In each cell of an \(n \times n\) grid, one of the numbers \(0\), \(1,\) or \(2\) must be written. Determine all positive integers \(n\) for which there exists a way to fill the \(n \times n\) grid such that, when calculating the sum of the numbers in each row and each column, the numbers \(1, 2, \ldots, 2n\) are obtained in some order.

Let $n$ be a positive integer such that the sum of its positive divisors is at least $2022n$. Prove that $n$ has at least $2022$ distinct prime factors.

Write $2017$ following numbers on the blackboard: $-\frac{1008}{1008}, -\frac{1007}{1008}, ..., -\frac{1}{1008}, 0,\frac{1}{1008},\frac{2}{1008}, ... ,\frac{1007}{1008},\frac{1008}{1008}$ . One processes some steps as: erase two arbitrary numbers $x, y$ on the blackboard and then write on it the number $x + 7xy + y$. After $2016$ steps, there is only one number. The last one on the blackboard is (A): $-\frac{1}{1008}$ (B): $0$ (C): $\frac{1}{1008}$ (D): $-\frac{144}{1008}$ (E): None of the above

Let $H$ be the orthocenter of the acute triangle $ABC$. Let $BB'$ and $CC'$ be altitudes of the triangle ($B^{\prime} \in AC$, $C^{\prime} \in AB$). A variable line $\ell$ passing through $H$ intersects the segments $[BC']$ and $[CB']$ in $M$ and $N$. The perpendicular lines of $\ell$ from $M$ and $N$ intersect $BB'$ and $CC'$ in $P$ and $Q$. Determine the locus of the midpoint of the segment $[ PQ]$. [i]Gheorghe Szolosy[/i]

At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group. Prove that at least 3/4 of all “F” marks were given to the same student.

Right $\vartriangle ABC$ has side lengths $6, 8$, and $10$. Find the positive integer $n$ such that the area of the region inside the circumcircle but outside the incircle of $\vartriangle ABC$ is $n\pi$. [img]https://cdn.artofproblemsolving.com/attachments/d/1/cb112332069c09a3b370343ca8a2ef21102fe2.png[/img]

Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of $\frac{gcd(m,n)}{lcm(m,n)}$ over all ordered pairs of positive integers $(m, n)$ where $mn = k$. If $S$ can be written in simplest form as $\frac{r}{s}$, compute $r + s$.

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions: $$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$ for all real numbers $ x,y,t $ with $ t\in [0,1] . $ Prove that: [b]a)[/b] $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $ [b]b)[/b] for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds. $$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$

Prove that there are real numbers $ a_1, a_2, ..$ such that: i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge; ii) f is a bijection of R to R; iii) f'(x) >0; iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.

Find all solution $(p,r)$ of the "Pythagorean-Euler Theorem" $$p^p+(p+1)^p+\cdots+(p+r)^p=(p+r+1)^p$$Where $p$ is a prime and $r$ is a positive integer. [i]Proposed by Li4 and Untro368[/i]

let $m,n$ be natural number with $m>n$ . find all such pairs of $(m,n) $ such that $gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $

The sequence $a_1, a_2,...$ is defined by the equalities $a_1 = 2, a_2 = 12$ and $a_{n+1} = 6a_n-a_{n-1}$ for every positive integer $n \ge 2$. Prove that no member of this sequence is equal to a perfect power (greater than one) of a positive integer.

Let \(a,b,c\) be positive real numbers such that \(ab+bc+ac = a+b+c\). Prove the following inequality: \[\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}.\] \\ \\ [i]Proposed by Dorlir Ahmeti.[/i]

Let $C$ be one of the two points of intersection of circles $\omega_1$ and $\omega_2$ with centers at points $O_1$ and $O_2$, respectively. The line $O_1O_2$ intersects the circles at points $A$ and $B$ as shown in the figure. Let $K$ be the second point of intersection of line $AC$ with circle $\omega_2$, $L$ be the second point of intersection of line $BC$ with circle $\omega_1$. Lines $AL$ and $BK$ intersect at point $D$. Prove that $AD=BD$. (Yurii Biletskyi) [img]https://cdn.artofproblemsolving.com/attachments/6/4/2cdccb43743fcfcb155e846a0e05ec79ba90e4.png[/img]

The non-negative numbers $x,y,z$ satisfy the relation $x + y+ z = 3$. Find the smallest possible numerical value and the largest possible numerical value for the expression $$E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} .$$

Let $ABCD$ be a convex quadrilateral. We assume that there exists a point $P$ inside the quadrilateral such that the areas of the triangles $ABP, BCP, CDP$, and $DAP$ are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.

In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization?

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Let $ a_1,a_2,...,a_{2003}\geq 0$, such that $ a_1\plus{}a_2\plus{}...\plus{}a_{2003}\equal{}2$ and $ a_1a_2\plus{}a_2a_3\plus{}...\plus{}a_{2003}a_1\equal{}1$. Determine the minimum and maximum value of $ a_1^2\plus{}a_2^2\plus{}...\plus{}a_{2003}^2$.

Find the number of permutations of the letters $AAAABBBCC$ where no letter is next to another letter of the same type. For example, count $ABCABCABA$ and $ABABCABCA$ but not $ABCCBABAA$.

Let $ABCDEF$ be a convex hexagon such that $\angle ACE = \angle BDF$ and $\angle BCA = \angle EDF$. Let $A_1=AC\cap FB$, $B_1=BD\cap AC$, $C_1=CE\cap BD$, $D_1=DF\cap CE$, $E_1=EA\cap DF$, and $F_1=FB\cap EA$. Suppose $B_1, C_1, D_1, F_1$ lie on the same circle $\Gamma$. The circumcircles of $\triangle BB_1F_1$ and $ED_1F_1$ meet at $F_1$ and $P$. The line $F_1P$ meets $\Gamma$ again at $Q$. Prove that $B_1D_1$ and $QC_1$ are parrallel. (Here, we use $l_1\cap l_2$ to denote the intersection point of lines $l_1$ and $l_2$)

Solve the system of equations $$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$

Find the distance $\overline{CF}$ in the diagram below where $ABDE$ is a square and angles and lengths are as given: [asy] markscalefactor=0.15; size(8cm); pair A = (0,0); pair B = (17,0); pair E = (0,17); pair D = (17,17); pair F = (-120/17,225/17); pair C = (17+120/17, 64/17); draw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle); label("$A$", A, S); label("$B$", B, S); label("$C$", C, dir(0)); label("$D$", D, N); label("$E$", E, N); label("$F$", F, W); label("$8$", (F+E)/2, NW); label("$15$", (F+A)/2, SW); label("$8$", (C+B)/2, SE); label("$15$", (D+C)/2, NE); draw(rightanglemark(E,F,A)); draw(rightanglemark(D,C,B)); [/asy] The length $\overline{CF}$ is of the form $a\sqrt{b}$ for integers $a, b$ such that no integer square greater than $1$ divides $b$. What is $a + b$?