Let $a, b, c$ be reals which satisfy $a+b+c+ab+bc+ac+abc=>7$, prove that $$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2}=>6$$

We call the sequence $d_1,....,d_n$ of natural numbers, not necessarily distinct, [b]covering[/b] if there exists arithmetic progressions like $c_1+kd_1$,....,$c_n+kd_n$ such that every natural number has come in at least one of them. We call this sequence [b]short[/b] if we can not delete any of the $d_1,....,d_n$ such that the resulting sequence be still covering. [b]a)[/b] Suppose that $d_1,....,d_n$ is a short covering sequence and suppose that we've covered all the natural numbers with arithmetic progressions $a_1+kd_1,.....,a_n+kd_n$, and suppose that $p$ is a prime number that $p$ divides $d_1,....,d_k$ but it does not divide $d_{k+1},....,d_n$. Prove that the remainders of $a_1,....,a_k$ modulo $p$ contains all the numbers $0,1,.....,p-1$. [b]b)[/b] Write anything you can about covering sequences and short covering sequences in the case that each of $d_1,....,d_n$ has only one prime divisor. [i]proposed by Ali Khezeli[/i]

1) Find a $4$-digit number $\overline{PERU}$ such that $\overline{PERU}=(P+E+R+U)^U$. Also prove that there is only one number satisfying this property.

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 \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Let $ABC$ be an equilateral triangle. Let $P$ be a point on the side $AC$ and $Q$ be a point on the side $AB$ so that both triangles $ABP$ and $ACQ$ are acute. Let $R$ be the orthocentre of triangle $ABP$ and $S$ be the orthocentre of triangle $ACQ$. Let $T$ be the point common to the segments $BP$ and $CQ$. Find all possible values of $\angle CBP$ and $\angle BCQ$ such that the triangle $TRS$ is equilateral.

Problems 17, 18, and 19 refer to the following: At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. They learn that a big concert is scheduled for the same night and attendance will be down $25\%$. How many recipes of cookies should they make for their smaller party? $ \text{(A)}\ 6\qquad\text{(B)}\ 8\qquad\text{(C)}\ 9\qquad\text{(D)}\ 10\qquad\text{(E)}\ 11 $

A rectangle $3 \times 5$ is divided into $15$ $1 \times 1$ cells. The middle $3$ cells that have no common points with the border of the rectangle are deleted. Is it possible to put in the remaining $12$ cells numbers $1, 2, \ldots, 12$ in some order, so that the sums of the numbers in the cells along each of the four sides of the rectangle are equal? [i]Proposed by Mariia Rozhkova[/i]

For a positive integer $n$ let $\sigma (n)$ be the sum of all its positive divisors. Find all positive integers $n$ such that the number $\frac{\sigma (n)}{n + 1}$ is an integer.

Thirty two pairs of identical twins are lined up in an $8\times 8$ formation. Prove that it is possible to choose $32 $ persons, one from each pair of twins, so that there is at least one chosen person in each row and in each column

How many positive integer cubes divide $ 3!\cdot 5!\cdot 7!$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 6$

Compute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$ and $5$ divides $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$.

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

Fix an integer $n \ge 2$ and let $A$ be an $n\times n$ array with $n$ cells cut out so that exactly one cell is removed out of every row and every column. A [i]stick [/i] is a $1\times k$ or $k\times 1$ subarray of $A$, where $k$ is a suitable positive integer. (a) Determine the minimal number of [i]sticks [/i] $A$ can be dissected into. (b) Show that the number of ways to dissect $A$ into a minimal number of [i]sticks [/i] does not exceed $100^n$. proposed by Palmer Mebane and Nikolai Beluhov [hide=a few comments]a variation of part a, was [url=https://artofproblemsolving.com/community/c6h1389637p7743073]problem 5[/url] a variation of part b, was posted [url=https://artofproblemsolving.com/community/c6h1389663p7743264]here[/url] this post was made in order to complete the post collection of RMM Shortlist 2017[/hide]

In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.

Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$

A circle $\Gamma$ and a line $\ell$ which does not intersect $\Gamma$ are given. Suppose $P, Q,R, S$ are variable points on circle $\Gamma$ such that the points $A = PQ\cap RS$ and $B = PS \cap QR$ lie on $\ell$. Prove that the circle on $AB$ as a diameter passes through two fixed points.

Find all positive integers $n$ such that $7^n + 147$ is a perfect square.

A triangle with a parallelogram inside was placed in a square. Prove that the area of a parallelogram is not more than a quarter of a square.

In a circle $\odot O$, radius $OA$ is perpendicular to radius $OB$. Chord $AC$ intersects $OB$ at $E$ so that the length of arc $AC$ is one-third the circumference of  $\odot O$. Point $D$ is chosen on $OB$ so that $CD \perp AB$. Suppose that segment $AC$ is $2$ units longer than segment $OD$. What is the length of segment $AC$?

Find all pairs of positive integers $(x,n) $ that are solutions of the equation $3 \cdot 2^x +4 =n^2$.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.

Let $ t$ be a positive number. Draw two tangent lines from the point $ (t, \minus{} 1)$ to the parabpla $ y \equal{} x^2$. Denote $ S(t)$ the area bounded by the tangents line and the parabola. Find the minimum value of $ \frac {S(t)}{\sqrt {t}}$.

Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$

Prove that all real numbers $x \ne -1$, $y \ne -1$ with $xy = 1$ satisfy the following inequality: $$\left(\frac{2+x}{1+x}\right)^2 + \left(\frac{2+y}{1+y}\right)^2 \ge \frac92$$ (Karl Czakler)