Let $a\circ b=a+b-ab$. Find all triples $(x,y,z)$ of integers such that \[(x\circ y)\circ z +(y\circ z)\circ x +(z\circ x)\circ y=0\]

How to hang a picture? What a strange question? It's simple. We take a piece of rope, attach its ends to the picture frame on the back side, then drive it into the wall. nail and throw a rope over the nail. The picture is hanging. If you pull out the nail, then, of course, it will fall. But Professor No wonder acted differently. At first, he attached the rope to the painting in the same way, only he took it a little longer. Then he hammered two nails into the wall nearby and threw a rope over these nails in a special way. The painting hangs on these nails, but if you pull out any nail, the painting will fall. Moreover, the professor claims that he can hang a painting on three nails so that the painting hangs on all three, but if any nail is pulled out, the painting will fall. You have two tasks: indicate how you can hang the picture in the right way on a) two nails; b) three nails.

A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.

If the circumradius of a regular $n$-gon is $1$ and the ratio of its perimeter over its area is $\dfrac{4\sqrt 3}{3}$, what is $n$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $

Let $a,b,k$ be positive integers such that $gcd(a,b)^2+lcm(a,b)^2+a^2b^2=2020^k$ Prove that $k$ is an even number.

In triangle $ABC$, the incircle touches sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. Let $X$ be the feet of the altitude of the vertex $D$ on side $EF$ of triangle $DEF$. Prove that $AX,BY$ and $CZ$ are concurrent on the Euler line of the triangle $DEF$.

There are a lot of different real numbers written on the board. It turned out that for each two numbers written, their product was also written. What is the largest possible number of numbers written on the board?

In a parallelogram $ABCD$, points $E$ and $F$ are the midpoints of $AB$ and $BC$, respectively, and $P$ is the intersection of $EC$ and $FD$. Prove that the segments $AP,BP,CP$ and $DP$ divide the parallelogram into four triangles whose areas are in the ratio $1 : 2 : 3 : 4$.

Determine if there exist positive integer pairs $(m,n)$, such that (i) the greatest common divisor of m and $n$ is $1$, and $m \le 2007$, (ii) for any $k=1,2,..., 2007$, $\big[\frac{nk}{m}\big]=\big[\sqrt2 k\big]$ . (Here $[x]$ stands for the greatest integer less than or equal to $x$.)

Given the equation \[ y^4 \plus{} 4y^2x \minus{} 11y^2 \plus{} 4xy \minus{} 8y \plus{} 8x^2 \minus{} 40x \plus{} 52 \equal{} 0,\] find all real solutions.

$ M,N,P,Q,R,S $ are the midpoints of the sides $ AB,BC,CD,DE,EF,FA $ of a convex hexagon $ ABCDEF. $ [b]a)[/b] Show that with the segments $ MQ,NR,PS, $ it can be formed a triangle. [b]b)[/b] Show that a triangle formed with the segments $ MQ,NR,PS $ is right if and only if ether $ MQ\perp NR $ or $ MQ\perp PS $ or $ PS\perp RN. $ [i]Vasile Pop[/i]

For how many positive integer values of $n$ are both $\frac{n}{3}$ and $3n$ three-digit whole numbers? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 27\qquad \textbf{(D)}\ 33\qquad \textbf{(E)}\ 34$

Integers $1, 2, 3, ... ,n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers?

Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$. (i) Prove that $CK$ is the angle bisector of $\angle ACB$. (ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.

Let $a > 1$ and $r > 1$ be real numbers. (a) Prove that if $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ is a function satisfying the conditions (i) $f (x)^{2}\leq ax^{r}f (\frac{x}{a})$ for all $x > 0$, (ii) $f (x) < 2^{2000}$ for all $x < \frac{1}{2^{2000}}$, then $f (x) \leq x^{r}a^{1-r}$ for all $x > 0$. (b) Construct a function $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ satisfying condition (i) such that for all $x > 0, f (x) > x^{r}a^{1-r}$ .

Prove that for all all non-negative real numbers $a,b,c$ with $a^2+b^2+c^2=1$ \[\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.\]

Assume integer $m \geq 2.$ There are $3m$ people in a meeting, any two of them either shake hands with each other once or not.We call the meeting "$n$-interesting", only if there exists $n(n\leq 3m-1)$ people of them, the time everyone of whom shakes hands with other $3m-1$ people is exactly $1,2,\cdots,n,$ respectively. If in any "$n$-interesting" meeting, there exists $3$ people of them who shake hands with each other, find the minimum value of $n.$

Let be given points $A,B$ on a circle and let $P$ be a variable point on that circle. Let point $M$ be determined by $P$ as the point that is either on segment $PA$ with $AM=MP+PB$ or on segment $PB$ with $AP+MP=PB$. Find the locus of points $M$.

For a positive real number $\alpha$, define \[S(\alpha)=\{ \lfloor n\alpha\rfloor \; \vert \; n=1,2,3,\cdots \}.\] Prove that $\mathbb{N}$ cannot be expressed as the disjoint union of three sets $S(\alpha)$, $S(\beta)$, and $S(\gamma)$.

Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.

Let $\omega$ be the circumscribed circle of triangle $ABC$, and let $\omega'$ 'be the circle tangent to the side $BC$ and the extensions of the sides $AB$ and $AC$. The common tangents to the circles $\omega$ and $\omega'$ intersect the line $BC$ at points $D$ and $E$. Prove that $\angle BAD = \angle CAE$.

We number both the rows and the columns of an $8 \times 8$ chessboard with the numbers $1$ to $8$. Some grains of rice are placed on each square, in such a way that the number of grains on each square is equal to the product of the row and column numbers of the square. How many grains of rice are there on the entire chessboard?

If $n$ is a positive integer such that $8n+1$ is a perfect square, then $\textbf{(A)}~n\text{ must be odd}$ $\textbf{(B)}~n\text{ cannot be a perfect square}$ $\textbf{(C)}~n\text{ cannot be a perfect square}$ $\textbf{(D)}~\text{None of the above}$

[u]Round 6 [/u] [b]p16.[/b] Let $a$ be a solution to $x^3 -x +1 = 0$. Find $a^6 -a^2 +2a$. [b]p17.[/b] For a positive integer $n$, $\phi (n)$ is the number of positive integers less than $n$ that are relatively prime to $n$. Compute the sum of all $n$ for which $\phi (n) = 24$. [b]p18.[/b] Let $x$ be a positive integer such that $x^2 \equiv 57$ (mod $59$). Find the least possible value of $x$. [u]Round 7[/u] [b]p19.[/b] In the diagram below, find the number of ways to color each vertex red, green, yellow or blue such that no two vertices of a triangle have the same color. [img]https://cdn.artofproblemsolving.com/attachments/1/e/01418af242c7e2c095a53dd23e997b8d1f3686.png[/img] [b]p20.[/b] In a set with $n$ elements, the sum of the number of ways to choose $3$ or $4$ elements is a multiple of the sumof the number of ways to choose $1$ or $2$ elements. Find the number of possible values of $n$ between $4$ and $120$ inclusive. [b]p21.[/b] In unit square $ABCD$, let $\Gamma$ be the locus of points $P$ in the interior of $ABCD$ such that $2AP < BP$. The area of $\Gamma$ can be written as $\frac{a\pi +b\sqrt{c}}{d}$ for integers $a,b,c,d$ with $c$ squarefree and $gcd(a,b,d) = 1$. Find $1000000a +10000b +100c +d$. [u]Round 8 [/u] [b]p22.[/b] Ephram, GammaZero, and Orz walk into a bar. Each write some permutation of the letters “LMT” once, then concatenate their permutations one after the other (i.e. LTMTLMTLM would be a possible string, but not LLLMMMTTT). Suppose that the probability that the string “LMT” appears in that order among the new $9$-character string can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p23.[/b] In $\vartriangle ABC$ with side lengths $AB = 27$, $BC = 35$, and $C A = 32$, let $D$ be the point at which the incircle is tangent to $BC$. The value of $\frac{\sin \angle C AD }{\sin\angle B AD}$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p24.[/b] Let $A$ be the greatest possible area of a square contained in a regular hexagon with side length $1$. Let B be the least possible area of a square that contains a regular hexagon with side length $1$. The value of $B-A$ can be expressed as $a\sqrt{b}-c$ for positive integers $a$, $b$, and $c$ with $b$ squarefree. Find $10000a +100b +c$. [u]Round 9[/u] [b]p25.[/b] Estimate how many days before today this problem was written. If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{2} \right| \right \rfloor , 0 \right)$ points. [b]p26.[/b] Circle $\omega_1$ is inscribed in unit square $ABCD$. For every integer $1 < n \le 10,000$, $\omega_n$ is defined as the largest circle which can be drawn inside $ABCD$ that does not overlap the interior of any of $\omega_1$,$\omega_2$, $...$,$\omega_{n-1}$ (If there are multiple such $\omega_n$ that can be drawn, one is chosen at random). Let r be the radius of ω10,000. Estimate $\frac{1}{r}$ . If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{200} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Answer with a positive integer less than or equal to $20$. We will compare your response with the response of every other team that answered this problem. When two equal responses are compared, neither team wins. When two unequal responses $A > B$ are compared, $A$ wins if $B | A$, and $B$ wins otherwise. If your team wins n times, you will receive $\left \lfloor \frac{n}{2} \right \rfloor$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167135p28823324]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $n$ such that $$\phi(n) + \sigma(n) = 2n + 8.$$