiation he knows that $f$ is initially a natural number smaller than $400$. Also is known to him that the thalarone frequency increases by $1$ over the course of every minute. Spock can do a harmonic phase resonance at the beginning and every full minute thereafter Generate a feedback loop, whereby he can determine gcd (f, a), where a is a natural number is less than $100$, which he can freely choose each time. Show that he is, provided he is skillful after the six possible measurements, the initial thalarone frequency is unambiguous can determine. [hide=original wording]Spock m¨ochte die Thalaron-Frequenz f einer faszinierenden Quantenanomalie herausfinden, welche in etwas mehr als fu¨nf Minuten kollabieren wird. Durch Beobachtung der resultierenden geod¨atischen Strahlung weiß er, dass f anfangs eine natu¨rliche Zahl kleiner als 400 ist. Auch ist ihm bekannt, dass sich die Thalaron-Frequenz im Laufe jeder Minute um 1 erh¨oht. Spock kann zu Beginn und jede ganze Minute danach durch harmonische Phasenresonanz eine Feedbackschleife erzeugen, wodurch er ggT(f, a) bestimmen kann, wobei a eine natu¨rliche Zahl kleiner als 100 ist die er jedes mal frei w¨ahlen kann. Zeige, dass er, sofern er sich geschickt anstellt, nach den sechs ihm m¨oglichen Messungen die anf¨angliche Thalaron-Frequenz eindeutigbestimmen kann.[/hide]

Suppose that we have a set $S$ of $756$ arbitrary integers between $1$ and $2008$ ($1$ and $2008$ included). Prove that there are two distinct integers $a$ and $b$ in $S$ such that their sum $a + b$ is divisible by $8$.

A tribe called Ababab uses only letters $A$ and $B$, and they create words according to the following rules: (1) $A$ is a word; (2) if $w$ is a word, then $ww$ and $w\overline{w}$ are also words, where $\overline{w}$ is obtained from $w$ by replacing all letters $A$ with $B$ and all letters $B$ with $A$ ( $xy$ denotes the concatenation of $x$ and $y$) (3) all words are created by rules (1) and (2). Prove that any two words with the same number of letters differ exactly in half of their letters.

Using following five figures, can a parallelepiped be constructed, whose side lengths are all integers larger than $1$ and has volume $1990$ ? (In the figure, every square represents a unit cube.) \[\text{Squares are the same and all are } \Huge{1 \times 1}\] [asy] import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt)); label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle); [/asy]

Merlin wants to buy a magical box, which happens to be an $n$-dimensional hypercube with side length $1$ cm. The box needs to be large enough to fit his wand, which is $25.6$ cm long. What is the minimal possible value of $n$? [i] Proposed by Evan Chen [/i]

a) Let $f_1,f_2,\ldots,f_n: \mathbb{R} \to \mathbb{R}$ be periodic functions such that the function $f: \mathbb{R} \to \mathbb{R},$ $f=f_1+f_2+\ldots+f_n$ has finite limit at $\infty.$ Prove that $f$ is constant. b) If $a_1,a_2,a_3$ are real numbers such that $a_1 \cos(a_1x) + a_2 \cos (a_2x) + a_3 \cos(a_3x) \ge 0$ for every $x \in \mathbb{R},$ then $a_1a_2a_3=0.$

Given integer $m\geq2$,$x_1,...,x_m$ are non-negative real numbers,prove that:$$(m-1)^{m-1}(x_1^m+...+x_m^m)\geq(x_1+...+x_m)^m-m^mx_1...x_m$$and please find out when the equality holds.

Let $ABC$ be a triangle, with $\angle A=\alpha,\angle B=\beta$ and $\angle C=\gamma$. Prove that \[\sum_{\text{cyc}}\tan \frac{\alpha}{2}\tan\frac{\beta}{2}\cot\frac{\gamma}{2}\geqslant\sqrt{3}.\][i]Proposed by R. Regimov (Azerbaijan)[/i]

A $50 \times 50$ square board is tiled by the tetrominoes of the following three types: [img]https://cdn.artofproblemsolving.com/attachments/2/9/62c0bce6356ea3edd8a2ebfe0269559b7527f1.png[/img] Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling. (Folklore)

In acute triangle $ABC$,$AC<BC$,$M$ is midpoint of $AB$ and $\Omega$ is it's circumcircle.Let $C^\prime$ be antipode of $C$ in $\Omega$. $AC^\prime$ and $BC^\prime$ intersect with $CM$ at $K,L$,respectively.The perpendicular drawn from $K$ to $AC^\prime$ and perpendicular drawn from $L$ to $BC^\prime$ intersect with $AB$ and each other and form a triangle $\Delta$.Prove that circumcircles of $\Delta$ and $\Omega$ are tangent.(M.Kungozhin)

Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\] holds. Determine all cases of equality. [i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]

Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, $2016=13+2003$). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of? $ \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}$\\ $\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$\\ $\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}$\\ $\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}$\\ $\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$