Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\ $\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\ $\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\ $\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true? \\ \\ [i] (Bogdan Blaga)[/i]

Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $ Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.

In a Pythagorean triangle all sides are longer than 5. Is it possible that (a) all three sides are prime numbers, (b) exactly two sides are prime numbers. (Note: We call a triangle "Pythagorean", if it is a right-angled triangle where all sides are positive integers.)

Let $O$ be the circumcenter of a triangle$ ABC$. Let $M$ be the midpoint of $AO$. The $BO$ and $CO$ intersect the altitude $AD$ at points $E$ and $F$,respectively. Let $O1$ and$ O2$ be the circumcenters of the triangle ABE and $ACF$, respectively. Prove that M lies on $O1O2$.

Let $(G,\cdot)$ be a group with no elements of order 4, and let $f:G\rightarrow G$ be a group morphism such that $f(x)\in\{x,x^{-1}\}$, for all $x\in G$. Prove that either $f(x)=x$ for all $x\in G$, or $f(x)=x^{-1}$ for all $x\in G$.

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that \[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]

Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$.

Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a \minus{} b}{b \minus{} c} \plus{} \frac {a \minus{} d}{d \minus{} c} \equal{} 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) \minus{} f(b)}{f(b) \minus{} f(c)} \plus{} \frac {f(a) \minus{} f(d)}{f(d) \minus{} f(c)} \equal{} 0.$ Prove that function $ f$ is linear

There are villager $0$, villager $1$, . . . , villager $2015$ i.e. $2016$ people in the village. You are villager $0$. Each villager is either honest or liar. You don’t know each villager is honest or liar, but you know you are honest and the number of liar is equal or smaller than integer $T$. The villagers became to write one letter without fail from one day. For integers $1 \le n \le 2015$, there are set integers $1 < k_n < 2015$. The letter villager $i$ wrote in day $n$ of the morning is delivered to villager $i + k_n$ if villager $i$ is honest, or villager $i - k_n$ if villager $i$ is liar in day $n$ of the evening. If $i - j$ is divisible by $2016$, villager $i$ and $j$ point same villager. Villagers don’t know $k_n$, but sender is told when letters are received. Villager can write anything on a letter, and each villager receives letters from any villagers a sufficient number of times after enough time. i.e. there are $n$ satisfying $k = k_n$ infinitely for each integer $1 \le k \le 2015$. You want to know the honest persons of this village. You can gather all villagers just once and instruct in one day of noon. The honest person obeys your instruction but the liar person not always obeys and he or she writes on a letter anything possible. One day from your instruction for a while, you could determine all honest persons of this village. Find the maximum value of $T$ such that it is possible to do this if you instruct appropriate regardless of the villagers who are honest or liar.

Triangle $\vartriangle ABC$ has $AB = 13$, $BC = 14$, and $CA = 15$. $\vartriangle ABC$ has incircle $\gamma$ and circumcircle $\omega$. $\gamma$ has center at $I$. Line $AI$ is extended to hit $\omega$ at $P$. What is the area of quadrilateral $ABPC$?

Prove that for every natural number $ n $ the inequality holds $$ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}} > \sqrt{n-1}.$$

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.

Let $ABCD$ a tetrahedron and $M$ a variable point on the face $BCD$. The line perpendicular to $(BCD)$ in $M$ . intersects the planes$ (ABC)$, $(ACD)$, and $(ADB)$ in $M_1$, $M_2$, and $M_3$. Show that the sum $MM_1 + MM_2 + MM_3$ is constant if and only if the perpendicular dropped from $A$ to $(BCD)$ passes through the centroid of triangle $BCD$.

Suppose there are $n$ boxes in a row and place $n$ balls in them one in each. The balls are colored red, blue or green. In how many ways can we place the balls subject to the condition that any box $B$ has at least one adjacent box having a ball of the same color as the ball in $B$? [Assume that balls in each color are available abundantly.]

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Express $$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$$ as a rational number.

Two unit circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. $M$ is an arbitrary point of $\omega_1$, $N$ is an arbitrary point of $\omega_2$. Two unit circles $\omega_3$ and $\omega_4$ pass through both points $M$ and $N$. Let $C$ be the second common point of $\omega_1$ and $\omega_3$, and $D$ be the second common point of $\omega_2$ and $\omega_4$. Prove that $ACBD$ is a parallelogram.

We are given convex quadrilateral $ABCD$. Each of its sides is divided into $N$ line segments of equal length. The points of division of side $AB$ are connected with the points of division of side $CD$ by straight lines (which we call the first set of straight lines), and the points of division of side BC are connected with the points of division of side $DA$ by straight lines (which we call the second set of straight lines) as shown in the diagram, which illustrates the case $N = 4$. This forms $N^2$ smaller quadrilaterals. From these we choose $N$ quadrilaterals in such a way that any two are at least divided by one line from the first set and one line from the second set. Prove that the sum of the areas of these chosen quadrilaterals is equal to the area of $ABCD$ divided by $N$. (A Andjans, Riga) [img]http://4.bp.blogspot.com/-8Qqk4r68nhU/XVco29-HzzI/AAAAAAAAKgo/UY8mXxg7tD0OrS6bEnoAw7Vuf31BuOE8wCK4BGAYYCw/s1600/TOT%2B1980%2BSpring%2BJ4.png[/img]

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

Which of the following represents the force corresponding to the given potential? [asy] // Code by riben size(400); picture pic; // Rectangle draw(pic,(0,0)--(22,0)--(22,12)--(0,12)--cycle); label(pic,"-15",(2,0),S); label(pic,"-10",(5,0),S); label(pic,"-5",(8,0),S); label(pic,"0",(11,0),S); label(pic,"5",(14,0),S); label(pic,"10",(17,0),S); label(pic,"15",(20,0),S); label(pic,"-2",(0,2),W); label(pic,"-1",(0,4),W); label(pic,"0",(0,6),W); label(pic,"1",(0,8),W); label(pic,"2",(0,10),W); label(pic,rotate(90)*"F (N)",(-2,6),W); label(pic,"x (m)",(11,-2),S); // Tick Marks draw(pic,(2,0)--(2,0.3)); draw(pic,(5,0)--(5,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(11,0)--(11,0.3)); draw(pic,(14,0)--(14,0.3)); draw(pic,(17,0)--(17,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(2,12)--(2,11.7)); draw(pic,(5,12)--(5,11.7)); draw(pic,(8,12)--(8,11.7)); draw(pic,(11,12)--(11,11.7)); draw(pic,(14,12)--(14,11.7)); draw(pic,(17,12)--(17,11.7)); draw(pic,(20,12)--(20,11.7)); draw(pic,(22,2)--(21.7,2)); draw(pic,(22,4)--(21.7,4)); draw(pic,(22,6)--(21.7,6)); draw(pic,(22,8)--(21.7,8)); draw(pic,(22,10)--(21.7,10)); // Paths path A=(0,6)--(5,6)--(5,4)--(11,4)--(11,8)--(17,8)--(17,6)--(22,6); path B=(0,6)--(5,6)--(5,2)--(11,2)--(11,10)--(17,10)--(17,6)--(22,6); path C=(0,6)--(5,6)--(5,5)--(11,5)--(11,7)--(17,7)--(17,6)--(22,6); path D=(0,6)--(5,6)--(5,7)--(11,7)--(11,5)--(17,5)--(17,6)--(22,6); path E=(0,6)--(5,6)--(5,8)--(11,8)--(11,4)--(17,4)--(17,6)--(22,6); draw(A); label("(A)",(9.5,-3),4*S); draw(shift(35*right)*B); label("(B)",(45.5,-3),4*S); draw(shift(20*down)*C); label("(C)",(9.5,-23),4*S); draw(shift(35*right)*shift(20*down)*D); label("(D)",(45.5,-23),4*S); draw(shift(40*down)*E); label("(E)",(9.5,-43),4*S); add(pic); picture pic2=shift(35*right)*pic; picture pic3=shift(20*down)*pic; picture pic4=shift(35*right)*shift(20*down)*pic; picture pic5=shift(40*down)*pic; add(pic2); add(pic3); add(pic4); add(pic5); [/asy]

In the convex pentagon $ABCDE$, the following equalities hold: $\angle CDE=90^\circ$, $AC=AD$, and $BD=BE$. Prove that triangle $ABD$ and quadrilateral $ABCE$ have the same area.

Compute the sum of all integers $n$ such that $n^2-3000$ is a perfect square.

Quadrilateral $ABCD$ is given with $AD \nparallel BC$. The midpoints of $AD$ and $BC$ are denoted by $M$ and $N$, respectively. The line $MN$ intersects the diagonals $AC$ and $BD$ in points $K$ and $L$, respectively. Prove that the circumcircles of the triangles $AKM$ and $BNL$ have common point on the line $AB$.( Proposed by Emil Stoyanov ) [img]http://estoyanov.net/wp-content/uploads/2015/09/est.png[/img]

Let $\vartriangle ABC$ be an equilateral triangle with side length $1$ and let $\Gamma$ the circle tangent to $AB$ and $AC$ at $B$ and $C$, respectively. Let $P$ be on side $AB$ and $Q$ be on side $AC$ so that $PQ // BC$, and the circle through $A, P$, and $Q$ is tangent to $\Gamma$ . If the area of $\vartriangle APQ$ can be written in the form $\frac{\sqrt{a}}{b}$ for positive integers $a$ and $b$, where $a$ is not divisible by the square of any prime, find $a + b$.